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y(v) -6 b(anishes) 44 b(for) p Fl 43 w(u) p Fm 43 w(=) g(0) f(and) g(is) f(p) s(erio) s(dic) h(with) f(p) s(erio) s (d) h(2) p Fl(\031) p Fm 46 w(in) g(the) p Fl 43 w(x) p Fm 42 w(v) -6 b(ariable.) 72 b(W) -8 b(e) 43 b(pro) m(v) m(e,) j(under) e(a) 0 3016 y(nonresonance) c(condition) f(whic) m(h) g(is) g (ful\014lled) g(for) g(most) p Fl 37 w(g) p Fm 4 w('s,) f(that) h(for) f (an) m(y) h(in) m(teger) p Fl 39 w(M) p Fm 49 w(there) g(exists) 0 3136 y(a) 48 b(canonical) g(transformation) f(that) h(puts) h(the) g (Hamiltonian) d(in) i(Birkho\013) f(normal) g(form) h(up) h(to) f(a) 0 3255 y(reminder) e(of) g(order) p Fl 47 w(M) p Fm 11 w(.) 82 b(The) 47 b(canonical) f(transformation) f(is) h(w) m(ell) g (de\014ned) i(in) e(a) g(neigh) m(b) s(ourho) s(o) s(d) 0 3375 y(of) h(the) h(origin) e(of) h(a) g(Sob) s(olev) g(t) m(yp) s(e) g (phase) h(space) g(of) g(su\016cien) m(tly) f(high) g(order.) 86 b(Some) 47 b(dynamical) 0 3494 y(consequences) 37 b(are) e(obtained.) 50 b(The) 36 b(tec) m(hnique) g(of) f(pro) s(of) g(is) g(applicable) g(to) g(quite) f(general) i(equations) 0 3614 y(in) d(one) h(space) g (dimension.) p Fn 0 4418 a(1.) 115 b(In) m(tro) s(duction) p Fm 199 4690 a(Consider) 34 b(a) e(\014nite) h(dimensional) f (Hamiltonian) e(system) i(ha) m(ving) h(an) g(elliptic) f(equilibrium) f (p) s(oin) m(t;) 0 4810 y(it) k(is) h(w) m(ell) g(kno) m(wn) g(that) g (there) h(exist) e(canonical) h(co) s(ordinates) g(in) g(whic) m(h) h (the) g(Hamiltonian) c(tak) m(es) j(the) 0 4929 y(form) p Fl 1731 5059 a(H) p Fm 35 w(=) p Fl 28 w(h) p Fi 2011 5074 a(0) p Fm 2079 5059 a(+) p Fl 22 w(f) p Fm 0 5258 a(where) p Fl 1428 5438 a(h) p Fi 1485 5453 a(0) p Fm 1530 5438 a(\() p Fl(p;) 17 b(q) p Fm 4 w(\)) 26 b(=) p Fh 1882 5343 a(X) p Fk 1936 5556 a(j) p Fl 2043 5438 a(!) p Fk 2105 5453 a(j) p Fl 2159 5360 a(p) p Fi 2209 5324 a(2) p Fk 2209 5386 a(j) p Fm 2276 5360 a(+) p Fl 22 w(q) p Fi 2423 5324 a(2) p Fk 2419 5386 a(j) p 2159 5415 310 4 v Fm 2289 5506 a(2) p Fl 2513 5438 a(;) p Fm 1223 w(\(1) p Fl(:) p Fm(1\)) 1807 5778 y(20.03.02) p 90 rotate dyy eop %%Page: 2 2 2 1 bop Fm 0 60 a(2) p Fg 1685 w(D.) 33 b(Bam) m(busi) p Fm 0 299 a(and) p Fl 24 w(f) p Fm 11 w(\() p Fl(p;) 17 b(q) p Fm 4 w(\)) 27 b(=) p Fj 28 w(O) p Fm 3 w(\() p Fj(k) p Fm -1 w(\() p Fl(p;) 17 b(q) p Fm 4 w(\)) p Fj(k) p Fi 1037 249 a(3) p Fm 1082 299 a(\).) 40 b(Moreo) m(v) m(er,) 26 b(if) e(the) h(frequencies) h(are) e(nonresonan) m(t,) i(then) f(for) f (an) m(y) g(in) m(teger) p Fl 0 418 a(M) p Fj 41 w(\025) p Fm 30 w(4) 34 b(there) i(exists) e(a) h(canonical) f(transformation) p Fj 34 w(T) p Fm 60 w(\(de\014ned) i(in) f(a) f(neigh) m(b) s(orho) s(o) s(d) i(of) e(the) h(origin\)) 0 538 y(whic) m(h) f(puts) g(the) g (system) e(in) h(Birkho\013) g(normal) e(form) i(up) h(to) f(order) p Fl 33 w(M) p Fm 11 w(,) g(i.e.) 43 b(suc) m(h) 35 b(that) p Fl 1533 781 a(H) p Fj 30 w(\016) 22 b(T) p Fm 53 w(=) p Fl 28 w(h) p Fi 1987 796 a(0) p Fm 2055 781 a(+) p Fl 22 w(Z) p Fm 29 w(+) p Fj 23 w(R) p Fm 1329 w(\(1) p Fl(:) p Fm(2\)) 0 1037 y(where) p Fl 35 w(Z) p Fm 41 w(dep) s(ends) 36 b(only) e(on) g(the) h(action) f(v) -6 b(ariables) p Fl 34 w(I) p Fk 2074 1052 a(j) p Fm 2145 1037 a(=) 29 b(\() p Fl(p) p Fi 2340 1001 a(2) p Fk 2340 1063 a(j) p Fm 2408 1037 a(+) p Fl 23 w(q) p Fi 2556 1001 a(2) p Fk 2552 1063 a(j) p Fm 2601 1037 a(\)) p Fl(=) p Fm(2,) k(and) p Fj 35 w(R) p Fm 29 w(=) p Fj 30 w(O) p Fm 3 w(\() p Fj(k) p Fm(\() p Fl(p;) 17 b(q) p Fm 4 w(\)) p Fj(k) p Fk 3656 987 a(M) p Fm 3745 1037 a(\)) 34 b(is) g(a) 0 1156 y(remainder) f(whic) m(h) h(is) f(small) f (close) h(to) g(the) h(origin.) 199 1277 y(In) 44 b(the) h(presen) m(t) g(pap) s(er) g(w) m(e) f(in) m(v) m(estigate) g(the) g(existence) h(of) f(suc) m(h) h(a) f(transformation) f(for) h(some) 0 1397 y(in\014nite) 33 b(dimensional) g(Hamiltonian) d(systems) j(describing) h(nonlinear) f(partial) f(di\013eren) m(tial) i(equations) 0 1516 y(\(PDEs\).) 43 b(As) 34 b(a) f(mo) s(del) f(problem) h(w) m(e) g (consider) h(the) g(nonlinear) g(w) m(a) m(v) m(e) f(equation) p Fl 1460 1759 a(u) p Fk 1517 1774 a(tt) p Fj 1604 1759 a(\000) p Fl 23 w(u) p Fk 1761 1774 a(xx) p Fm 1878 1759 a(+) p Fl 23 w(g) p Fm 4 w(\() p Fl(x;) 17 b(u) p Fm(\)) 26 b(=) i(0) p Fl 33 w(;) p Fm 1255 w(\(0) p Fl(:) p Fm(1\)) 0 2002 y(with) d(Diric) m(hlet) g(b) s(oundary) h(conditions) g(on) f([0) p Fl(;) 17 b(\031) p Fm 4 w(],) 24 b(where) p Fl 27 w(g) p Fm 28 w(is) h(a) g(function) i(whic) m(h) f(extends) g(analytically) 0 2122 y(to) 33 b(the) g(region) p Fj 34 w(j) p Fl(u) p Fj(j) p Fl 27 w(<) 28 b(R) p Fi 914 2137 a(1) p Fm 991 2122 a(and) p Fj 34 w(j) p Fm(Im) p Fl 19 w(x) p Fj(j) p Fl 27 w(<) g(\033) p Fi 1626 2137 a(1) p Fm 1671 2122 a(,) 33 b(where) p Fl 34 w(R) p Fi 2096 2137 a(1) p Fl 2140 2122 a(;) 17 b(\033) p Fi 2242 2137 a(1) p Fm 2319 2122 a(are) 33 b(p) s(ositiv) m(e) g(n) m(um) m(b) s(ers,) h(and) g (ful\014lls) p Fl 607 2365 a(g) p Fm 4 w(\() p Fl(x;) p Fm 17 w(0\)) 25 b(=) j(0) p Fl 33 w(;) 116 b(g) p Fm 4 w(\() p Fl(x) p Fm 21 w(+) 23 b(2) p Fl(\031) t(;) 17 b(u) p Fm(\)) 26 b(=) p Fl 28 w(g) p Fm 4 w(\() p Fl(x;) 17 b(u) p Fm(\)) p Fl 31 w(;) 116 b(g) p Fm 4 w(\() p Fj(\000) p Fl(x;) p Fj 17 w(\000) p Fl(u) p Fm(\)) 26 b(=) p Fj 28 w(\000) p Fl(g) p Fm 4 w(\() p Fl(x;) 17 b(u) p Fm(\)) p Fl 32 w(:) p Fm 402 w(\(1) p Fl(:) p Fm(3\)) 0 2608 y(W) -8 b(e) 28 b(will) f(pro) m(v) m(e) i(that) e(if) p Fl 28 w(g) p Fm 31 w(ful\014lls) h(a) g(suitable) g(nonresonance) i (condition) e(\(whic) m(h) g(has) h(full) f(measure) g(in) g(a) 0 2727 y(sense) i(that) e(will) g(b) s(e) h(made) f(precise\),) i(then) g (for) f(an) m(y) p Fl 28 w(M) p Fj 38 w(\025) p Fm 28 w(4) g(and) g(an) m(y) g(p) s(ositiv) m(e) f(\(small\)) p Fl 26 w(R) p Fm 29 w(there) i(exists) 0 2847 y(a) e(canonical) g (transformation) f(whic) m(h) i(puts) f(the) h(Hamiltonian) c(of) k (the) f(system) f(in) h(the) h(form) e(\(1.2\).) 41 b(The) 0 2966 y(transformation) 35 b(is) h(w) m(ell) g(de\014ned) i(in) e(an) h (op) s(en) f(ball) g(of) g(radius) p Fl 37 w(R) p Fm 37 w(of) g(Sob) s(olev{t) m(yp) s(e) g(phase) h(spaces) g(of) 0 3086 y(su\016cien) m(tly) k(high) f(order,) i(and) f(the) f(norm) g(of) g(the) h(hamiltonian) d(v) m(ector) i(\014eld) h(of) p Fj 41 w(R) p Fm 40 w(is) f(smaller) f(than) p Fl 0 3205 a(C) 7 b(R) p Fk 155 3169 a(M) p Fm 276 3205 a(in) 34 b(the) f(same) g(ball,) f(with) p Fl 33 w(C) p Fm 41 w(a) h(p) s(ositiv) m(e) f(constan) m(t) i(indep) s(enden) m(t) i(of) p Fl 33 w(R) p Fm 1 w(.) 199 3326 y(As) 41 b(a) g(corollary) e(w) m(e) i (obtain) g(an) f(estimate) g(of) h(high) g(Sob) s(olev) f(norms) g(of) h (the) h(solution) e(whic) m(h) i(is) 0 3446 y(v) -6 b(alid) 33 b(for) g(v) m(ery) g(long) g(times) g(\(in) g(the) g(spirit) g(of) h ([8]\)) e(and) i(also) f(a) g(v) m(ery) g(precise) h(qualitativ) m(e) e (description) 0 3565 y(of) g(the) g(dynamics) e(for) i(suc) m(h) h (times.) 43 b(In) 31 b(particular) h(w) m(e) g(obtain) f(\(in) g(the) h (spirit) f(of) h([15,16]\)) d(that) i(all) g(the) 0 3685 y(actions) p Fl 38 w(I) p Fk 387 3700 a(j) p Fm 466 3685 a(of) 38 b(the) g(linearized) g(system) f(are) h(appro) m(ximately) d (constan) m(ts) k(of) f(motion) e(for) i(times) e(of) i(the) 0 3804 y(order) c(of) f(the) h(norm) e(of) i(the) f(initial) f(datum) g (to) h(the) h(p) s(o) m(w) m(er) p Fj 34 w(\000) p Fl(M) p Fm 11 w(.) 199 3985 y(W) -8 b(e) 36 b(recall) e(that) h(p) s (erturbation) g(theory) g(for) g(Hamiltonian) e(PDEs) i(has) g (attracted) g(great) g(in) m(terest) 0 4105 y(during) f(the) g(last) f (\014fteen) i(y) m(ears,) f(and) g(in) f(particular) h(KAM) g(theory) f (has) h(b) s(een) h(partially) d(extended) i(b) m(y) 0 4224 y(pro) m(ving) 25 b(that) f(most) f(of) i(the) p Ff 26 w(\014nite) i(dimensional) p Fm 24 w(in) m(v) -6 b(arian) m(t) 25 b(tori) f(of) h(an) f(in) m(tegrable) h(\(e.g.) 41 b(linear\)) 24 b(system) 0 4344 y(surviv) m(e) 48 b(a) g(small) f(p) s (erturbation) i([12,13,11,7,9,18,19,10].) 83 b(Ho) m(w) m(ev) m(er,) 52 b(it) c(is) g(clear) g(that,) k(although) 0 4463 y(v) m(ery) 45 b(in) m(teresting,) k(the) e(solutions) e(lying) g(on) h(these) h(tori) e(corresp) s(ond) i(to) e(exceptional) h(initial) e(data.) 0 4583 y(Concerning) 49 b(solutions) e(starting) h(outside) g(suc) m(h) h (tori) e(only) g(a) h(few) g(results) h(are) f(kno) m(wn) g ([1,2,6,3,8].) 0 4702 y(In) g(particular,) j(the) d(pap) s(ers) g ([1,2]) e(deal) i(with) g(p) s(erturbations) g(of) g(resonan) m(t) g (systems) f(and) i(pro) m(v) m(e) f(a) 0 4822 y(long) 36 b(time) f(stabilit) m(y) g(prop) s(ert) m(y) i(of) f(p) s(erio) s(dic) h (solutions) f(or) g(\014nite) h(dimensional) e(tori) h(in) g(the) h (top) s(ology) 0 4941 y(induced) j(b) m(y) f(the) f(energy) h(norm;) h (suc) m(h) g(results) f(are) g(based) g(on) g(resonan) m(t) g(a) m(v) m (eraging) f(com) m(bined) h(with) 0 5061 y(other) k(tec) m(hninques.) 73 b(The) 43 b(pap) s(ers) g([6,3]) e(deal) i(with) f(p) s(erturbations) h (of) f(nonresonan) m(t) i(systems) e(and) 0 5181 y(their) 34 b(results) h(allo) m(w) f(to) f(con) m(trol,) i(for) f(long) g(times,) f (the) i(dynamics) e(corresp) s(onding) j(to) e(all) f(initial) g(data) 0 5300 y(in) 25 b(a) h(ball) e(of) i(a) f(phase) i(space) f(of) g(smo) s (oth) e(functions;) 29 b(ho) m(w) m(ev) m(er) d(the) g(top) s(ology) e (in) h(whic) m(h) h(the) g(dynamics) f(is) 0 5420 y(con) m(trolled) j (is) f(m) m(uc) m(h) h(w) m(eak) m(er) f(than) h(that) f(of) h(the) f (original) f(phase) j(space) f(\(see) g(remark) e(2.4\).) 41 b(Suc) m(h) 29 b(latter) 0 5539 y(results) 37 b(are) f(obtained) g(b) m (y) g(exploiting) f(the) i(linear) f(stabilit) m(y) e(of) j(\014nite) f (dimensional) g(tori.) 51 b(Finally) 35 b(w) m(e) 1807 5778 y(20.03.02) p 90 rotate dyy eop %%Page: 3 3 3 2 bop Fg 1247 60 a(Birkho\013) 33 b(normal) f(form) g(for) i(PDE's) p Fm 1246 w(3) 0 299 y(quote) h(the) i(remark) -6 b(able) 35 b(pap) s(er) h([8]) f(where) h(a) g(stabilit) m(y) e(prop) s(ert) m(y) i (of) g(most) f(small) f(amplitude) h(in\014nite) 0 418 y(dimensional) 27 b(tori) f(of) i(some) e(nonlinear) i(Sc) m(hr\177) -50 b(odinger) 29 b(equations) e(is) g(obtained.) 43 b(This) 27 b(result) h(is) f(strongly) 0 538 y(related) 33 b(to) g(the) h(main) e (one) h(pro) m(v) m(ed) h(here.) 199 658 y(In) 47 b(the) f(presen) m(t) i(pap) s(er) f(w) m(e) f(obtain) g(a) g(precise) h(qualitativ) m(e) d (description) j(of) g(all) e(the) h(solutions) 0 777 y(starting) 41 b(in) g(a) g(ball) g(of) g(a) g(Sob) s(olev) g(t) m(yp) s (e) g(phase) i(space,) h(in) d(particular) g(w) m(e) g(obtain) g(a) g (con) m(trol) h(of) f(the) 0 897 y(\(phase) 29 b(space\)) h(norm) e(of) h(the) g(solution,) g(and) g(moreo) m(v) m(er) f(w) m(e) h(pro) m(v) m (e) g(that) f(eac) m(h) i(solution) e(remains) g(close,) 0 1016 y(in) 34 b(a) g(top) s(ology) e(sligh) m(tly) h(w) m(eak) m(er) i (than) f(that) g(of) g(the) h(phase) g(spase,) g(to) e(an) h (in\014nite) h(dimensional) e(torus) 0 1136 y(\(see) 39 b(corollary) f(2.3\).) 60 b(W) -8 b(e) 39 b(p) s(oin) m(t) g(out) g (that) f(the) i(tec) m(hnique) g(of) f(pro) s(of) g(of) g(the) g (presen) m(t) i(pap) s(er) e(is) g(v) m(ery) 0 1255 y(general,) c (while) f(that) g(of) h([8]) e(\(whic) m(h) i(is) f(completely) f (di\013eren) m(t) j(from) e(ours\)) g(exploits) g(some) g(particular) 0 1375 y(features) 40 b(of) f(the) g(NLS.) g(Moreo) m(v) m(er,) h(w) m(e) f(think) g(that) f(the) h(approac) m(h) h(of) f(the) g(presen) m(t) i (pap) s(er) e(could) g(b) s(e) 0 1494 y(the) c(starting) f(p) s(oin) m (t) g(for) h(further) g(dev) m(elopmen) m(ts) g(in) g(the) g(same) f(w) m(a) m(y) g(as) g(Birkho\013) g(normal) f(form) h(is) g(for) 0 1614 y(\014nite) g(dimensional) e(systems) h(\(follo) m(wing) f(for) i (example) e([13,14,17]\).) 199 1733 y(The) 26 b(pap) s(er) h(is) e (organized) h(as) g(follo) m(ws:) 40 b(in) 26 b(sect.) g(2) f(w) m(e) h (giv) m(e) g(a) f(precise) i(statemen) m(t) e(of) h(the) g(results) g (for) 0 1853 y(the) 34 b(equation) g(\(0.1\),) e(in) i(sect.) g(3) g(w) m(e) g(presen) m(t) i(the) e(idea) g(of) g(the) g(pro) s(of) h(and) f (w) m(e) g(discuss) i(some) d(p) s(ossible) 0 1973 y(extensions.) 65 b(In) 40 b(sect.) g(4) g(w) m(e) g(state) g(our) g(abstract) g(normal) f (form) g(theorem.) 63 b(In) 40 b(sect.) h(5) e(w) m(e) i(giv) m(e) e (the) 0 2092 y(pro) s(of) 32 b(of) g(suc) m(h) i(an) e(abstract) f (result.) 44 b(This) 32 b(section) h(is) f(divided) g(in) f(three) i (subsection:) 45 b(in) 32 b(the) g(\014rst) g(one) 0 2212 y(w) m(e) j(study) f(the) h(prop) s(erties) f(of) h(the) f(class) h (of) f(functions) h(allo) m(w) m(ed) g(as) f(nonlinearities) g(b) m(y) g (our) g(theory;) g(in) 0 2331 y(the) 39 b(second) h(one) f(w) m(e) g (study) g(the) g(b) s(eha) m(viour) g(of) g(suc) m(h) h(functions) g (under) g(canonical) e(transformations) 0 2451 y(generated) k(b) m(y) f (functions) i(of) e(the) g(same) g(class;) k(in) c(the) g(third) g (subsection) i(w) m(e) e(iterativ) m(ely) f(construct) 0 2570 y(the) 32 b(normalizing) e(transformation.) 42 b(Sect.) i(6) 31 b(is) g(dev) m(oted) h(to) f(the) h(application) f(to) g(the) h (nonlinear) f(w) m(a) m(v) m(e) 0 2690 y(equation) 44 b(\(0.1\)) f(and) h(is) g(divided) h(in) f(t) m(w) m(o) g(subsections:) 67 b(in) 44 b(the) h(\014rst) f(one) h(w) m(e) f(giv) m(e) g(equation) g (\(0.1\)) 0 2809 y(the) c(form) f(needed) j(for) e(the) g(application) g (of) g(the) g(abstract) g(theorem;) i(in) e(the) h(second) g (subsection) g(w) m(e) 0 2929 y(pro) m(v) m(e) d(that) f(for) g(most) g (v) -6 b(alues) 37 b(of) h(a) f(suitable) h(parameter) e(the) i (frequencies) h(ful\014ll) f(the) g(nonresonance) 0 3049 y(condition) 33 b(needed) i(to) e(apply) g(the) h(theorem.) p Fn 0 3646 a(2.) 115 b(Statemen) m(t) 35 b(of) j(the) f(results) p Fm 199 3915 a(T) -8 b(o) 33 b(start) g(with) g(w) m(e) g(precise) i (our) e(assumptions.) 44 b(W) -8 b(rite) p Fl 33 w(g) p Fm 36 w(as) p Fl 491 4154 a(g) p Fm 4 w(\() p Fl(x;) 17 b(u) p Fm(\)) 26 b(=) p Fl 28 w(g) p Fi 959 4169 a(0) p Fm 1003 4154 a(\() p Fl(x) p Fm(\)) p Fl(u) p Fm 22 w(+) g(\026) p Fl -54 w(g) p Fm 4 w(\() p Fl(x;) 17 b(u) p Fm(\)) p Fl 31 w(;) 116 b(g) p Fi 1828 4169 a(0) p Fm 1873 4154 a(\() p Fl(x) p Fm(\)) 27 b(=) p Fl 28 w(m) p Fm 22 w(+) p Fl 23 w(V) p Fm 22 w(\() p Fl(x) p Fm(\)) p Fl 33 w(;) p Fm 119 w(\026) p Fl -53 w(g) p Fm 3 w(\() p Fl(x;) 17 b(u) p Fm(\)) 26 b(=) p Fj 29 w(O) p Fm 3 w(\() p Fj(j) p Fl(u) p Fj(j) p Fi 3395 4113 a(2) p Fm 3438 4154 a(\)) 0 4394 y(where) p Fl 33 w(m) p Fm 33 w(is) 33 b(the) g(a) m(v) m(erage) g (o) m(v) m(er) f([) p Fj(\000) p Fl(\031) t(;) 17 b(\031) p Fm 4 w(]) 31 b(of) p Fl 32 w(g) p Fi 1738 4409 a(0) p Fm 1815 4394 a(and) p Fl 33 w(V) p Fm 55 w(has) i(zero) g(a) m(v) m (erage.) 44 b(Consider) 33 b(no) m(w) g(the) g(Sturm) 0 4513 y(Liouville) f(problem) p Fj 976 4633 a(\000) p Fl(@) p Fk 1106 4648 a(xx) p Fl 1201 4633 a(') p Fk 1266 4648 a(j) p Fm 1331 4633 a(+) p Fl 22 w(V) 23 b(') p Fk 1576 4648 a(j) p Fm 1645 4633 a(=) p Fl 29 w(\026) p Fk 1811 4648 a(j) p Fl 1853 4633 a(') p Fk 1918 4648 a(j) p Fl 1993 4633 a(;) 116 b(') p Fk 2202 4648 a(j) p Fm 2244 4633 a(\(0\)) 27 b(=) p Fl 28 w(') p Fk 2569 4648 a(j) p Fm 2611 4633 a(\() p Fl(\031) p Fm 4 w(\)) g(=) h(0) 33 b(;) 771 b(\(2) p Fl(:) p Fm(1\)) 0 4822 y(it) 29 b(is) g(w) m(ell) g (kno) m(wn) h(that,) f(since) p Fl 30 w(V) p Fm 52 w(is) g(analytic) f (and) i(symmetric,) e(the) h(solutions) p Fj 30 w(f) p Fl(') p Fk 3158 4837 a(j) p Fj 3200 4822 a(g) p Fk 3250 4837 a(j) p Fe 4 w(\025) p Fi(1) p Fm 3423 4822 a(of) g(\(2.1\)) f (form) 0 4941 y(an) 33 b(orthogonal) f(basis) h(of) p Fl 33 w(L) p Fi 1061 4905 a(2) p Fm 1138 4941 a(that) g(w) m(e) g(will) e(assume) i(to) g(b) s(e) g(normalized,) e(and) j(that) e(they) h(are) g (analytic) 0 5061 y(and) 39 b(sk) m(ewsymmetric.) p Ff 57 w(We) g(assume) i(that) f(the) g(eigenvalues) p Fl 40 w(\026) p Fk 2447 5076 a(j) p Ff 2529 5061 a(ar) -5 b(e) 40 b(distinct) p Fm 37 w(\(the) e(case) h(of) g(m) m(ultiple) 0 5181 y(eigen) m(v) -6 b(alues) 24 b(can) f(also) f(b) s(e) h(dealt) g (with) f(v) m(ery) h(easily) -8 b(,) 23 b(but) g(the) h(statemen) m(t) e (is) g(more) g(complicate\).) 39 b(Remark) 0 5300 y(that) 33 b(the) g(eigen) m(v) -6 b(alues) p Fl 35 w(\026) p Fk 963 5315 a(j) p Fm 1038 5300 a(de\014ne) 35 b(the) e(frequencies) i(of) f(oscillation) e(of) h(the) h(linear) f(system) f(b) m(y) p Fl 1616 5539 a(!) p Fk 1678 5554 a(j) p Fm 1748 5539 a(:=) p Fh 1881 5459 a(p) p 1980 5459 312 4 v Fl 1980 5539 a(\026) p Fk 2040 5554 a(j) p Fm 2104 5539 a(+) p Fl 23 w(m) i(:) p Fm 1411 w(\(2) p Fl(:) p Fm(2\)) 1807 5778 y(20.03.02) p 90 rotate dyy eop %%Page: 4 4 4 3 bop Fg 0 60 a(4) 1685 b(D.) 33 b(Bam) m(busi) p Fm 199 299 a(It) g(is) g(w) m(ell) g(kno) m(wn) h(that) f(\(0.1\)) f(is) h (Hamiltonian) e(with) i(Hamiltonian) e(function) j(giv) m(en) f(b) m(y) p Fl 1700 523 a(H) p Fm 36 w(=) p Fl 28 w(h) p Fi 1981 538 a(0) p Fm 2048 523 a(+) p Fl 23 w(f) 44 b(;) p Fm 1495 w(\(2) p Fl(:) p Fm(3\)) 0 747 y(where) p Fl 567 906 a(h) p Fi 624 921 a(0) p Fm 669 906 a(\() p Fl(v) t(;) 17 b(u) p Fm(\)) 26 b(:=) p Fh 1060 770 a(Z) p Fk 1160 795 a(\031) p Fi 1115 996 a(0) p Fm 1242 838 a([) p Fl(v) p Fm 4 w(\() p Fl(x) p Fm(\)]) p Fi 1485 802 a(2) p Fm 1551 838 a(+) c([) p Fl(u) p Fk 1735 853 a(x) p Fm 1785 838 a(\() p Fl(x) p Fm(\)]) p Fi 1948 802 a(2) p Fm 2014 838 a(+) p Fl 23 w(V) p Fm 22 w(\() p Fl(x) p Fm(\)[) p Fl(u) p Fm(\() p Fl(x) p Fm(\)]) p Fi 2577 802 a(2) p Fm 2642 838 a(+) p Fl 23 w(m) p Fm([) p Fl(u) p Fm(\() p Fl(x) p Fm(\)]) p Fi 3077 802 a(2) p 1242 883 1879 4 v Fm 2156 974 a(2) p Fl 3133 906 a(dx) 33 b(;) 705 1177 y(f) p Fm 11 w(\() p Fl(u) p Fm(\)) 27 b(:=) p Fh 1060 1042 a(Z) p Fk 1160 1066 a(\031) p Fi 1115 1268 a(0) p Fl 1230 1177 a(G) p Fm(\() p Fl(x;) 17 b(u) p Fm(\() p Fl(x) p Fm(\)\)) p Fl(dx) p Fm 32 w(;) p Fl 0 1416 a(v) p Fm 46 w(:=) p Fl 42 w(u) p Fk 298 1431 a(t) p Fm 375 1416 a(is) 42 b(the) g(momen) m(tum) e(conjugated) j(to) p Fl 41 w(u) p Fm 42 w(and) p Fl 42 w(G) p Fm(\() p Fl(x;) 17 b(u) p Fm(\)) 41 b(is) h(suc) m(h) h(that) p Fl 42 w(@) p Fk 3118 1431 a(u) p Fl 3170 1416 a(G) p Fm 42 w(=) j(\026) p Fl -53 w(g) p Fm 3 w(.) 70 b(T) -8 b(o) 41 b(de\014ne) 0 1535 y(formally) 36 b(the) j(phase) h(space) f(consider) h(the) e(Sob) s (olev) g(space) p Fl 40 w(H) p Fk 2438 1499 a(s) p Fm 2519 1535 a(of) g(the) h(functions) p Fl 40 w(u) p Fj 36 w(2) p Fl 37 w(L) p Fi 3513 1499 a(2) p Fm 3596 1535 a(ha) m(ving) p Fl 38 w(s) p Fm 0 1655 a(w) m(eak) 34 b(deriv) -6 b(ativ) m(es) 35 b(in) p Fl 34 w(L) p Fi 933 1619 a(2) p Fm 977 1655 a(.) 48 b(F) -8 b(or) 34 b(an) m(y) g(in) m(teger) p Fl 35 w(s) p Fj 29 w(\025) p Fm 30 w(2) h(w) m(e) f(de\014ne) i(the) f(phase) h(space) p Fj 35 w(F) p Fk 3248 1670 a(s) p Fm 3325 1655 a(as) e(the) h(space) g (of) 0 1774 y(the) f(functions) g(\() p Fl(u;) 17 b(v) p Fm 4 w(\)) p Fj 26 w(2) 29 b(F) p Fk 1026 1789 a(s) p Fm 1095 1774 a(:=) p Fl 28 w(H) p Fk 1319 1738 a(s) p Fj 1384 1774 a(\010) p Fl 22 w(H) p Fk 1574 1738 a(s) p Fe(\000) p Fi(1) p Fm 1752 1774 a(ful\014lling) k(the) h(compatibilit) m (y) d(conditions) p Fl 1072 2045 a(u) p Fi 1129 2004 a(\(2) p Fk(j) p Fi 4 w(\)) p Fm 1273 2045 a(\(0\)) c(=) p Fl 28 w(u) p Fi 1590 2004 a(\(2) p Fk(j) p Fi 4 w(\)) p Fm 1734 2045 a(\() p Fl(\031) p Fm 4 w(\)) f(=) j(0) p Fl 33 w(;) p Fm 115 w(0) p Fj 28 w(\024) p Fl 28 w(j) p Fj 34 w(\024) p Fl 2606 1978 a(s) p Fj 21 w(\000) p Fm 23 w(1) p 2606 2022 219 4 v 2690 2114 a(2) p Fl 2869 2045 a(;) 1077 2264 y(v) p Fi 1129 2223 a(\(2) p Fk(j) p Fi 4 w(\)) p Fm 1273 2264 a(\(0\)) e(=) p Fl 28 w(v) p Fi 1585 2223 a(\(2) p Fk(j) p Fi 4 w(\)) p Fm 1729 2264 a(\() p Fl(\031) p Fm 4 w(\)) f(=) i(0) p Fl 33 w(;) p Fm 116 w(0) p Fj 28 w(\024) p Fl 28 w(j) p Fj 33 w(\024) p Fl 2602 2196 a(s) p 2600 2241 50 4 v Fm 2600 2332 a(2) p Fj 2684 2264 a(\000) p Fm 23 w(1) p Fl 33 w(;) p Fm 3764 2148 a(\(2) p Fl(:) p Fm(4\)) 0 2521 y(\(the) j(upp) s(er) i(index) e (in) g(paren) m(theses) j(denoting) d(deriv) -6 b(ativ) m(e) 31 b(of) h(the) f(corresp) s(onding) h(order\).) 44 b(W) -8 b(e) 32 b(endo) m(w) p Fj 0 2641 a(F) p Fk 72 2656 a(s) p Fm 147 2641 a(b) m(y) i(the) f(norm) p Fj 857 2916 a(k) p Fm(\() p Fl(u;) 17 b(v) p Fm 4 w(\)) p Fj(k) p Fi 1188 2866 a(2) p Fe 1188 2946 a(F) p Fd 1245 2956 a(s) p Fm 1316 2916 a(:=) p Fh 1449 2780 a(Z) p Fk 1548 2805 a(\031) p Fi 1504 3007 a(0) p Fh 1619 2835 a(\002) p Fm 1660 2916 a(\() p Fl(@) p Fk 1758 2875 a(s) p Fe(\000) p Fi(1) p Fk 1752 2940 a(x) p Fl 1902 2916 a(v) p Fm 4 w(\)) p Fi 1993 2875 a(2) p Fm 2060 2916 a(+) 22 b(\() p Fl(@) p Fk 2257 2875 a(s) 2251 2940 y(x) p Fl 2301 2916 a(u) p Fm(\)) p Fi 2397 2875 a(2) p Fm 2464 2916 a(+) p Fl 22 w(u) p Fi 2620 2875 a(2) p Fm 2687 2916 a(+) p Fl 23 w(v) p Fi 2839 2875 a(2) p Fh 2883 2835 a(\003) p Fl 2941 2916 a(dx) p Fm 33 w(;) 0 3196 y(w) m(e) 34 b(will) e(denote) i(b) m(y) p Fl 33 w(B) p Fk 870 3211 a(s) p Fm 912 3196 a(\() p Fl(R) p Fm 1 w(\)) f(the) g(ball) g(of) g(radius) p Fl 34 w(R) p Fm 34 w(in) p Fj 33 w(F) p Fk 2176 3211 a(s) p Fm 2251 3196 a(cen) m(tered) i(at) e(the) h(origin.) p Fn 0 3355 a(Remark) 145 b(2.1.) p Fg 32 w(It) 31 b(is) g(w) m(ell) h (kno) m(wn) f(that) g(the) h(Cauc) m(h) m(y) g(problem) f(for) h (equation) f(\(0.1\)) g(is) g(lo) s(cally) f(w) m(ell) 0 3475 y(p) s(osed) k(in) p Fj 33 w(F) p Fk 468 3490 a(s) p Fg 510 3475 a(,) f(and) h(that) f(it) g(is) g(also) g(globally) e(w) m (ell) i(p) s(osed) h(for) g(small) d(enough) k(initial) c(data.) p Fm 199 3634 a(T) -8 b(o) 33 b(in) m(tro) s(duce) g(the) h(normal) d(mo) s(des) h(and) i(the) f(linear) g(actions) g(consider) g(the) h (expansion) f(of) p Fl 33 w(u) p Fm 33 w(and) p Fl 0 3754 a(v) p Fm 37 w(on) g(the) h(eigenfunctions) p Fl 35 w(') p Fk 1111 3769 a(j) p Fm 1153 3754 a(:) p Fl 1047 4001 a(u) p Fm(\() p Fl(x) p Fm(\)) 27 b(=) p Fh 1371 3907 a(X) p Fk 1374 4121 a(j) p Fe 4 w(\025) p Fi(1) p Fl 1532 4001 a(u) p Fk 1589 4016 a(j) p Fl 1631 4001 a(') p Fk 1696 4016 a(j) p Fm 1738 4001 a(\() p Fl(x) p Fm(\)) p Fl 32 w(;) 116 b(v) p Fm 4 w(\() p Fl(x) p Fm(\)) 27 b(=) p Fh 2368 3907 a(X) p Fk 2371 4121 a(j) p Fe 4 w(\025) p Fi(1) p Fl 2529 4001 a(v) p Fk 2577 4016 a(j) p Fl 2619 4001 a(') p Fk 2684 4016 a(j) p Fm 2726 4001 a(\() p Fl(x) p Fm(\)) p Fl 33 w(;) p Fm 0 4328 a(and) 34 b(de\014ne) h(the) p Fl 33 w(j) p Fm 6 w({th) e(action) g(of) g(the) h(linearized) f(system) g(b) m(y) p Fl 1409 4621 a(I) p Fk 1453 4636 a(j) p Fj 1523 4621 a(\021) p Fl 28 w(I) p Fk 1672 4636 a(j) p Fm 1714 4621 a(\() p Fl(u;) 17 b(v) p Fm 4 w(\)) 26 b(:=) p Fl 2117 4543 a(v) p Fi 2169 4507 a(2) p Fk 2165 4569 a(j) p Fm 2236 4543 a(+) p Fl 22 w(!) p Fi 2401 4507 a(2) p Fk 2397 4569 a(j) p Fl 2445 4543 a(u) p Fi 2502 4507 a(2) p Fk 2502 4569 a(j) p 2117 4598 431 4 v Fm 2255 4689 a(2) p Fl(!) p Fk 2367 4704 a(j) p Fm 199 4902 a(Ha) m(ving) 37 b(\014xed) g(t) m(w) m(o) g(p) s(ositiv) m(e) g(n) m (um) m(b) s(ers) p Fl 38 w(m) p Fi 1846 4917 a(1) p Fl 1925 4902 a(<) e(m) p Fi 2124 4917 a(2) p Fm 2169 4902 a(,) j(suc) m(h) g(that) p Fl 37 w(\026) p Fk 2743 4917 a(j) p Fm 2810 4902 a(+) p Fl 25 w(m) p Fi 2999 4917 a(1) p Fl 3078 4902 a(>) p Fm 35 w(0) f(for) g(all) p Fl 37 w(j) p Fj 39 w(\025) p Fm 35 w(1) g(w) m(e) 0 5021 y(will) 32 b(consider) i(the) g(parameter) p Fl 33 w(m) p Fm 33 w(as) g(v) -6 b(arying) 32 b(in) h(the) h(in) m(terv) -6 b(al) p Fj 33 w(I) p Fm 35 w(:=) 28 b([) p Fl(m) p Fi 2807 5036 a(1) p Fl 2852 5021 a(;) 17 b(m) p Fi 2984 5036 a(2) p Fm 3028 5021 a(].) 199 5141 y(W) -8 b(e) 34 b(ha) m(v) m(e) g(the) f(follo) m(wing) p Fn 0 5300 a(Theorem) 190 b(2.2.) p Fg 41 w(Fix) 41 b(a) g(p) s(ositiv) m(e) g(in) m(teger) p Fl 42 w(M) p Fj 52 w(\025) p Fm 41 w(4) p Fg(,) i(then) f(there) h (exists) e(a) g(subset) p Fj 43 w(J) 59 b(\032) 42 b(I) p Fg 49 w(with) 0 5420 y(measure) c(equal) g(to) p Fl 38 w(m) p Fi 875 5435 a(2) p Fj 946 5420 a(\000) p Fl 26 w(m) p Fi 1136 5435 a(1) p Fg 1219 5420 a(and) h(a) f(p) s(ositiv) m(e) f(\(large\)) h(in) m(teger) p Fl 38 w(s) p Fe 2587 5435 a(\003) p Fg 2671 5420 a(with) g(the) h(follo) m(wing) e(prop) s (erties:) 0 5539 y(for) 44 b(an) m(y) p Fl 43 w(m) p Fj 45 w(2) h(J) p Fg 61 w(there) g(exists) e(a) g(p) s(ositiv) m(e) p Fl 43 w(R) p Fe 1832 5554 a(\003) p Fg 1921 5539 a(and,) j(for) d(an) m (y) p Fl 44 w(R) i(<) g(R) p Fe 2835 5554 a(\003) p Fg 2924 5539 a(there) f(exists) f(an) h(analytic) p Fm 1807 5778 a(20.03.02) p 90 rotate dyy eop %%Page: 5 5 5 4 bop Fg 1247 60 a(Birkho\013) 33 b(normal) f(form) g(for) i(PDE's) p Fm 1246 w(5) p Fg 0 299 a(canonical) 44 b(transformation) p Fj 43 w(T) p Fk 1192 314 a(R) p Fm 1304 299 a(:) p Fl 46 w(B) p Fk 1454 314 a(s) p Fc 1492 324 a(\003) p Fm 1538 299 a(\() p Fl(R) q(=) p Fm(3\)) p Fj 44 w(!) p Fl 46 w(B) p Fk 2059 314 a(s) p Fc 2097 324 a(\003) p Fm 2143 299 a(\() p Fl(R) p Fm 1 w(\)) p Fg 43 w(whic) m(h) h(puts) g (the) g(Hamiltonian) c(\(2.3\)) i(in) 0 418 y(Birkho\013) 32 b(normal) g(form) h(up) h(to) e(order) p Fl 34 w(R) p Fk 1563 382 a(M) p Fg 1651 418 a(,) h(namely) f(suc) m(h) i(that) p Fm 1378 612 a(\() p Fl(h) p Fi 1474 627 a(0) p Fm 1541 612 a(+) p Fl 23 w(f) p Fm 11 w(\)) p Fj 21 w(\016) 22 b(T) p Fk 1887 627 a(R) p Fm 1980 612 a(=) p Fl 28 w(h) p Fi 2142 627 a(0) p Fm 2210 612 a(+) p Fl 22 w(Z) p Fm 29 w(+) p Fj 23 w(R) p Fg 0 806 a(with) p Fl 26 w(Z) p Fg 33 w(a) k(function) g(that) g(dep) s(ends) i(only) d(on) i(the) f (actions) p Fl 26 w(I) p Fk 2248 821 a(j) p Fg 2290 806 a(,) h(and) p Fj 26 w(R) p Fg 26 w(an) g(analytic) d(functional) j(ha) m (ving) f(a) 0 926 y(Hamiltonian) 21 b(v) m(ector) i(\014eld) p Fl 24 w(X) p Fe 1141 941 a(R) p Fg 1237 926 a(whic) m(h) h(is) g (analytic) e(as) h(a) g(function) i(from) p Fl 22 w(B) p Fk 2839 941 a(s) p Fc 2877 951 a(\003) p Fm 2923 926 a(\() p Fl(R) q(=) p Fm(3\)) p Fg 22 w(to) p Fj 23 w(F) p Fk 3384 941 a(s) p Fc 3422 951 a(\003) p Fg 3468 926 a(.) 40 b(Moreo) m(v) m(er,) 0 1045 y(for) c(an) m(y) h(in) m(teger) p Fl 36 w(s) p Fj 33 w(\025) p Fl 33 w(s) p Fe 916 1060 a(\003) p Fg 997 1045 a(there) g(exists) f(a) g(p) s(ositiv) m(e) g (constan) m(t) p Fl 37 w(C) p Fk 2467 1060 a(s) p Fg 2546 1045 a(suc) m(h) h(that) f(the) h(transformation) p Fj 35 w(T) p Fk 3903 1060 a(R) p Fg 0 1165 a(and) g(the) f(v) m(ector) g (\014eld) p Fl 36 w(X) p Fe 974 1180 a(R) p Fg 1083 1165 a(are) g(analytic) f(also) h(as) g(maps) g(from) p Fl 35 w(B) p Fk 2540 1180 a(s) p Fm 2582 1165 a(\() p Fl(R) q(=C) p Fk 2819 1180 a(s) p Fm 2861 1165 a(\)) p Fg 35 w(to) p Fj 36 w(F) p Fk 3132 1180 a(s) p Fg 3210 1165 a(and) h(the) f(follo) m (wing) 0 1284 y(estimates) c(hold) p Fm 1235 1380 a(sup) p Fe 1006 1469 a(k) p Fi(\() p Fk(u;v) p Fi 3 w(\)) p Fe(k) p Fc 1263 1491 a(F) p Fd 1312 1501 a(s) p Fe 1356 1469 a(\024) p Fk(R=C) p Fd 1576 1479 a(s) p Fj 1631 1380 a(k) p Fm(\() p Fl(u;) 17 b(v) p Fm 4 w(\)) p Fj 21 w(\000) 22 b(T) p Fk 2087 1395 a(R) p Fm 2153 1380 a(\() p Fl(u;) 17 b(v) p Fm 4 w(\)) p Fj(k) p Fe 2433 1410 a(F) p Fd 2490 1420 a(s) p Fj 2562 1380 a(\024) p Fl 28 w(C) p Fk 2738 1395 a(s) p Fl 2781 1380 a(R) p Fi 2858 1339 a(2) p Fl 2935 1380 a(;) p Fm 1636 1640 a(sup) p Fe 1407 1728 a(k) p Fi(\() p Fk(u;v) p Fi 3 w(\)) p Fe(k) p Fc 1664 1750 a(F) p Fd 1713 1760 a(s) p Fe 1757 1728 a(\024) p Fk(R=C) p Fd 1977 1738 a(s) p Fj 2032 1640 a(k) p Fl(X) p Fe 2165 1655 a(R) p Fj 2238 1640 a(k) p Fe 2288 1670 a(F) p Fd 2345 1680 a(s) p Fj 2416 1640 a(\024) p Fl 28 w(C) p Fk 2592 1655 a(s) p Fl 2635 1640 a(R) p Fk 2712 1599 a(M) p Fe 7 w(\000) p Fi(1) p Fl 2935 1640 a(:) p Fm 3764 1557 a(\(2) p Fl(:) p Fm(5\)) 199 2049 y(Consider) 40 b(no) m(w) g(an) f(initial) f(datum) h(\() 6 b(\026) p Fl -56 w(u;) p Fm 20 w(\026) p Fl -53 w(v) p Fm 3 w(\)) p Fj 38 w(2) 38 b(F) p Fk 2040 2064 a(s) p Fm 2121 2049 a(for) i(the) g(nonlinear) f(w) m(a) m(v) m(e) h(equation) f(\(0.1\)) f (and) 0 2169 y(de\014ne) k(the) e(in\014nite) h(dimensional) e(torus) i (on) f(whic) m(h) h(all) e(the) i(actions) f(coincide) h(with) f(the) g (actions) g(of) 0 2288 y(\() 6 b(\026) p Fl -56 w(u;) p Fm 20 w(\026) p Fl -53 w(v) p Fm 3 w(\),) 33 b(namely) p Fn 903 2408 a(T) p Fi 983 2423 a(0) p Fm 1055 2408 a(:=) p Fj 28 w(f) p Fm(\() p Fl(u;) 17 b(v) p Fm 4 w(\)) p Fj 26 w(2) 29 b(F) p Fk 1663 2423 a(s) p Fm 1766 2408 a(:) p Fl 60 w(I) p Fk 1898 2423 a(j) p Fm 1940 2408 a(\() p Fl(u;) 17 b(v) p Fm 4 w(\)) 26 b(=) p Fl 28 w(I) p Fk 2347 2423 a(j) p Fm 2389 2408 a(\() 6 b(\026) p Fl -56 w(u;) p Fm 20 w(\026) p Fl -53 w(v) p Fm 3 w(\)) p Fl 33 w(;) p Fj 50 w(8) p Fl(j) p Fj 33 w(\025) p Fm 28 w(1) p Fj(g) p Fm 0 2577 a(Then) 34 b(w) m(e) g(ha) m(v) m(e) g(the) f (follo) m(wing) p Fn 0 2736 a(Corollary) 129 b(2.3.) p Fg 28 w(Fix) p Fl 27 w(M) p Fj 38 w(\025) p Fm 28 w(4) p Fg 28 w(and) p Fl 28 w(m) p Fj 28 w(2) 28 b(J) p Fg 18 w(,) h(\014x) f(also) p Fl 27 w(s) p Fj 28 w(\025) p Fl 28 w(s) p Fe 2394 2751 a(\003) p Fg 2439 2736 a(,) h(then) f(there) h (exist) e(p) s(ositiv) m(e) h(constan) m(ts) p Fl 0 2856 a(R) p Fk 76 2871 a(s) p Fg 118 2856 a(,) p Fl 49 w(C) p Fk 266 2871 a(s) p Fg 354 2856 a(suc) m(h) 47 b(that) f(the) g(follo) m (wing) f(holds) h(true:) 69 b(let) p Fl 46 w(\020) p Fm 7 w(\() p Fl(t) p Fm(\)) p Fj 47 w(\021) p Fm 49 w(\() p Fl(u) p Fm(\() p Fl(t) p Fm(\)) p Fl(;) 17 b(v) p Fm 4 w(\() p Fl(t) p Fm(\)\)) p Fg 44 w(b) s(e) 46 b(the) g(solution) f (of) h(the) 0 2975 y(Cauc) m(h) m(y) 34 b(problem) f(for) g(equation) g (\(0.1\)) f(with) h(initial) f(datum) p Fm 32 w(\() 6 b(\026) p Fl -56 w(u;) p Fm 20 w(\026) p Fl -53 w(v) p Fm 3 w(\)) p Fj 28 w(2) 28 b(F) p Fk 2797 2990 a(s) p Fg 2872 2975 a(if) p Fl 1540 3169 a(") p Fm 28 w(:=) p Fj 28 w(k) p Fm(\() 6 b(\026) p Fl -56 w(u) o(;) p Fm 21 w(\026) p Fl -54 w(v) p Fm 4 w(\)) p Fj(k) p Fe 2077 3199 a(F) p Fd 2134 3209 a(s) p Fj 2205 3169 a(\024) p Fl 28 w(R) p Fk 2386 3184 a(s) p Fg 0 3363 a(then) 34 b(for) f(all) g(times) p Fl 32 w(t) p Fg 33 w(with) p Fj 1685 3505 a(j) p Fl(t) p Fj(j) 27 b(\024) p Fm 2071 3437 a(1) p 1921 3482 351 4 v Fl 1921 3573 a(C) p Fk 1992 3588 a(s) p Fl 2035 3573 a(") p Fk 2081 3544 a(M) p Fe 7 w(\000) p Fi(3) p Fm 3764 3505 a(\(2) p Fl(:) p Fm(6\)) p Fg 0 3717 a(the) 34 b(follo) m(wing) e(estimates) h(hold) p Fj 1069 3963 a(k) p Fl(\020) p Fm 7 w(\() p Fl(t) p Fm(\)) p Fj(k) p Fe 1333 3992 a(F) p Fd 1390 4002 a(s) p Fj 1461 3963 a(\024) p Fm 29 w(2) p Fl(C) p Fk 1688 3978 a(s) p Fl 1730 3963 a(") h(;) p Fj 1966 3895 a(j) p Fl -1 w(I) p Fk 2037 3910 a(j) p Fm 2079 3895 a(\() p Fl(t) p Fm(\)) p Fj 22 w(\000) p Fl 22 w(I) p Fk 2358 3910 a(j) p Fm 2400 3895 a(\(0\)) p Fj(j) p 1966 3940 590 4 v Fl 2215 4031 a(") p Fi 2261 4002 a(2) p Fj 2595 3963 a(\024) p Fl 2719 3895 a(C) p Fk 2790 3910 a(s) p 2712 3940 130 4 v Fl 2712 4031 a(j) p Fi 2759 4002 a(2) p Fk(s) p Fl 2853 3963 a(") p Fm 865 w(\(2) p Fl(:) p Fm(7\)) p Fg 0 4209 a(moreo) m(v) m(er,) 29 b(for) g(all) p Fl 29 w(s) p Fe 784 4173 a(0) p Fl 839 4209 a(<) f(s) p Fj 14 w(\000) p Fm 14 w(1) p Fl(=) p Fm(2) p Fg(,) p Fl 30 w(s) p Fe 1351 4173 a(0) p Fj 1406 4209 a(\025) p Fm 29 w(2) p Fg(,) h(there) h(exists) f(a) g(constan) m(t) p Fl 30 w(C) p Fk 2689 4224 a(s;s) p Fc 2789 4204 a(0) p Fg 2849 4209 a(suc) m(h) i(that,) e(up) h(to) f(the) g(times) 0 4329 y(\(2.6\)) j(one) i(has) p Fl 1465 4449 a(d) p Fk 1517 4464 a(s) p Fc 1555 4444 a(0) p Fm 1586 4449 a(\() p Fl(\020) p Fm 7 w(\() p Fl(t) p Fm(\)) p Fl(;) p Fn 17 w(T) p Fi 1915 4464 a(0) p Fm 1958 4449 a(\)) p Fj 28 w(\024) p Fl 28 w(C) p Fk 2201 4464 a(s;s) p Fc 2301 4444 a(0) p Fl 2332 4449 a(") p Fi 2378 4407 a(5) p Fk(=) p Fi(4) p Fm 3764 4449 a(\(2) p Fl(:) p Fm(8\)) p Fg 0 4618 a(where) p Fl 34 w(d) p Fk 340 4633 a(s) p Fc 378 4613 a(0) p Fm 409 4618 a(\() p Fl(:;) 17 b(:) p Fm(\)) p Fg 32 w(denotes) 34 b(the) g(distance) g(in) p Fj 33 w(F) p Fk 1728 4633 a(s) p Fc 1766 4613 a(0) p Fg 1797 4618 a(.) p Fm 199 4837 a(W) -8 b(e) 41 b(p) s(oin) m(t) f(out) f(that) h(the) g(\014rst) h(of) f(the) g(estimates) g(\(2.7\)) e(holds) j(also) e(for) i(the) f(nonlinear) g(Klein{) 0 4956 y(Gordon) 32 b(equation) f(\(corresp) s(onding) i(to) e(the) h(case) p Fl 32 w(V) p Fm 50 w(=) c(0\)) j(with) h(p) s(erio) s(dic) f(b) s (oundary) h(conditions) g(\(see) 0 5076 y(corollary) g(6.7) g(b) s(elo) m(w\),) h(and) h(also) f(the) g(second) i(of) e(\(2.7\)) f(has) i(an) f (anologous) g(in) g(this) h(case.) p Fn 0 5235 a(Remark) 166 b(2.4.) p Fg 36 w(Corollary) 35 b(2.3) g(has) i(to) e(b) s(e) i (confron) m(ted) g(with) f(the) g(main) f(result) i(of) f([6].) 52 b(In) 36 b([6]) f(the) 0 5355 y(estimates) 42 b(\(2.7\)) g(where) i (pro) m(v) m(en) g(for) g(solutions) f(corresp) s(onding) h(to) f (initial) f(data) h(in) p Fj 43 w(F) p Fk 3432 5370 a(s) p Fi(+) p Fk(s) p Fb 3569 5380 a(1) p Fg 3612 5355 a(,) i(with) p Fl 43 w(") p Fg 0 5474 a(de\014ned) 35 b(as) e(the) p Fj 34 w(F) p Fk 709 5489 a(s) p Fi(+) p Fk(s) p Fb 846 5499 a(1) p Fg 922 5474 a(norm) g(of) g(the) h(initial) d(datum,) i (and) p Fl 33 w(s) p Fi 2339 5489 a(1) p Fg 2417 5474 a(a) g(large) p Fl 33 w(M) p Fg 43 w(dep) s(enden) m(t) j(n) m(um) m(b) s(er.) p Fm 1807 5778 a(20.03.02) p 90 rotate dyy eop %%Page: 6 6 6 5 bop Fm 0 60 a(6) p Fg 1685 w(D.) 33 b(Bam) m(busi) p Fn 0 299 a(3.) 115 b(Idea) 37 b(of) i(the) e(pro) s(of) h(and) g (discussion) p Fm 199 570 a(W) -8 b(e) 45 b(b) s(egin) g(b) m(y) f (discussing) i(the) f(renonresonance) h(condition.) 78 b(T) -8 b(o) 44 b(this) g(end) h(consider) h(a) e(\014nite) 0 689 y(sequence) 38 b(of) f(frequencies) p Fl 38 w(!) p Fj 37 w(\021) c(f) p Fl(!) p Fk 1367 704 a(j) p Fj 1409 689 a(g) p Fk 1459 653 a(n) 1459 715 y(j) p Fi 4 w(=1) p Fm 1602 689 a(,) k(and) g(recall) f(that) g(the) h(standard) g (nonresonance) h(condition) 0 809 y(used) 33 b(in) f(order) h(to) e (pro) m(v) m(e) i(Birkho\013) 8 b('s) 30 b(theorem) i(in) g(a) g (\014nite) g(dimensional) f(con) m(text) h(is) g(\\for) g(an) m(y) g (in) m(teger) p Fl 0 928 a(r) p Fj 30 w(\025) p Fm 29 w(3) f(there) h(exists) g(a) f(p) s(ositiv) m(e) p Fl 31 w(\016) p Fk 1281 943 a(r) p Fm 1357 928 a(suc) m(h) j(that) p Fj 31 w(j) p Fl(!) p Fj 22 w(\001) p Fn 19 w(k) p Fj(j) 28 b(\025) p Fl 28 w(\016) p Fk 2219 943 a(r) p Fm 2263 928 a(,) p Fj 32 w(8) p Fn 10 w(k) p Fj 28 w(2) p Fn 29 w(Z) p Fk 2641 892 a(n) p Fj 2705 928 a(nf) p Fm(0) p Fj(g) p Fm 31 w(with) p Fj 31 w(j) p Fn(k) p Fj(j) g(\024) p Fl 28 w(r) p Fm 3 w(".) 43 b(It) 31 b(can) h(b) s(e) 0 1048 y(easily) k(seen) h(that) f(in) g(general) h(suc) m(h) g(a) f (condition) h(is) f(not) g(satis\014ed) h(in) f(the) h(in\014nite) f (dimensional) g(case) p Fl 0 1167 a(n) p Fm 27 w(=) p Fj 29 w(1) p Fm(,) c(so) h(it) g(has) h(to) f(b) s(e) g(mo) s(di\014ed) g(in) g(order) h(to) f(deal) g(with) g(PDEs.) 199 1289 y(T) -8 b(o) 33 b(this) g(end) h(remark) e(that,) h(giv) m(en) p Fn 33 w(k) p Fm 34 w(as) g(ab) s(o) m(v) m(e,) g(one) h(has) p Fn 1354 1533 a(k) p Fj 23 w(\001) p Fl 22 w(!) p Fj 31 w(\021) 28 b(\006) p Fl(!) p Fk 1824 1548 a(i) p Fb 1852 1558 a(1) p Fj 1919 1533 a(\006) p Fl 22 w(!) p Fk 2080 1548 a(i) p Fb 2108 1558 a(2) p Fj 2175 1533 a(\006) p Fl 22 w(:::) p Fj 21 w(\006) p Fl 23 w(!) p Fk 2541 1548 a(i) p Fd 2569 1558 a(r) p Fm 0 1777 a(with) g(a) g(suitable) h(c) m (hoice) h(of) e(the) h(signs) g(and) g(of) g(the) g(indices) p Fl 29 w(i) p Fi 2295 1792 a(1) p Fl 2340 1777 a(;) 17 b(:::;) g(i) p Fk 2548 1792 a(r) p Fm 2590 1777 a(,) 29 b(and) g(consider) h(the) f(nonresonance) 0 1897 y(condition) p Fj 1240 2021 a(j) p Fl(!) p Fk 1330 2036 a(i) p Fb 1358 2046 a(1) p Fj 1424 2021 a(\006) p Fl 23 w(!) p Fk 1586 2036 a(i) p Fb 1614 2046 a(2) p Fj 1680 2021 a(\006) p Fl 23 w(:::) p Fj 21 w(\006) p Fl 22 w(!) p Fk 2046 2036 a(i) p Fd 2074 2046 a(r) p Fj 2120 2021 a(j) e(\025) p Fl 2431 1954 a(\015) p 2292 1999 336 4 v 2292 2090 a(i) p Fk 2326 2056 a(\034) p Fi 2326 2116 a(1) p Fl 2376 2090 a(i) p Fk 2410 2056 a(\034) p Fi 2410 2116 a(2) p Fl 2460 2090 a(:::i) p Fk 2578 2061 a(\034) 2578 2114 y(r) p Fm 2700 2021 a(:) 1036 b(\(3) p Fl(:) p Fm(1\)) 0 2276 y(w) m(e) 40 b(will) e(pro) m(v) m(e) i(that,) h(pro) m(vided) p Fl 40 w(\034) p Fm 50 w(is) f(su\016cien) m(tly) g(large) f(\(dep) s (ending) i(on) p Fl 39 w(r) p Fm 3 w(\),) g(the) e(frequencies) j (\(2.2\)) 0 2396 y(ful\014ll) 27 b(\(3.1\)) f(if) p Fl 27 w(m) p Fm 27 w(is) h(in) g(a) f(set) h(with) g(complemen) m(t) f(ha) m(ving) g(measure) h(of) g(the) h(order) f(of) p Fl 27 w(\015) p Fi 3270 2359 a(1) p Fk(=r) p Fm 3421 2396 a(\(see) g(theorem) 0 2515 y(6.16\).) 61 b(W) -8 b(e) 40 b(p) s(oin) m(t) f(out) g(that) g (condition) h(\(3.1\)) e(is) h(ful\014lled) h(with) f(large) g (probabilit) m(y) g(in) g(v) m(ery) g(general) 0 2635 y(mo) s(dels.) h(F) -8 b(or) 24 b(example,) h(this) f(is) g(true) h(if) p Fl 24 w(!) p Fj 32 w(2) p Fm 28 w([0) p Fl(;) p Fm 17 w(1]) p Fe 1901 2598 a(1) p Fm 2008 2635 a(with) f([0) p Fl(;) p Fm 17 w(1]) p Fe 2427 2598 a(1) p Fm 2534 2635 a(endo) m(w) m(ed) i(b) m(y) e(the) h(pro) s(duct) g(measure.) p Ff 199 2756 a(Condition) 38 b(\(3.1\)) g(makes) g(cle) -5 b(ar) 39 b(that) g(very) f(smal) 5 b(l) 40 b(smal) 5 b(l{denominators) 39 b(always) h(involve) e(many) 0 2875 y(times) d(lar) -5 b(ge) 36 b(indexes) p Fm 32 w(\(or) d(man) m(y) f (times) g(a) h(large) g(index\).) 199 2997 y(Recall) g(no) m(w) g(that) f(small) f(denominators) h(app) s(ear) h(in) g(solving) e(the) i (homological) e(equation.) 43 b(More) 0 3116 y(precisely) -8 b(,) 28 b(ha) m(ving) e(in) m(tro) s(duced) i(v) -6 b(ariables) p Fj 26 w(f) p Fm(\() p Fl(\030) p Fk 1767 3131 a(j) p Fl 1808 3116 a(;) 17 b(\021) p Fk 1902 3131 a(j) p Fm 1944 3116 a(\)) p Fj(g) p Fk 2033 3150 a(j) p Fe 4 w(\025) p Fi(1) p Fm 2203 3116 a(in) 27 b(whic) m(h) g(the) g(op) s(erator) f(of) h(P) m(oisson) g(brac) m(k) m(ets) 0 3236 y(with) p Fl 32 w(h) p Fi 283 3251 a(0) p Fm 360 3236 a(is) 32 b(diagonal,) f(the) h (algorithm) e(of) i(Birkho\013) f(normal) g(form) g(in) m(tro) s(duces) i(a) f(denominator) f(of) h(the) 0 3355 y(form) p Fl 996 3480 a(!) p Fk 1058 3495 a(k) p Fb 1100 3505 a(1) p Fm 1166 3480 a(+) p Fl 23 w(!) p Fk 1328 3495 a(k) p Fb 1370 3505 a(2) p Fm 1436 3480 a(+) p Fl 23 w(:::) p Fm 21 w(+) p Fl 22 w(!) p Fk 1802 3495 a(k) p Fd 1844 3505 a(n) p Fb 1888 3520 a(1) p Fj 1960 3480 a(\000) p Fl 22 w(!) p Fk 2121 3495 a(j) p Fb 2154 3505 a(1) p Fj 2220 3480 a(\000) p Fl 23 w(!) p Fk 2382 3495 a(j) p Fb 2415 3505 a(2) p Fj 2481 3480 a(\000) p Fl 22 w(:::) p Fj 22 w(\000) p Fl 22 w(!) p Fk 2847 3495 a(j) p Fd 2880 3505 a(n) p Fb 2924 3520 a(2) p Fm 0 3675 a(in) h(order) h(to) f (eliminate) e(from) i(the) g(nonlinearit) m(y) g(a) g(monomial) e(of) i (the) h(form) p Fl 1356 3919 a(f) p Fk 1405 3934 a(k) r(;j) p Fl 1547 3919 a(\030) p Fk 1591 3934 a(k) p Fb 1633 3944 a(1) p Fl 1677 3919 a(\030) p Fk 1721 3934 a(k) p Fb 1763 3944 a(2) p Fl 1807 3919 a(:::\030) p Fk 1935 3934 a(k) p Fd 1977 3944 a(n) p Fb 2021 3959 a(1) p Fl 2069 3919 a(\021) p Fk 2118 3934 a(j) p Fb 2151 3944 a(1) p Fl 2195 3919 a(\021) p Fk 2244 3934 a(j) p Fb 2277 3944 a(2) p Fl 2321 3919 a(:::\021) p Fk 2454 3934 a(j) p Fd 2487 3944 a(n) p Fb 2531 3959 a(2) p Fm 3764 3919 a(\(3) p Fl(:) p Fm(2\)) 0 4163 y(where) p Fl 46 w(f) p Fk 349 4178 a(k) r(;j) p Fj 505 4163 a(2) p Fn 47 w(C) p Fm 44 w(is) 45 b(a) g(complex) e(co) s(e\016cien) m(t.) 79 b(The) 46 b(main) d(p) s(oin) m(t) i(is) f(that,) j(due) f(to) e(the) h (lo) s(calit) m(y) e(of) 0 4283 y(the) 32 b(nonlinearit) m(y) f(in) g (\(0.1\),) f(the) i(v) m(ector) f(\014eld) h(of) g(monomial) c(ha) m (ving) j(a) h(non) m(v) -6 b(anishing) 32 b(co) s(e\016cien) m(t) p Fl 32 w(f) p Fk 3859 4298 a(k) r(;j) p Fm 0 4402 a(turns) 37 b(out) g(to) f(b) s(e) h(small) e(when) j(at) e(least) h(three) g(of) g (the) g(indexes) p Fl 37 w(k) p Fk 2556 4417 a(l) p Fm 2587 4402 a(,) p Fl 37 w(j) p Fk 2693 4417 a(l) p Fm 2760 4402 a(are) g(large) f(and) h(the) h(sequences) p Fl 0 4522 a(\030) p Fk 44 4537 a(j) p Fl 86 4522 a(;) 17 b(\021) p Fk 180 4537 a(j) p Fm 254 4522 a(deca) m(y) 34 b(fast) g(enough) g(with) p Fl 33 w(j) p Fm 39 w(\(see) g(lemma) d(5.6) h(and) i(prop) s(osition) f(5.8) f(b) s(elo) m(w\).) 199 4643 y(The) 26 b(smallness) f(of) g(monomials) d(in) m(v) m(olving) i (man) m(y) g(v) -6 b(ariables) 25 b(with) g(large) g(index) g(allo) m (ws) f(to) h(p) s(erform) 0 4763 y(a) k(spatial) g(analogue) g(of) g (the) h(ultra) m(violet) e(cuto\013) h(curren) m(tly) h(used) h(in) e (KAM) h(theory) -8 b(,) 29 b(namely) f(a) h(momen-) 0 4882 y(tum) k(cuto\013:) 46 b(Precisely) 34 b(w) m(e) g(\014x) g(a) f (large) p Fl 34 w(N) p Fm 45 w(and) h(eliminate) f(from) g(the) h (nonlinearit) m(y) g(only) f(monomials) 0 5002 y(whic) m(h) 44 b(con) m(tain) g(at) f(most) f(t) m(w) m(o) h(v) -6 b(ariables) 43 b(with) g(index) h(larger) f(than) p Fl 43 w(N) p Fm 11 w(,) j(the) e(others) f(b) s(eing) h(already) 0 5121 y(small.) d(In) 31 b(order) g(to) e(do) i(that) e(w) m(e) i(assume) f (that) g(the) g(small) f(denominators) h(one) g(has) h(to) f(con) m (trol) g(satisfy) 0 5241 y(a) j(diophan) m(tine) h(t) m(yp) s(e) f (condition) h(stronger) f(than) h(\(3.1\),) d(namely) h(that) h(there) h (exists) p Fl 33 w(\013) p Fm 33 w(suc) m(h) h(that) p Fj 1133 5511 a(j) p Fl -1 w(!) p Fk 1222 5526 a(i) p Fb 1250 5536 a(1) p Fj 1316 5511 a(\006) p Fl 23 w(!) p Fk 1478 5526 a(i) p Fb 1506 5536 a(2) p Fj 1572 5511 a(\006) p Fl 23 w(:::) p Fj 21 w(\006) p Fl 23 w(!) p Fk 1939 5526 a(i) p Fd 1967 5536 a(r) p Fj 2034 5511 a(\006) p Fl 23 w(!) p Fk 2196 5526 a(j) p Fj 2260 5511 a(\006) p Fl 23 w(!) p Fk 2422 5526 a(k) p Fj 2471 5511 a(j) 27 b(\025) p Fl 2688 5443 a(\015) p 2643 5488 148 4 v 2643 5579 a(N) p Fk 2734 5550 a(\013) p Fm 3764 5511 a(\(3) p Fl(:) p Fm(3\)) 1807 5778 y(20.03.02) p 90 rotate dyy eop %%Page: 7 7 7 6 bop Fg 1247 60 a(Birkho\013) 33 b(normal) f(form) g(for) i(PDE's) p Fm 1246 w(7) 0 299 y(where) e(the) f(indexes) p Fl 31 w(i) p Fk 838 314 a(l) p Fm 900 299 a(are) f(smaller) g(than) p Fl 31 w(N) p Fm 41 w(and) p Fl 32 w(k) p Fj 30 w(\025) p Fl 28 w(j) k(>) 28 b(N) p Fm 11 w(.) 43 b(This) 31 b(is) f(the) i (condition) e(w) m(e) h(will) f(pro) m(v) m(e) 0 418 y(to) h(ha) m(v) m(e) g(full) h(measure) f(\(see) h(theorem) e(6.5) h (b) s(elo) m(w\).) 43 b(W) -8 b(e) 31 b(p) s(oin) m(t) g(out) g(that) g (a) g(related) g(diophan) m(tine) i(t) m(yp) s(e) 0 538 y(condition) h(w) m(as) g(already) f(used) i(in) f([6]) f(b) m(y) h (Bourgain,) f(and) h(that) f(a) h(momen) m(tum) d(cuto\013) j(where) h (already) 0 658 y(used) f(in) g(di\013eren) m(t) g(con) m(texts) f(in) h ([5,11].) 199 780 y(W) -8 b(e) 44 b(also) f(p) s(oin) m(t) h(out) f (that) g(the) h(prop) s(erties) g(that) f(allo) m(w) g(to) g(pro) m(v) m (e) h(that) f(monomials) e(in) m(v) m(olving) 0 899 y(v) -6 b(ariables) 31 b(with) g(large) g(index) h(ha) m(v) m(e) g(a) f(small) f (v) m(ector) h(\014eld) h(are) f(\(i\)) g(the) h(regularit) m(y) e(of) i (the) g(nonlinearit) m(y) -8 b(,) 0 1019 y(\(ii\)) 33 b(its) h(lo) s(calit) m(y) e(and) i(\(iii\)) f(the) h(fact) h(that) e (the) i(eigenfunctions) g(of) f(the) h(linearized) f(problem) g(are) g (\\w) m(ell) 0 1138 y(lo) s(calized) f(with) h(resp) s(ect) g(to) f (the) h(F) -8 b(ourier) 34 b(basis".) 45 b(T) -8 b(ec) m(hnically) 34 b(w) m(e) g(are) f(able) h(to) f(obtain) h(the) g(pro) s(of) f(b) m(y) 0 1258 y(a) c(tric) m(k) f(whic) m(h) i(consists) f(in) g(em) m(b) s (edding) h(the) f(dynamical) e(system) h(\(0.1\)) g(in) m(to) h(a) f (larger) h(one,) h(essen) m(tially) 0 1377 y(in) k(the) g(same) g(w) m (a) m(y) g(as) g(one) g(em) m(b) s(eds) h(p) s(erio) s(dic) f(sk) m (ewsymmetric) e(functions) j(in) g(the) f(spaces) h(of) g(p) s(erio) s (dic) 0 1497 y(functions) 45 b(in) e(order) h(to) f(use) h(the) g(F) -8 b(ourier) 43 b(expansions) i(on) e(the) h(exp) s(onen) m(tials) g(whic) m(h) g(is) f(simpler) g(to) 0 1617 y(manipulate) 32 b(than) i(the) f (sin) h(expansion.) 199 1739 y(W) -8 b(e) 26 b(conclude) g(b) m(y) f (remarking) e(that) i(our) g(approac) m(h) h(applies) f(directly) g (also) f(to) h(the) g(case) g(of) h(the) f(non-) 0 1858 y(linear) 35 b(Klein{Gordon) h(equation) e(with) h(p) s(erio) s(dic) g (b) s(oundary) h(conditions) f(\(see) h(corollary) e(6.7) g(b) s(elo) m (w\),) 0 1978 y(and) i(also) f(to) g(quite) g(general) h(equations) f (in) g(one) h(space) h(dimension.) 50 b(F) -8 b(or) 35 b(example) f(it) h(can) h(b) s(e) g(used) g(in) 0 2097 y(the) e(case) g(of) g(the) g(nonlinear) g(Sc) m(hr\177) -50 b(odinger) 36 b(equation) d(with) g(or) h(without) f(p) s(oten) m (tial,) g(and) h(in) g(particular) 0 2217 y(w) m(e) 39 b(exp) s(ect) f(that) g(it) g(allo) m(ws) g(to) f(re{obtain) i(the) f (results) h(of) g([8].) 58 b(Concerning) 39 b(p) s(ossible) g (extensions) g(to) 0 2336 y(systems) g(with) g(more) g(than) h(one) g (space) g(dimension) g(w) m(e) g(remark) e(that) h(the) h(nonresonance) h(condition) 0 2456 y(\(3.1\)) c(still) g(holds,) i(and) f(also) g(the) g(smallness) f(prop) s(ert) m(y) i(of) f(terms) f(with) g(large) h (index) g(holds) g(for) h(these) 0 2576 y(systems,) 28 b(ho) m(w) m(ev) m(er) h(equation) f(\(3.3\)) f(is) h(t) m(ypical) f (of) h(systems) g(in) g(one) g(space) h(dimension) f(and) g(w) m(e) h (ha) m(v) m(e) f(no) 0 2695 y(ideas) 34 b(on) f(the) h(p) s(ossibilit) m (y) e(of) h(obtaining) g(some) g(results) h(under) g(w) m(eak) m(er) g (nonresonance) h(assumptions.) p Fn 0 3452 a(4.) 115 b(Statemen) m(t) 35 b(of) j(the) f(abstract) g(result) p Fm 199 3723 a(Denote) 559 3698 y(\026) p Fn 549 3723 a(Z) p Fm 41 w(:=) p Fn 41 w(Z) p Fj 10 w(nf) p Fm(0) p Fj(g) p Fm(;) 44 b(de\014ne) e(the) g(space) p Fl 42 w(`) p Fi 1949 3687 a(2) p Fk 1949 3748 a(s) p Fm 2034 3723 a(of) f(the) h(complex) e(sequences) p Fl 43 w(\030) p Fj 45 w(\021) h(f) p Fl(\030) p Fk 3494 3738 a(j) p Fj 3535 3723 a(g) p Fk 3585 3747 a(j) p Fe 4 w(2) p Fi 3684 3729 a(\026) p Fa 3676 3747 a(Z) p Fm 3777 3723 a(suc) m(h) 0 3843 y(that) p Fj 1282 3970 a(k) p Fl(\030) p Fj 5 w(k) p Fi 1430 3920 a(2) p Fk 1430 4000 a(`) p Fb 1463 3980 a(2) p Fd 1463 4020 a(s) p Fm 1534 3970 a(:=) p Fh 1668 3875 a(X) p Fk 1667 4099 a(j) p Fe 4 w(2) p Fi 1766 4081 a(\026) p Fa 1758 4099 a(Z) p Fm 1813 3970 a(\(1) 22 b(+) p Fl 22 w(j) p Fi 2070 3929 a(2) p Fk(s) p Fm 2152 3970 a(\)) p Fj 17 w(j) p Fl -1 w(\030) p Fk 2279 3985 a(j) p Fj 2321 3970 a(j) p Fi 2349 3920 a(2) p Fl 2421 3970 a(<) p Fj 28 w(1) p Fl 33 w(;) p Fm 0 4280 a(and) 34 b(the) f(symplectic) g(spaces) p Fj 1103 4526 a(P) p Fk 1172 4541 a(s) p Fm 1242 4526 a(:=) p Fl 28 w(`) p Fi 1417 4485 a(2) p Fk 1417 4551 a(s) p Fj 1484 4526 a(\010) p Fl 22 w(`) p Fi 1625 4485 a(2) p Fk 1625 4551 a(s) p Fj 1697 4526 a(3) p Fl 28 w(\020) p Fj 35 w(\021) p Fh 1975 4446 a(\000) p Fj 2021 4526 a(f) p Fl(\030) p Fk 2115 4541 a(j) p Fj 2156 4526 a(g) p Fk 2206 4550 a(j) p Fe 4 w(2) p Fi 2305 4532 a(\026) p Fa 2297 4550 a(Z) p Fl 2356 4526 a(;) p Fj 17 w(f) p Fl(\021) p Fk 2500 4541 a(j) p Fj 2542 4526 a(g) p Fk 2592 4550 a(j) p Fe 4 w(2) p Fi 2691 4532 a(\026) p Fa 2683 4550 a(Z) p Fh 2743 4446 a(\001) p Fl 2838 4526 a(;) p Fm 0 4796 a(endo) m(w) m(ed) 44 b(b) m(y) e(the) g(symplectic) f(form) p Fl 42 w(i) p Fh 1535 4721 a(P) p Fk 1640 4826 a(j) p Fe 4 w(2) p Fi 1739 4808 a(\026) p Fa 1731 4826 a(Z) p Fl 1807 4796 a(d\021) p Fk 1908 4811 a(j) p Fj 1978 4796 a(^) p Fl 29 w(d\030) p Fe 2169 4811 a(\000) p Fk(j) p Fl 2314 4796 a(:) p Fm 42 w(In) p Fj 42 w(P) p Fk 2586 4811 a(s) p Fm 2671 4796 a(w) m(e) h(will) f(use) i(the) f(norm) p Fj 42 w(k) p Fl -1 w(\020) p Fj 7 w(k) p Fi 3804 4746 a(2) p Fk 3804 4826 a(s) p Fm 3891 4796 a(=) p Fj 0 4944 a(k) p Fl(\030) p Fj 5 w(k) p Fi 148 4894 a(2) p Fk 148 4974 a(`) p Fb 181 4954 a(2) p Fd 181 4994 a(s) p Fm 249 4944 a(+) p Fj 24 w(k) p Fl(\021) p Fj 4 w(k) p Fi 503 4894 a(2) p Fk 503 4974 a(`) p Fb 536 4954 a(2) p Fd 536 4994 a(s) p Fm 580 4944 a(,) 36 b(and) h(will) e(denote) h(b) m (y) p Fj 37 w(B) p Fk 1562 4959 a(s) p Fm 1605 4944 a(\() p Fl(R) p Fi 1721 4908 a(\() p Fk(s) p Fi(\)) p Fm 1825 4944 a(\)) f(the) h(ball) g(of) g(radius) p Fl 36 w(R) p Fi 2765 4908 a(\() p Fk(s) p Fi(\)) p Fm 2906 4944 a(cen) m(tered) h (at) f(the) g(origin) f(of) p Fj 0 5064 a(P) p Fk 69 5079 a(s) p Fm 112 5064 a(.) 47 b(F) -8 b(or) 34 b(the) h(Hamiltonian) d (v) m(ector) i(\014eld) h(of) g(a) f(Hamiltonian) e(function) p Fl 35 w(H) p Fm 42 w(w) m(e) j(will) e(use) j(the) e(notation) p Fl 0 5183 a(X) p Fk 83 5198 a(H) p Fm 158 5183 a(,) f(so) g(that) g (the) h(Hamilton) d(equations) i(are) 1801 5157 y(_) p Fl 1781 5183 a(\020) p Fm 35 w(=) p Fl 28 w(X) p Fk 2048 5198 a(H) p Fm 2123 5183 a(\() p Fl(\020) p Fm 7 w(\).) 43 b(Explicitly) 32 b(they) h(are) g(giv) m(en) g(b) m(y) p Fl 1256 5415 a(d) p 1238 5459 88 4 v 1238 5551 a(dt) 1338 5482 y(\030) p Fk 1382 5497 a(j) p Fm 1451 5482 a(=) p Fl 28 w(i) 1634 5415 y(@) 6 b(H) p 1602 5459 213 4 v 1602 5551 a(@) g(\021) p Fe 1710 5566 a(\000) p Fk(j) p Fl 1860 5482 a(;) 2034 5415 y(d) p 2016 5459 88 4 v 2016 5551 a(dt) 2115 5482 y(\021) p Fk 2164 5497 a(j) p Fm 2234 5482 a(=) p Fj 29 w(\000) p Fl(i) 2492 5415 y(@) g(H) p 2463 5459 207 4 v 2463 5551 a(@) g(\030) p Fe 2566 5566 a(\000) p Fk(j) p Fl 2715 5482 a(:) p Fm 1807 5778 a(20.03.02) p 90 rotate dyy eop %%Page: 8 8 8 7 bop Fg 0 60 a(8) 1685 b(D.) 33 b(Bam) m(busi) p Fm 0 299 a(In) 38 b(order) h(to) f(\014x) g(ideas) g(one) h(can) f(think) g (of) p Fl 38 w(\030) p Fk 1727 314 a(j) p Fm 1807 299 a(as) g(the) p Fl 39 w(j) p Fm 6 w(-th) g(F) -8 b(ourier) 38 b(co) s(e\016cien) m(t) h(on) f(the) g(basis) h(of) f(the) 0 418 y(exp) s(onen) m(tials) 33 b(of) h(a) f(smo) s(oth) f(function,) i (and) g(similarly) c(for) p Fl 34 w(\021) p Fk 2343 433 a(j) p Fm 2385 418 a(.) 199 539 y(Consider) k(a) f(Hamiltonian) e(of) j (the) f(form) p Fl 1717 781 a(H) p Fm 35 w(:=) p Fl 28 w(h) p Fi 2025 796 a(0) p Fm 2093 781 a(+) p Fl 22 w(f) p Fm 1523 w(\(4) p Fl(:) p Fm(1\)) 0 1023 y(with) p Fl 1458 1145 a(h) p Fi 1515 1160 a(0) p Fm 1560 1145 a(\() p Fl(\030) 5 b(;) 17 b(\021) p Fm 4 w(\)) 26 b(:=) p Fh 1945 1051 a(X) p Fk 1944 1275 a(j) p Fe 4 w(2) p Fi 2043 1257 a(\026) p Fa 2035 1275 a(Z) p Fl 2106 1145 a(!) p Fk 2168 1160 a(j) p Fl 2210 1145 a(\030) p Fk 2254 1160 a(j) p Fl 2296 1145 a(\021) p Fe 2345 1160 a(\000) p Fk(j) p Fl 2483 1145 a(;) p Fm 1253 w(\(4) p Fl(:) p Fm(2\)) 0 1450 y(and) j(real) p Fl 29 w(!) p Fk 441 1465 a(j) p Fm 483 1450 a('s;) h(the) f(functional) p Fl 29 w(f) p Fm 39 w(will) f(b) s(e) h(assumed) g(to) f(b) s(e) i(smo) s(oth) d(in) i(a) f(sense) j(that) d(will) g(b) s(e) h(sp) s(eci\014ed) 0 1570 y(in) k(a) g(while) g(and) h(to) f(ha) m(v) m(e) h(a) f(zero) g (of) h(order) f(three) h(at) f(the) h(origin.) p Fn 0 1710 a(De\014nition.) p Fg 30 w(Consider) f(a) f(p) s(olynomial) p Fl 30 w(h) p Fm(\() p Fl(\030) 5 b(;) 17 b(\021) p Fm 4 w(\)) p Fg 31 w(homogeneous) 33 b(of) g(degree) p Fl 33 w(n) p Fi 2976 1725 a(1) p Fg 3053 1710 a(in) p Fl 32 w(\030) p Fg 37 w(and) g(of) g(degree) p Fl 33 w(n) p Fi 3924 1725 a(2) p Fg 0 1830 a(in) p Fl 33 w(\021) p Fg 4 w(,) g(nalmely) p Fl 1226 1966 a(h) p Fm(\() p Fl(\030) 5 b(;) 17 b(\021) p Fm 4 w(\)) 26 b(=) p Fh 1639 1872 a(X) p Fk 1659 2086 a(k) r(;j) p Fl 1800 1966 a(h) p Fk 1857 1981 a(k) r(;j) p Fl 1967 1966 a(\030) p Fk 2011 1981 a(k) p Fb 2053 1991 a(1) p Fl 2096 1966 a(:::\030) p Fk 2224 1981 a(k) p Fd 2266 1991 a(n) p Fb 2310 2006 a(1) p Fl 2358 1966 a(\021) p Fk 2407 1981 a(j) p Fb 2440 1991 a(1) p Fl 2484 1966 a(:::\021) p Fk 2617 1981 a(j) p Fd 2650 1991 a(n) p Fb 2694 2006 a(2) p Fm 3764 1966 a(\(4) p Fl(:) p Fm(3\)) p Fg 0 2267 a(where) 34 b(w) m(e) g(denoted) p Fl 34 w(k) p Fm 31 w(=) 28 b(\() p Fl(k) p Fi 1090 2282 a(1) p Fl 1134 2267 a(;) 17 b(:::;) g(k) p Fk 1360 2282 a(n) p Fb 1409 2292 a(1) p Fm 1451 2267 a(\)) p Fg(,) p Fl 32 w(j) p Fm 34 w(=) 28 b(\() p Fl(j) p Fi 1810 2282 a(1) p Fl 1854 2267 a(;) 17 b(:::;) g(j) p Fk 2069 2282 a(n) p Fb 2118 2292 a(2) p Fm 2160 2267 a(\)) p Fg(.) 44 b(The) 34 b(quan) m(tit) m(y) p Fj 1058 2529 a(j) p Fl(h) p Fj(j) p Fk 1171 2559 a(\026) p Fm 1252 2529 a(:=) p Fh 1385 2434 a(X) p Fk 1390 2649 a(l) p Fe(2) p Fa(Z) p Fm 1546 2529 a(e) p Fk 1590 2488 a(\026) p Fe(j) p Fk(l) p Fe(j) p Fm 2127 2529 a(sup) p Fk 1733 2613 a(k) p Fb 1775 2623 a(1) p Fi 1814 2613 a(+) p Fk(:::) p Fi(+) p Fk(k) p Fd 2050 2623 a(n) p Fb 2094 2638 a(1) p Fi 2137 2613 a(+) p Fk(j) p Fb 2231 2623 a(1) p Fi 2270 2613 a(+) p Fk(:::) p Fi(+) p Fk(j) p Fd 2497 2623 a(n) p Fb 2541 2638 a(2) p Fi 2584 2613 a(=) p Fk(l) p Fj 2688 2529 a(j) p Fl(h) p Fk 2773 2544 a(k) r(;j) p Fj 2883 2529 a(j) p Fm 853 w(\(4) p Fl(:) p Fm(4\)) p Fg 0 2866 a(will) e(b) s(e) i(called) p Fl 33 w(\026) p Fg({mo) s(dulus) f(of) p Fl 33 w(h) p Fg(.) p Fn 0 3026 a(De\014nition.) p Fg 25 w(Consider) c(a) e(functional) p Fl 28 w(h) p Fm 29 w(=) p Fh 1697 2951 a(P) p Fk 1802 3056 a(n) p Fb 1851 3066 a(1) p Fk 1890 3056 a(;n) p Fb 1963 3066 a(2) p Fl 2023 3026 a(h) p Fk 2080 2990 a(n) p Fb 2129 3000 a(1) p Fk 2169 2990 a(;n) p Fb 2242 3000 a(2) p Fg 2313 3026 a(with) p Fl 28 w(h) p Fk 2592 2990 a(n) p Fb 2641 3000 a(1) p Fk 2680 2990 a(;n) p Fb 2753 3000 a(2) p Fg 2825 3026 a(homogenous) g(of) h(degrees) p Fl 29 w(n) p Fi 3896 3041 a(1) p Fg 3941 3026 a(,) p Fl 0 3146 a(n) p Fi 60 3161 a(2) p Fg 138 3146 a(in) p Fl 33 w(\030) p Fg 37 w(and) p Fl 34 w(\021) p Fg 37 w(resp) s(ectiv) m(ely) -8 b(.) 44 b(If) 34 b(the) f(quan) m(tit) m(y) p Fj 1298 3402 a(hj) p Fl(h) p Fj(ji) p Fk 1488 3352 a(\026) 1488 3432 y(R) p Fm 1598 3402 a(:=) p Fh 1759 3307 a(X) p Fk 1731 3516 a(n) p Fb 1780 3526 a(1) p Fk 1819 3516 a(;n) p Fb 1892 3526 a(2) p Fj 1947 3402 a(j) p Fl(h) p Fk 2032 3361 a(n) p Fb 2081 3371 a(1) p Fk 2121 3361 a(;n) p Fb 2194 3371 a(2) p Fj 2237 3402 a(j) p Fk 2265 3417 a(\026) p Fl 2318 3402 a(R) p Fk 2395 3361 a(n) p Fb 2444 3371 a(1) p Fi 2483 3361 a(+) p Fk(n) p Fb 2593 3371 a(2) p Fm 3764 3402 a(\(4) p Fl(:) p Fm(5\)) p Fg 0 3750 a(is) 39 b(\014nite) h(for) g(some) f(p) s(ositiv) m(e) p Fl 39 w(R) p Fg 1 w(,) p Fl 40 w(\026) p Fg(,) i(then) f(w) m(e) g (will) e(sa) m(y) i(that) p Fl 39 w(h) p Fg 40 w(is) f(of) h(class) p Fj 40 w(M) p Fk 3095 3702 a(\026) 3095 3780 y(R) p Fg 3199 3750 a(and) g(w) m(e) g(will) f(write) p Fl 0 3870 a(h) p Fj 28 w(2) 28 b(M) p Fk 299 3822 a(\026) 299 3899 y(R) p Fg 364 3870 a(.) p Fm 199 4010 a(W) -8 b(e) 32 b(will) f(assume) h(that) p Fl 31 w(f) p Fj 39 w(2) c(M) p Fk 1415 3962 a(\026) 1415 4040 y(R) p Fm 1480 4010 a(,) j(and) i(w) m (e) f(will) f(pro) m(v) m(e) h(that) f(in) h(the) g(case) h(of) f(the) g (nonlinear) g(w) m(a) m(v) m(e) 0 4130 y(equation) h(the) g(p) s (erturbation) h(is) f(of) h(class) p Fj 33 w(M) p Fk 1730 4082 a(\026) 1730 4159 y(R) p Fm 1795 4130 a(.) p Fn 0 4270 a(De\014nition.) p Fg 29 w(W) -8 b(e) 32 b(will) e(sa) m(y) h (that) g(a) g(sequence) i(of) f(frequencies) p Fj 33 w(f) p Fl(!) p Fk 2524 4285 a(j) p Fj 2566 4270 a(g) p Fk 2616 4294 a(j) p Fe 4 w(2) p Fi 2715 4276 a(\026) p Fa 2707 4294 a(Z) p Fg 2798 4270 a(is) f(strongly) g(non) h(v) -6 b(anishing) 31 b(if) 0 4390 y(for) i(an) m(y) p Fl 33 w(r) p Fj 30 w(\025) p Fm 28 w(3) p Fg 33 w(there) g(exists) g(a) f(p) s (ositiv) m(e) p Fl 33 w(\015) p Fg 37 w(and) i(a) e(real) p Fl 33 w(\013) p Fg 33 w(suc) m(h) i(that,) e(for) h(an) m(y) g (\(large\)) p Fl 31 w(N) p Fg 11 w(,) g(an) m(y) g(c) m(hoice) 0 4509 y(of) k(indexes) p Fl 37 w(i) p Fi 506 4524 a(1) p Fl 551 4509 a(;) 17 b(:::;) g(i) p Fk 759 4524 a(r) p Fl 801 4509 a(;) g(j;) g(k) p Fg 38 w(with) p Fj 36 w(j) p Fl(i) p Fi 1314 4524 a(1) p Fj 1359 4509 a(j) 33 b(\024) h(j) p Fl(i) p Fi 1593 4524 a(2) p Fj 1637 4509 a(j) f(\024) p Fl 34 w(:::) p Fj 32 w(\024) g(j) p Fl(i) p Fk 2097 4524 a(r) p Fj 2142 4509 a(j) f(\024) p Fl 34 w(N) 44 b(<) p Fj 34 w(j) p Fl(j) p Fj 6 w(j) 32 b(\024) h(j) p Fl(k) p Fj 3 w(j) p Fg 36 w(and) k(an) m(y) g(c) m(hoice) g(of) p Fl 37 w(r) p Fm 27 w(+) 25 b(1) p Fg 0 4629 a(signs) 34 b(one) f(has) h(that) p Fl 1238 4751 a(!) p Fk 1300 4766 a(i) p Fb 1328 4776 a(1) p Fj 1394 4751 a(\006) p Fl 23 w(!) p Fk 1556 4766 a(i) p Fb 1584 4776 a(2) p Fj 1650 4751 a(\006) p Fl 22 w(:::) p Fj 22 w(\006) p Fl 22 w(!) p Fk 2016 4766 a(i) p Fd 2044 4776 a(r) p Fj 2112 4751 a(\006) p Fl 22 w(!) p Fk 2273 4766 a(j) p Fj 2337 4751 a(\006) p Fl 23 w(!) p Fk 2499 4766 a(k) p Fj 2576 4751 a(6) p Fm(=) 28 b(0) p Fg 0 4944 a(implies) p Fj 1119 5086 a(j) p Fl -1 w(!) p Fk 1208 5101 a(i) p Fb 1236 5111 a(1) p Fj 1303 5086 a(\006) p Fl 22 w(!) p Fk 1464 5101 a(i) p Fb 1492 5111 a(2) p Fj 1558 5086 a(\006) p Fl 23 w(:::) p Fj 21 w(\006) p Fl 23 w(!) p Fk 1925 5101 a(i) p Fd 1953 5111 a(r) p Fj 2020 5086 a(\006) p Fl 23 w(!) p Fk 2182 5101 a(j) p Fj 2246 5086 a(\006) p Fl 23 w(!) p Fk 2408 5101 a(k) p Fj 2457 5086 a(j) f(\025) p Fl 2675 5019 a(\015) p 2629 5063 148 4 v 2629 5154 a(N) p Fk 2720 5126 a(\013) p Fl 2822 5086 a(:) p Fm 914 w(\(3) p Fl(:) p Fm(3\)) 199 5419 y(W) -8 b(e) 34 b(will) e(assume) h(that) g(the) h(frequencies) h(are) e (strongly) g(non) m(v) -6 b(anishing.) 199 5539 y(The) 34 b(last) f(de\014nition) h(w) m(e) f(need) i(is) e(that) g(of) g (Birkho\013) g(normal) e(form:) 1807 5778 y(20.03.02) p 90 rotate dyy eop %%Page: 9 9 9 8 bop Fg 1247 60 a(Birkho\013) 33 b(normal) f(form) g(for) i(PDE's) p Fm 1246 w(9) p Fn 0 299 a(De\014nition.) p Fg 33 w(Consider) i(a) f (functional) p Fl 35 w(Z) p Fm 38 w(=) p Fh 1751 224 a(P) p Fk 1856 329 a(n) p Fb 1905 339 a(1) p Fk 1944 329 a(;n) p Fb 2017 339 a(2) p Fl 2077 299 a(Z) p Fk 2152 263 a(n) p Fb 2201 273 a(1) p Fk 2241 263 a(;n) p Fb 2314 273 a(2) p Fg 2357 299 a(,) h(with) p Fl 35 w(Z) p Fk 2725 263 a(n) p Fb 2774 273 a(1) p Fk 2813 263 a(;n) p Fb 2886 273 a(2) p Fg 2965 299 a(homogeneous) f(of) g(degree) p Fl 0 418 a(n) p Fi 60 433 a(1) p Fg 138 418 a(in) p Fl 33 w(\030) p Fg 37 w(and) f(of) f(degree) p Fl 35 w(n) p Fi 1013 433 a(2) p Fg 1090 418 a(in) p Fl 34 w(\021) p Fg 4 w(.) 44 b(W) -8 b(rite) p Fl 370 632 a(Z) p Fk 445 591 a(n) p Fb 494 601 a(1) p Fk 533 591 a(;n) p Fb 606 601 a(2) p Fm 677 632 a(=) p Fh 1143 537 a(X) p Fk 783 752 a(k) p Fb 825 762 a(1) p Fk 864 752 a(;k) p Fb 930 762 a(2) p Fk 969 752 a(;:::;k) p Fd 1131 762 a(n) p Fb 1175 777 a(1) p Fk 1217 752 a(;j) p Fb 1274 762 a(1) p Fk 1313 752 a(;j) p Fb 1370 762 a(2) p Fk 1408 752 a(;:::;j) p Fd 1561 762 a(n) p Fb 1605 777 a(2) p Fl 1664 632 a(Z) p Fk 1739 584 a(n) p Fb 1788 594 a(1) p Fk 1827 584 a(;n) p Fb 1900 594 a(2) p Fk 1732 662 a(k) p Fb 1774 672 a(1) p Fk 1813 662 a(;k) p Fb 1879 672 a(2) p Fk 1918 662 a(;:::;k) p Fd 2080 672 a(n) p Fb 2124 687 a(1) p Fk 2166 662 a(;j) p Fb 2223 672 a(1) p Fk 2262 662 a(;:::;j) p Fd 2415 672 a(n) p Fb 2459 687 a(2) p Fl 2506 632 a(\030) p Fk 2550 647 a(k) p Fb 2592 657 a(1) p Fl 2636 632 a(\030) p Fk 2680 647 a(k) p Fb 2722 657 a(2) p Fl 2765 632 a(:::\030) p Fk 2893 647 a(k) p Fd 2935 657 a(n) p Fb 2979 672 a(1) p Fl 3027 632 a(\021) p Fk 3076 647 a(j) p Fb 3109 657 a(1) p Fl 3153 632 a(\021) p Fk 3202 647 a(j) p Fb 3235 657 a(2) p Fl 3280 632 a(:::\021) p Fk 3413 647 a(j) p Fd 3446 657 a(n) p Fb 3490 672 a(2) p Fm 3571 632 a(;) p Fl 0 945 a(Z) p Fg 40 w(will) 33 b(b) s(e) g(said) g(to) g(b) s(e) h (in) f(Birkho\013) f(normal) g(form) h(if) p Fl 254 1144 a(Z) p Fk 329 1096 a(n) p Fb 378 1106 a(1) p Fk 417 1096 a(;n) p Fb 490 1106 a(2) p Fk 322 1174 a(k) p Fb 364 1184 a(1) p Fk 403 1174 a(;k) p Fb 469 1184 a(2) p Fk 508 1174 a(;:::;k) p Fd 670 1184 a(n) p Fb 714 1199 a(1) p Fk 756 1174 a(;j) p Fb 813 1184 a(1) p Fk 852 1174 a(;:::;j) p Fd 1005 1184 a(n) p Fb 1049 1199 a(2) p Fj 1123 1144 a(6) p Fm(=) 28 b(0) g(=) p Fj -17 w(\)) p Fl 28 w(!) p Fk 1556 1159 a(k) p Fb 1598 1169 a(1) p Fm 1664 1144 a(+) p Fl 23 w(!) p Fk 1826 1159 a(k) p Fb 1868 1169 a(2) p Fm 1934 1144 a(+) p Fl 23 w(:::) p Fm 21 w(+) p Fl 23 w(!) p Fk 2301 1159 a(k) p Fd 2343 1169 a(n) p Fb 2387 1184 a(1) p Fj 2458 1144 a(\000) p Fl 23 w(!) p Fk 2620 1159 a(j) p Fb 2653 1169 a(1) p Fj 2719 1144 a(\000) p Fl 22 w(!) p Fk 2880 1159 a(j) p Fb 2913 1169 a(2) p Fj 2979 1144 a(\000) p Fl 23 w(:::) p Fj 21 w(\000) p Fl 23 w(!) p Fk 3346 1159 a(j) p Fd 3379 1169 a(n) p Fb 3423 1184 a(2) p Fm 3499 1144 a(=) g(0) p Fl 33 w(:) p Fm 199 1482 a(In) 23 b(the) f(application) g(to) g(equation) f(\(0.1\)) g(w) m(e) i(will) e(need) i(to) f(restrict) g(the) h(system) e(to) h (the) g(submanifold) 0 1602 y(of) 33 b(sk) m(ew{symmetric) f(functions) i(that) f(w) m(e) h(will) e(denote) i(b) m(y) p Fj 1045 1801 a(A) p Fm 27 w(:=) p Fj 28 w(f) p Fm(\() p Fl(\030) p Fk 1418 1816 a(k) p Fl 1466 1801 a(;) 17 b(\021) p Fk 1560 1816 a(k) p Fm 1609 1801 a(\)) 60 b(:) p Fl 61 w(\030) p Fk 1841 1816 a(k) p Fm 1917 1801 a(=) p Fl 28 w(\030) p Fe 2066 1816 a(\000) p Fk(k) p Fm 2211 1801 a(and) p Fl 33 w(\021) p Fk 2453 1816 a(k) p Fm 2530 1801 a(=) p Fl 29 w(\021) p Fe 2685 1816 a(\000) p Fk(k) p Fj 2796 1801 a(g) p Fl 50 w(;) p Fm 0 1999 a(and) 34 b(to) e(the) i(manifold) p Fj 1347 2119 a(R) p Fl(e) p Fm 29 w(:=) p Fj 28 w(f) p Fm -1 w(\() p Fl(\030) p Fk 1771 2134 a(k) p Fl 1820 2119 a(;) 17 b(\021) p Fk 1914 2134 a(k) p Fm 1962 2119 a(\)) 61 b(:) p Fl 61 w(\030) p Fe 2200 2078 a(\003) p Fk 2195 2144 a(k) p Fm 2272 2119 a(=) p Fl 28 w(\021) p Fe 2426 2134 a(\000) p Fk(k) p Fj 2538 2119 a(g) p Fm 0 2290 a(corresp) s(onding) 34 b(to) f(real) g(functions.) 199 2410 y(W) -8 b(e) 34 b(assume) f(that) 22 2529 y(\(1\)) 49 b(the) 30 b(frequencies) p Fl 31 w(!) p Fk 935 2544 a(j) p Fm 1006 2529 a(of) f(the) g(hamiltonian) f(system) g(\(4.1\)) g(are) h (strongly) f(non) m(v) -6 b(anishing) 30 b(with) f(some) 199 2649 y(indices) p Fl 34 w(\013;) 17 b(\015) p Fm 6 w(.) 22 2769 y(\(2\)) 49 b(there) 34 b(exists) 751 2743 y(\026) p Fl 729 2769 a(R) p Fi 805 2784 a(1) p Fl 878 2769 a(>) p Fm 28 w(0) f(and) p Fl 34 w(\026) 28 b(>) p Fm 28 w(0) 33 b(suc) m(h) i(that) e(the) h(function) p Fl 34 w(f) p Fm 44 w(is) f(of) h(class) p Fj 33 w(M) p Fk 3199 2721 a(\026) 3199 2798 y(R) p Fm 3298 2769 a(for) f(an) m(y) p Fl 34 w(R) p Fj 28 w(\024) p Fm 3870 2743 a(\026) p Fl 3848 2769 a(R) p Fi 3924 2784 a(1) p Fm 199 2888 a(and) h(ful\014lls) g (the) f(estimate) p Fj 1782 3008 a(hj) p Fl(f) p Fj 11 w(j) o(i) p Fk 1975 2958 a(\026) 1975 3038 y(R) p Fj 2084 3008 a(\024) p Fl 29 w(AR) p Fi 2342 2967 a(3) p Fm 199 3179 a(with) g(some) g(p) s(ositiv) m(e) p Fl 32 w(A) p Fm(.) p Fn 0 3338 a(Theorem) 188 b(4.1.) p Fg 42 w(Under) 41 b(the) h(ab) s(o) m(v) m(e) f(assumptions,) i(\014x) e (a) g(large) p Fl 40 w(M) p Fg 11 w(,) i(de\014ne) p Fl 42 w(s) p Fe 3252 3353 a(\003) p Fm 3339 3338 a(:=) e(4) p Fl(M) 11 b(\013) p Fm(\(2) p Fl(M) p Fm 11 w(\)) p Fg(;) 0 3458 y(then) 38 b(there) g(exist) f(constan) m(ts) p Fl 38 w(C) p Fg 7 w(,) p Fl 38 w(R) p Fe 1395 3473 a(\003) p Fg 1478 3458 a(suc) m(h) i(that) d(for) i(an) m(y) f(p) s(ositiv) m(e) p Fl 37 w(R) p Fi 2725 3422 a(\() p Fk(s) p Fc 2794 3432 a(\003) p Fi 2835 3422 a(\)) p Fg 2908 3458 a(ful\014lling) p Fl 37 w(R) p Fi 3379 3422 a(\() p Fk(s) p Fc 3448 3432 a(\003) p Fi 3489 3422 a(\)) p Fl 3560 3458 a(<) d(R) p Fe 3747 3473 a(\003) p Fg 3830 3458 a(the) 0 3578 y(follo) m(wing) 27 b(holds) h(true:) 42 b(There) 28 b(exists) g(a) f(canonical) h (transformation) p Fj 26 w(T) p Fm 53 w(:) p Fj 28 w(B) p Fk 2879 3593 a(s) p Fc 2917 3603 a(\003) p Fm 2963 3578 a(\() p Fl(R) p Fi 3079 3541 a(\() p Fk(s) p Fc 3148 3551 a(\003) p Fi 3189 3541 a(\)) p Fl 3225 3578 a(=) p Fm(3\)) p Fj 27 w(!) f(B) p Fk 3583 3593 a(s) p Fc 3621 3603 a(\003) p Fm 3668 3578 a(\() p Fl(R) p Fi 3784 3541 a(\() p Fk(s) p Fc 3853 3551 a(\003) p Fi 3894 3541 a(\)) p Fm 3930 3578 a(\)) p Fg 0 3697 a(suc) m(h) 35 b(that) e(the) g (transformed) h(Hamiltonian) c(has) k(the) g(form) p Fm 1398 3896 a(\() p Fl(h) p Fi 1494 3911 a(0) p Fm 1561 3896 a(+) p Fl 23 w(f) p Fm 11 w(\)) p Fj 21 w(\016) 22 b(T) p Fm 53 w(=) p Fl 28 w(h) p Fi 2122 3911 a(0) p Fm 2190 3896 a(+) p Fl 22 w(Z) p Fm 29 w(+) p Fj 23 w(R) p Fg 0 4095 a(with) p Fl 32 w(Z) p Fg 40 w(in) 32 b(Birkho\013) g(normal) f(form,) h(and) p Fj 33 w(R) p Fg 33 w(an) g(analytic) g(functional) h (ha) m(ving) f(a) g(Hamiltonian) f(v) m(ector) 0 4214 y(\014eld) 41 b(whic) m(h) h(is) f(analytic) f(as) h(a) f(function) i (from) p Fj 40 w(P) p Fk 1941 4229 a(s) p Fc 1979 4239 a(\003) p Fg 2066 4214 a(to) f(itself;) j(moreo) m(v) m(er) c(there) i (exists) f(a) f(p) s(ositiv) m(e) p Fl 40 w(c) p Fg 0 4334 a(suc) m(h) d(that) f(for) g(an) m(y) p Fl 36 w(s) p Fj 32 w(\025) p Fl 33 w(s) p Fe 1028 4349 a(\003) p Fg 1073 4334 a(,) h(and) f(an) m(y) p Fl 36 w(R) p Fi 1602 4298 a(\() p Fk(s) p Fi(\)) p Fj 1739 4334 a(\024) p Fl 32 w(R) p Fe 1924 4349 a(\003) p Fl 1970 4334 a(=c) p Fk 2063 4298 a(s) p Fg 2141 4334 a(the) h(transformation) p Fj 35 w(T) p Fg 61 w(and) g(the) f(v) m(ector) g(\014eld) p Fl 0 4454 a(X) p Fe 83 4469 a(R) p Fg 189 4454 a(are) d(analytic) g (also) f(as) i(maps) e(from) p Fj 33 w(B) p Fk 1618 4469 a(s) p Fm 1661 4454 a(\() p Fl(R) p Fi 1777 4417 a(\() p Fk(s) p Fi(\)) p Fm 1881 4454 a(\)) p Fg 33 w(to) p Fj 33 w(P) p Fk 2144 4469 a(s) p Fg 2220 4454 a(and) h(the) h(follo) m (wing) f(estimate) f(holds) p Fm 1339 4652 a(sup) p Fe 1226 4746 a(k) p Fk(\020) p Fe 5 w(k) p Fd 1349 4768 a(s) p Fe 1387 4746 a(\024) p Fk(R) p Fb 1509 4726 a(\() p Fd(s) p Fb(\)) p Fj 1619 4652 a(k) p Fl(\020) p Fj 29 w(\000) 22 b(T) p Fm 26 w(\() p Fl(\020) p Fm 7 w(\)) p Fj(k) p Fk 2099 4682 a(s) p Fj 2169 4652 a(\024) p Fl 29 w(C) p Fm 7 w(\() p Fl(c) p Fk 2435 4611 a(s) p Fl 2477 4652 a(R) p Fi 2554 4611 a(\() p Fk(s) p Fi(\)) p Fm 2659 4652 a(\)) p Fi 2698 4611 a(2) p Fg 0 4910 a(moreo) m(v) m(er) p Fm 1362 5058 a(sup) p Fe 1249 5151 a(k) p Fk(\020) p Fe 5 w(k) p Fd 1372 5174 a(s) p Fe 1410 5151 a(\024) p Fk(R) p Fb 1532 5131 a(\() p Fd(s) p Fb(\)) p Fj 1642 5058 a(k) p Fl -1 w(X) p Fe 1774 5073 a(R) p Fj 1847 5058 a(k) p Fk 1897 5088 a(s) p Fj 1967 5058 a(\024) p Fl 29 w(C) p Fh 2167 4947 a(\020) p Fl 2227 5058 a(c) p Fk 2270 5017 a(s) p Fl 2313 5058 a(R) p Fi 2390 5017 a(\() p Fk(s) p Fi(\)) p Fh 2494 4947 a(\021) p Fk 2553 4968 a(M) p Fl 2692 5058 a(:) p Fm 1044 w(\(4) p Fl(:) p Fm(6\)) p Fg 0 5320 a(Finally) -8 b(,) 38 b(if) g(the) h(v) m(ector) f (\014eld) h(of) p Fl 39 w(f) p Fg 48 w(lea) m(v) m(es) g(in) m(v) -6 b(arian) m(t) p Fj 38 w(A) p Fg 38 w(and/or) p Fj 39 w(R) p Fl(e) p Fg(,) 40 b(then) f(the) g(same) e(is) i(true) f(for) p Fj 39 w(T) p Fg 0 5440 a(and) c(for) f(the) h(v) m(ector) f(\014elds) h (of) p Fl 34 w(Z) p Fg 40 w(and) g(of) p Fj 33 w(R) p Fg(.) p Fm 1807 5778 a(20.03.02) p 90 rotate dyy eop %%Page: 10 10 10 9 bop Fm 0 60 a(10) p Fg 1660 w(D.) 33 b(Bam) m(busi) p Fn 0 299 a(5.) 115 b(Pro) s(of) 39 b(of) g(theorem) e(4.1.) p Fm 199 568 a(W) -8 b(e) 48 b(split) g(the) g(pro) s(of) g(in) g(three) h (parts.) 88 b(In) 48 b(the) g(\014rst) h(w) m(e) f(establish) g(some) g (prop) s(erties) g(of) g(the) 0 687 y(functions) 35 b(of) f(class) p Fj 34 w(M) p Fk 898 640 a(\026) 898 717 y(R) p Fm 963 687 a(,) f(in) h(the) g(second) h(w) m(e) f(pro) m(v) m(e) h(that) e (the) h(Lie) g(transform) f(of) h(a) g(function) g(of) g(class) p Fj 0 807 a(M) p Fk 120 759 a(\026) 120 836 y(R) p Fm 220 807 a(is) h(of) g(class) p Fj 35 w(M) p Fk 792 759 a(\026) 792 836 y(R) p Fc 852 816 a(0) p Fm 919 807 a(with) f(an) m(y) p Fl 35 w(R) p Fe 1414 771 a(0) p Fl 1472 807 a(<) d(R) p Fm 1 w(,) j(pro) m(vided) h(it) g(is) f(generated) i(b) m(y) f(a) g (function) h(of) f(class) p Fj 35 w(M) p Fk 3876 759 a(\026) 3876 836 y(R) p Fm 3941 807 a(;) 0 927 y(in) f(the) h(third) f (subsection) h(w) m(e) g(pro) m(v) m(e) g(the) f(iterativ) m(e) f (lemma) f(whic) m(h) j(constitute) g(the) f(main) f(step) i(of) f(the) 0 1046 y(pro) s(of,) f(and) h(deduce) h(theorem) e(4.1.) 199 1166 y(W) -8 b(e) 34 b(will) e(denote) p Fl 2073 1324 a(\032) p Fk 2125 1339 a(s) p Fm 2195 1324 a(:=) 2340 1257 y(2) p Fk 2390 1221 a(s) p Fi(+1) p 2340 1301 194 4 v Fl 2406 1392 a(\031) 2578 1324 y(;) p Fm 1363 1576 a(\006) p Fk 1435 1591 a(s) p Fm 1505 1576 a(:=) p Fh 1638 1440 a(s) p 1737 1440 808 4 v 1738 1481 a(X) p Fk 1737 1695 a(j) p Fe 4 w(2) p Fa(Z) p Fm 1883 1576 a(\(1) 22 b(+) p Fl 22 w(j) p Fi 2140 1547 a(2) p Fk(s) p Fm 2222 1576 a(\)e) p Fe 2305 1547 a(\000) p Fi(2) p Fk(\026) p Fe(j) p Fk(j) p Fe 4 w(j) p Fl 2578 1576 a(;) p Fm 0 1880 a(and) 34 b(remark) e(that) h(there) g(exists) g(a) h(p) s(ositiv) m(e) e(\() p Fl(\026) p Fm 33 w(dep) s(enden) m(t\)) j(constan) m(t) p Fl 34 w(c) p Fm 34 w(suc) m(h) f(that) 1818 2097 y(\006) p Fk 1890 2112 a(s) p Fl 1960 2097 a(<) 28 b(c) p Fk 2108 2056 a(s) p Fm 3703 2097 a(\(5) p Fl(:) p Fm(1\)) p Fl 32 w(:) p Fm 0 2315 a(Moreo) m(v) m(er) p Fl 46 w(C) p Fm 52 w(will) 45 b(denote) h(a) f(p) s(ositiv) m(e) g(\(large\)) f (constan) m(t) i(whose) g(v) -6 b(alue) 45 b(can) h(c) m(hange) g (through) g(the) 0 2434 y(pap) s(er,) 39 b(and) f(whic) m(h) g(is) g (indep) s(enden) m(t) i(of) d(the) h(relev) -6 b(an) m(t) 38 b(quan) m(tities.) 57 b(F) -8 b(or) 37 b(analytic) f(functions) p Fl 39 w(f) p Fm 11 w(,) i(that) 0 2554 y(ha) m(v) m(e) c(an) f (analytic) f(v) m(ector) h(\014eld) h(w) m(e) g(will) e(write) p Fj 1271 2799 a(k) p Fl(X) p Fk 1404 2814 a(f) p Fj 1455 2799 a(k) p Fk 1505 2749 a(R) p Fb 1565 2719 a(\() p Fd(s) p Fb(\)) p Fk 1505 2832 a(s) p Fm 1690 2799 a(:=) 141 b(sup) p Fe 1823 2892 a(k) p Fk(\020) p Fe 5 w(k) p Fd 1945 2914 a(s) p Fe 1984 2892 a(\024) p Fk(R) p Fb 2106 2872 a(\() p Fd(s) p Fb(\)) p Fj 2215 2799 a(k) p Fl(X) p Fk 2348 2814 a(f) p Fm 2399 2799 a(\() p Fl(\020) p Fm 7 w(\)) p Fj(k) p Fk 2578 2832 a(s) p Fl 2670 2799 a(:) p Fm 0 3107 a(Ha) m(ving) 42 b(\014xed) i(some) e(in) m(teger) p Fl 44 w(N) p Fm 11 w(,) j(w) m(e) f(will) e(denote) i(b) m(y) f(\() p Fl(q) p Fk 2281 3122 a(j) p Fl 2323 3107 a(;) 17 b(p) p Fk 2418 3122 a(j) p Fm 2460 3107 a(\)) p Fe 2499 3125 a(j) p Fk(j) p Fe 4 w(j\024) p Fk(N) p Fm 2765 3107 a(=) 44 b(\() p Fl(\030) p Fk 2969 3122 a(j) p Fl 3011 3107 a(;) 17 b(\021) p Fk 3105 3122 a(j) p Fm 3146 3107 a(\)) p Fe 3185 3125 a(j) p Fk(j) p Fe 4 w(j\024) p Fk(N) p Fm 3407 3107 a(,) 46 b(the) d(\014rst) p Fl 44 w(N) p Fm 0 3227 a(canonical) 33 b(v) -6 b(ariables) 33 b(and) h(b) m(y) g(\() p Fl(Q) p Fk 1301 3242 a(j) p Fl 1342 3227 a(;) 17 b(P) p Fk 1451 3242 a(j) p Fm 1492 3227 a(\)) p Fe 1531 3245 a(j) p Fk(j) p Fe 4 w(j) p Fk(>N) p Fm 1781 3227 a(=) 28 b(\() p Fl(\030) p Fk 1969 3242 a(j) p Fl 2010 3227 a(;) 17 b(\021) p Fk 2104 3242 a(j) p Fm 2146 3227 a(\)) p Fe 2185 3245 a(j) p Fk(j) p Fe 4 w(j) p Fk(>N) p Fm 2440 3227 a(the) 34 b(remaining) e(ones.) p Fg 0 3844 a(5.1) h(Prop) s (erties) g(of) g(the) h(functions) g(of) g(class) p Fj 33 w(M) p Fk 1823 3796 a(\026) 1823 3874 y(R) p Fm 199 4113 a(The) d(main) d(results) j(of) f(this) g(sub{section) h(are) f (prop) s(osition) f(5.3) g(and) h(5.8;) g(the) g(h) m(urried) i(reader) e(can) 0 4233 y(go) j(directly) g(to) f(them) h(and) h(lea) m(v) m(e) f (the) h(rest) f(of) h(the) f(subsection) i(for) e(a) g(subsequen) m(t) j (reading.) 199 4352 y(Giv) m(en) e(a) f(homogeneous) g(p) s(olynomial) p Fl 31 w(h) p Fm 34 w(of) g(degree) p Fl 34 w(n) p Fi 2269 4367 a(1) p Fm 2347 4352 a(in) p Fl 33 w(\030) p Fm 38 w(and) g(of) h(degree) p Fl 34 w(n) p Fi 3222 4367 a(2) p Fm 3300 4352 a(in) p Fl 33 w(\021) p Fm 4 w(,) f(write) p Fl 1205 4584 a(h) p Fm(\() p Fl(\030) 5 b(;) 17 b(\021) p Fm 4 w(\)) 26 b(=) p Fh 1618 4490 a(X) p Fk 1642 4704 a(k) r(;i) p Fl 1778 4584 a(h) p Fk 1835 4599 a(k) r(;i) p Fl 1937 4584 a(\030) p Fk 1981 4599 a(k) p Fb 2023 4609 a(1) p Fl 2066 4584 a(:::\030) p Fk 2194 4599 a(k) p Fd 2236 4609 a(n) p Fb 2280 4624 a(1) p Fl 2328 4584 a(\021) p Fk 2377 4599 a(i) p Fb 2405 4609 a(1) p Fl 2450 4584 a(:::\021) p Fk 2583 4599 a(i) p Fd 2611 4609 a(n) p Fb 2655 4624 a(2) p Fm 2736 4584 a(;) 3764 4629 y(\(5) p Fl(:) p Fm(2\)) 0 4905 y(denote) 34 b(b) m(y) 1470 5024 y($) p Fl(i) p Fm($) k(:=) p Fl 28 w(i) p Fi 1809 5039 a(1) p Fm 1876 5024 a(+) p Fl 22 w(i) p Fi 2009 5039 a(2) p Fm 2076 5024 a(+) p Fl 23 w(:::) p Fm 21 w(+) p Fl 23 w(i) p Fk 2415 5039 a(n) p Fb 2464 5049 a(1) p Fm 0 5204 a(the) c(sum) e(of) i(the) f(comp) s(onen) m(ts) h(of) f(a) g(m) m(ultiindex) p Fl 33 w(i) p Fm 33 w(and) h(de\014ne) h(the) e (functional) 1001 5409 y(\024) p Fl 1000 5436 a(h) p Fm 1 w(\() p Fl(\030) 5 b(;) 17 b(\021) p Fm 4 w(\)) 25 b(:=) p Fh 1441 5341 a(X) p Fk 1465 5556 a(k) r(;i) p Fm 1602 5436 a(e) p Fe 1646 5395 a(\000) p Fk(\026) p Fe(j) p Fi 10 w($) p Fk(k) p Fi 2 w($) 10 b(+) g($) p Fk(i) p Fi($) p Fe 10 w(j) p Fl 2141 5436 a(\030) p Fk 2185 5451 a(k) p Fb 2227 5461 a(1) p Fl 2271 5436 a(:::\030) p Fk 2399 5451 a(k) p Fd 2441 5461 a(n) p Fb 2485 5476 a(1) p Fl 2532 5436 a(\021) p Fk 2581 5451 a(i) p Fb 2609 5461 a(1) p Fl 2654 5436 a(:::\021) p Fk 2787 5451 a(i) p Fd 2815 5461 a(n) p Fb 2859 5476 a(2) p Fl 2941 5436 a(;) p Fm 795 w(\(5) p Fl(:) p Fm(3\)) 1807 5778 y(20.03.02) p 90 rotate dyy eop %%Page: 11 11 11 10 bop Fg 1222 60 a(Birkho\013) 33 b(normal) f(form) g(for) i(PDE's) 1222 b(11) p Fm 0 299 a(whic) m(h) 37 b(clearly) e(dep) s(ends) j(only) d(the) i(v) -6 b(alues) 36 b(of) p Fl 36 w(n) p Fi 1868 314 a(1) p Fm 1913 299 a(,) p Fl 36 w(n) p Fi 2037 314 a(2) p Fm 2082 299 a(,) g(in) g(\(5.2\).) 51 b(The) 36 b(in) m(terest) h(of) f(this) g(functional) h(is) 0 418 y(that) i(it) f(is) h(v) m(ery) f(easy) h(to) g(estimate) f(the) h (norm) f(of) h(its) g(v) m(ector) g(\014eld,) i(since) 2928 392 y(\024) p Fl 2927 418 a(h) p Fm 40 w(can) e(b) s(e) g(written) g (in) g(the) 0 538 y(form) 1051 663 y(\024) p Fl 1050 689 a(h) p Fm 1 w(\() p Fl(\030) 5 b(;) 17 b(\021) p Fm 4 w(\)) 25 b(=) 1625 622 y(1) p 1476 666 349 4 v 1476 778 a(\(2) p Fl(\031) p Fm 4 w(\)) p Fd 1677 711 a(n) p Fc(\000) p Fb(1) p 1675 727 132 4 v 1724 767 a(2) p Fh 1852 553 a(Z) p Fk 1952 578 a(\031) p Fe 1908 780 a(\000) p Fk(\031) p Fl 2041 689 a(F) p Fm 14 w(\() p Fl(x) p Fm(\)) p Fl(\030) p Fm 5 w(\() p Fl(x) p Fm(\)) p Fk 2438 648 a(n) p Fb 2487 658 a(1) p Fl 2529 689 a(\021) p Fm 4 w(\() p Fl(x) p Fm(\)) p Fk 2717 648 a(n) p Fb 2766 658 a(2) p Fl 2809 689 a(dx) p Fm 846 w(\(5) p Fl(:) p Fm(4\)) 0 925 y(where) p Fl 34 w(n) p Fm 28 w(=) p Fl 28 w(n) p Fi 541 940 a(1) p Fm 607 925 a(+) p Fl 23 w(n) p Fi 767 940 a(2) p Fm 845 925 a(and) 33 b(w) m(e) h(de\014ned) p Fl 98 1142 a(\030) p Fm 5 w(\() p Fl(x) p Fm(\)) 26 b(:=) 525 1075 y(1) p 453 1119 194 4 v Fj 453 1139 a(p) p 536 1139 111 4 v Fm 536 1222 a(2) p Fl(\031) p Fh 676 1048 a(X) p Fk 675 1272 a(j) p Fe 4 w(2) p Fi 774 1254 a(\026) p Fa 766 1272 a(Z) p Fl 837 1142 a(\030) p Fk 881 1157 a(j) p Fm 923 1142 a(e) p Fk 967 1101 a(ij) t(x) p Fl 1116 1142 a(;) 116 b(\021) p Fm 4 w(\() p Fl(x) p Fm(\)) 27 b(:=) 1691 1075 y(1) p 1620 1119 194 4 v Fj 1620 1139 a(p) p 1703 1139 111 4 v Fm 1703 1222 a(2) p Fl(\031) p Fh 1842 1048 a(X) p Fk 1841 1272 a(j) p Fe 4 w(2) p Fi 1940 1254 a(\026) p Fa 1932 1272 a(Z) p Fl 2004 1142 a(\021) p Fk 2053 1157 a(j) p Fm 2095 1142 a(e) p Fk 2139 1101 a(ij) t(x) p Fl 2288 1142 a(;) 116 b(F) p Fm 14 w(\() p Fl(x) p Fm(\)) 27 b(:=) 2888 1075 y(1) p 2817 1119 194 4 v Fj 2817 1139 a(p) p 2900 1139 111 4 v Fm 2900 1222 a(2) p Fl(\031) p Fh 3039 1048 a(X) p Fk 3038 1272 a(j) p Fe 4 w(2) p Fi 3137 1254 a(\026) p Fa 3129 1272 a(Z) p Fm 3201 1142 a(e) p Fe 3245 1101 a(\000) p Fk(\026) p Fe(j) p Fk(j) p Fe 4 w(j) p Fm 3445 1142 a(e) p Fk 3489 1101 a(ij) t(x) p Fl 3638 1142 a(:) p Fm 98 w(\(5) p Fl(:) p Fm(5\)) p Fn 0 1475 a(Lemma) 153 b(5.1.) p Fg 33 w(One) 35 b(has) p Fj 1527 1594 a(k) p Fl(X) p Fk 1660 1609 a(h) p Fj 1711 1594 a(k) p Fk 1761 1544 a(R) 1761 1624 y(s) p Fj 1854 1594 a(\024) 28 b(j) p Fl(h) p Fj(j) p Fk 2072 1609 a(\026) p Fj 2142 1594 a(k) p Fl(X) p Fi 2276 1604 a(\024) p Fk 2275 1622 a(h) p Fj 2326 1594 a(k) p Fk 2376 1544 a(R) 2376 1627 y(s) p Fn 0 1917 a(Pro) s(of.) p Fm 26 w(W) -8 b(e) 24 b(write) h(expliciptely) e(the) i(v) m(ector) g (\014eld) g(of) p Fl 24 w(h) p Fm(,) i(to) d(this) g(end) i(w) m(e) f (use) g(the) g(form) f(\(5.2\).) 40 b(Consider) 0 2036 y(the) p Fl 34 w(\021) p Fk 221 2051 a(t) p Fm 289 2036 a(comp) s(onen) m(t) 34 b(of) p Fl 33 w(X) p Fk 995 2051 a(h) p Fm 1046 2036 a(,) f(it) g(is) g(giv) m(en) g(\(eccept) h(for) g (an) f(irrelev) -6 b(an) m(t) 33 b(sign\)) g(b) m(y) p Fl 866 2269 a(i) 954 2201 y(@) 6 b(h) p 912 2246 200 4 v 912 2337 a(@) g(\030) p Fe 1015 2352 a(\000) p Fk(t) p Fm 1152 2269 a(=) p Fl 28 w(n) p Fi 1317 2284 a(1) p Fh 1497 2174 a(X) p Fk 1378 2388 a(k) p Fb 1420 2398 a(2) p Fk 1459 2388 a(;:::;k) p Fd 1621 2398 a(n) p Fb 1665 2413 a(1) p Fk 1707 2388 a(;i) p Fl 1776 2269 a(h) p Fe 1833 2284 a(\000) p Fk(t;k) p Fb 1991 2294 a(2) p Fk 2031 2284 a(;:::;k) p Fd 2193 2294 a(n) p Fb 2237 2309 a(1) p Fk 2279 2284 a(;i) p Fl 2336 2269 a(\030) p Fk 2380 2284 a(k) p Fb 2422 2294 a(2) p Fl 2466 2269 a(:::\030) p Fk 2594 2284 a(k) p Fd 2636 2294 a(n) p Fb 2680 2309 a(1) p Fl 2727 2269 a(\021) p Fk 2776 2284 a(i) p Fb 2804 2294 a(1) p Fl 2849 2269 a(:::\021) p Fk 2982 2284 a(i) p Fd 3010 2294 a(n) p Fb 3054 2309 a(2) p Fm 0 2563 a(and) 34 b(the) f(norm) g(of) g(the) p Fl 34 w(\021) p Fm 37 w(comp) s(onen) m(t) g(of) p Fl 34 w(X) p Fk 1703 2578 a(h) p Fm 1787 2563 a(is) g(giv) m(en) g(b) m(y) p Fh 724 2783 a(X) p Fk 781 2992 a(t) p Fh 885 2797 a(\000) p Fm 931 2878 a(1) 22 b(+) p Fj 22 w(j) p Fl(t) p Fj(j) p Fi 1194 2836 a(2) p Fk(s) p Fh 1276 2797 a(\001) 1338 2673 y(\014) 1338 2733 y(\014) 1338 2793 y(\014) 1338 2853 y(\014) 1338 2912 y(\014) 1338 2972 y(\014) p Fl 1371 2878 a(n) p Fi 1431 2893 a(1) p Fh 1611 2783 a(X) p Fk 1492 2997 a(k) p Fb 1534 3007 a(2) p Fk 1574 2997 a(;:::;k) p Fd 1736 3007 a(n) p Fb 1780 3022 a(1) p Fk 1822 2997 a(;i) p Fl 1890 2878 a(h) p Fe 1947 2893 a(\000) p Fk(t;k) p Fb 2105 2903 a(2) p Fk 2145 2893 a(;:::;k) p Fd 2307 2903 a(n) p Fb 2351 2918 a(1) p Fk 2393 2893 a(;i) p Fl 2450 2878 a(\030) p Fk 2494 2893 a(k) p Fb 2536 2903 a(2) p Fl 2580 2878 a(:::\030) p Fk 2708 2893 a(k) p Fd 2750 2903 a(n) p Fb 2794 2918 a(1) p Fl 2842 2878 a(\021) p Fk 2891 2893 a(i) p Fb 2919 2903 a(1) p Fl 2963 2878 a(:::\021) p Fk 3096 2893 a(i) p Fd 3124 2903 a(n) p Fb 3168 2918 a(2) p Fh 3217 2673 a(\014) 3217 2733 y(\014) 3217 2793 y(\014) 3217 2853 y(\014) 3217 2912 y(\014) 3217 2972 y(\014) p Fi 3250 2698 a(2) p Fj 674 3267 a(\024) p Fh 779 3096 a(") p Fm 837 3267 a(sup) p Fk 837 3351 a(t;k) r(;i) p Fh 1004 3182 a(\014) 1004 3242 y(\014) p Fl 1037 3267 a(h) p Fk 1094 3282 a(t;k) p Fb 1190 3292 a(2) p Fk 1229 3282 a(;:::;k) p Fd 1391 3292 a(n) p Fb 1435 3307 a(1) p Fk 1478 3282 a(;i) p Fh 1535 3182 a(\014) 1535 3242 y(\014) p Fm 1584 3267 a(e) p Fk 1628 3226 a(\026) p Fe(j) p Fk(k) p Fb 1742 3236 a(2) p Fi 1782 3226 a(+) p Fk(:::) p Fi(+) p Fk(k) p Fd 2018 3236 a(n) p Fb 2062 3251 a(1) p Fi 2105 3226 a(+) 10 b($) p Fk(i) p Fi($) g(+) p Fk(t) p Fe(j) p Fh 2414 3096 a(#) p Fi 2472 3117 a(2) p Fh 2533 3096 a(") 2592 3172 y(X) p Fk 2648 3381 a(t) p Fh 2752 3186 a(\000) p Fm 2798 3267 a(1) 22 b(+) p Fj 22 w(j) p Fl(t) p Fj(j) p Fi 3061 3226 a(2) p Fk(s) p Fh 3143 3186 a(\001) p Fj 3217 3267 a(\002) p Fh 748 3465 a(0) 748 3644 y(@) p Fl 835 3665 a(n) p Fi 895 3680 a(1) p Fh 1075 3571 a(X) p Fk 956 3785 a(k) p Fb 998 3795 a(2) p Fk 1037 3785 a(;:::;k) p Fd 1199 3795 a(n) p Fb 1243 3810 a(1) p Fk 1285 3785 a(;i) p Fm 1354 3665 a(e) p Fe 1398 3624 a(\000) p Fk(\026) p Fe(j) p Fk(k) p Fb 1574 3634 a(2) p Fi 1614 3624 a(+) p Fk(:::) p Fi(+) p Fk(k) p Fd 1850 3634 a(n) p Fb 1894 3649 a(1) p Fi 1937 3624 a(+) 10 b($) p Fk(i) p Fi($) p Fe 10 w(\000) p Fk(t) p Fe(j) p Fh 2264 3581 a(\014) 2264 3640 y(\014) p Fl 2297 3665 a(\030) p Fk 2341 3680 a(k) p Fb 2383 3690 a(2) p Fl 2427 3665 a(:::\030) p Fk 2555 3680 a(k) p Fd 2597 3690 a(n) p Fb 2641 3705 a(1) p Fl 2688 3665 a(\021) p Fk 2737 3680 a(i) p Fb 2765 3690 a(1) p Fl 2810 3665 a(:::\021) p Fk 2943 3680 a(i) p Fd 2971 3690 a(n) p Fb 3015 3705 a(2) p Fh 3063 3581 a(\014) 3063 3640 y(\014) 3096 3465 y(1) 3096 3644 y(A) p Fi 3184 3486 a(2) p Fh 3228 3435 a(3) 3228 3610 y(7) 3228 3674 y(5) p Fm 0 3995 a(but) 38 b(the) f(\014rst) h(square) f(brac) m (k) m(et) h(is) f(smaller) f(than) p Fj 37 w(j) p Fl(h) p Fj(j) p Fk 2046 4010 a(\026) p Fm 2100 3995 a(,) h(while) h(the) f (second) i(one) e(is) g(con) m(trolled) h(b) m(y) f(the) 0 4114 y(norm) 32 b(of) i(the) p Fl 33 w(\021) p Fm 38 w(comp) s(onen) m(t) f(of) p Fl 33 w(X) p Fi 1338 4124 a(\024) p Fk 1337 4142 a(h) p Fm 1389 4114 a(.) 43 b(Indeed,) 35 b(the) f(norm) e(of) h(the) p Fl 34 w(\021) p Fm 37 w(comp) s(onen) m (t) g(of) p Fl 34 w(X) p Fi 3322 4124 a(\024) p Fk 3321 4142 a(h) p Fm 3405 4114 a(is) g(giv) m(en) h(b) m(y) p Fh 533 4343 a(X) p Fk 589 4552 a(t) p Fh 693 4357 a(\000) p Fm 739 4437 a(1) 22 b(+) p Fj 22 w(j) p Fl(t) p Fj(j) p Fi 1002 4396 a(2) p Fk(s) p Fh 1084 4357 a(\001) 1146 4233 y(\014) 1146 4293 y(\014) 1146 4353 y(\014) 1146 4413 y(\014) 1146 4472 y(\014) 1146 4532 y(\014) p Fl 1179 4437 a(n) p Fi 1239 4452 a(1) p Fh 1419 4343 a(X) p Fk 1301 4557 a(k) p Fb 1343 4567 a(2) p Fk 1382 4557 a(;:::;k) p Fd 1544 4567 a(n) p Fb 1588 4582 a(1) p Fk 1630 4557 a(;i) p Fm 1699 4437 a(e) p Fe 1743 4396 a(\000) p Fk(\026) p Fe(j) p Fk(k) p Fb 1919 4406 a(2) p Fi 1959 4396 a(+) p Fk(:::) p Fi(+) p Fk(k) p Fd 2195 4406 a(n) p Fb 2239 4421 a(1) p Fi 2282 4396 a(+) 10 b($) p Fk(i) p Fi($) p Fe 10 w(\000) p Fk(t) p Fe(j) p Fl 2592 4437 a(\030) p Fk 2636 4452 a(k) p Fb 2678 4462 a(2) p Fl 2721 4437 a(:::\030) p Fk 2849 4452 a(k) p Fd 2891 4462 a(n) p Fb 2935 4477 a(1) p Fl 2983 4437 a(\021) p Fk 3032 4452 a(i) p Fb 3060 4462 a(1) p Fl 3105 4437 a(:::\021) p Fk 3238 4452 a(i) p Fd 3266 4462 a(n) p Fb 3310 4477 a(2) p Fh 3358 4233 a(\014) 3358 4293 y(\014) 3358 4353 y(\014) 3358 4413 y(\014) 3358 4472 y(\014) 3358 4532 y(\014) p Fi 3391 4258 a(2) p Fm 0 4732 a(taking) 42 b(real) h(p) s(ositiv) m(e) p Fl 43 w(\030) 5 b(;) 17 b(\021) p Fm 46 w(one) 44 b(obtains) f(the) h(expression) g(in) g(the) f (ab) s(o) m(v) m(e) h(square) f(brac) m(k) m(et.) 75 b(Similar) 0 4852 y(inequalities) 31 b(clearly) h(hold) g(for) g(the) h (other) f(comp) s(onen) m(ts) g(of) h(the) f(v) m(ector) g(\014eld) h (of) p Fl 32 w(h) p Fm(.) 44 b(One) 33 b(th) m(us) g(obtains) p Fh 631 5005 a(\020) p Fj 691 5116 a(j) p Fl -1 w(h) p Fj(j) p Fk 803 5146 a(\026) p Fj 873 5116 a(k) p Fl(X) p Fi 1007 5126 a(\024) p Fk 1006 5144 a(h) p Fj 1057 5116 a(k) p Fk 1107 5066 a(R) 1107 5149 y(s) p Fh 1173 5005 a(\021) p Fi 1232 5026 a(2) p Fm 1304 5116 a(=) p Fj 29 w(j) p Fl -1 w(h) p Fj(j) p Fi 1522 5066 a(2) p Fk 1522 5146 a(\026) p Fm 1708 5116 a(sup) p Fk 1592 5199 a(R) p Fb 1652 5209 a(1) p Fi 1691 5199 a(+) p Fk(R) p Fb 1812 5209 a(2) p Fi 1851 5199 a(=) p Fk(R) p Fh 1989 4945 a( ) 2068 4975 y(\022) 2142 5001 y(\015) 2142 5061 y(\015) 2142 5121 y(\015) p Fl 2197 5116 a(X) p Fk 2288 5068 a(\030) p Fi 2281 5141 a(\024) p Fk 2280 5160 a(h) p Fh 2331 5001 a(\015) 2331 5061 y(\015) 2331 5121 y(\015) p Fk 2387 5026 a(R) p Fb 2447 5036 a(1) p Fk 2387 5186 a(s) p Fh 2490 4975 a(\023) p Fi 2564 4996 a(2) p Fm 2631 5116 a(+) p Fh 2730 4975 a(\022) 2804 5001 y(\015) 2804 5061 y(\015) 2804 5121 y(\015) p Fl 2859 5116 a(X) p Fk 2950 5068 a(\021) p Fi 2943 5141 a(\024) p Fk 2942 5160 a(h) p Fh 2998 5001 a(\015) 2998 5061 y(\015) 2998 5121 y(\015) p Fk 3053 5026 a(R) p Fb 3113 5036 a(2) p Fk 3053 5186 a(s) p Fh 3157 4975 a(\023) p Fi 3230 4996 a(2) p Fh 3275 4945 a(!) p Fj 642 5455 a(\025) p Fh 747 5284 a( ) p Fm 942 5455 a(sup) p Fk 826 5538 a(R) p Fb 886 5548 a(1) p Fi 925 5538 a(+) p Fk(R) p Fb 1046 5548 a(2) p Fi 1085 5538 a(=) p Fk(R) p Fh 1223 5314 a(\022) 1297 5340 y(\015) 1297 5400 y(\015) 1297 5460 y(\015) p Fl 1352 5455 a(X) p Fk 1443 5407 a(\030) 1435 5485 y(h) p Fh 1486 5340 a(\015) 1486 5400 y(\015) 1486 5460 y(\015) p Fk 1542 5365 a(R) p Fb 1602 5375 a(1) p Fk 1542 5524 a(s) p Fh 1645 5314 a(\023) p Fi 1719 5335 a(2) p Fm 1786 5455 a(+) p Fh 1885 5344 a(\020) p Fj 1945 5455 a(k) p Fl(X) p Fk 2086 5407 a(\021) 2078 5485 y(h) p Fj 2133 5455 a(k) p Fk 2183 5401 a(R) p Fb 2243 5411 a(2) p Fk 2183 5490 a(s) p Fh 2287 5344 a(\021) p Fi 2347 5365 a(2) p Fh 2391 5284 a(!) p Fm 2498 5455 a(=) p Fh 2603 5344 a(\020) p Fj 2663 5455 a(k) p Fl -1 w(X) p Fk 2795 5470 a(h) p Fj 2847 5455 a(k) p Fk 2896 5404 a(R) 2896 5484 y(s) p Fh 2962 5344 a(\021) p Fi 3021 5365 a(2) p Fm 1807 5778 a(20.03.02) p 90 rotate dyy eop %%Page: 12 12 12 11 bop Fm 0 60 a(12) p Fg 1660 w(D.) 33 b(Bam) m(busi) p Fm 0 299 a(whic) m(h) h(is) f(the) h(thesis.) p 3891 225 78 4 v 3891 295 4 70 v 3965 295 V 3891 299 78 4 v Fn 0 458 a(Lemma) 153 b(5.2.) p Fg 33 w(Consider) 34 b(the) g(function) p Fm 1711 432 a(\024) p Fl 1710 458 a(h) p Fg 34 w(as) f(ab) s(o) m(v) m(e,) g(one) h(has) p Fj 1479 675 a(k) p Fl(X) p Fi 1613 685 a(\024) p Fk 1612 703 a(h) p Fj 1663 675 a(k) p Fk 1713 625 a(R) 1713 708 y(s) p Fj 1806 675 a(\024) p Fl 28 w(n) p Fm(\006) p Fk 2043 690 a(s) p Fm 2085 675 a(\() p Fl(\032) p Fk 2176 690 a(s) p Fl 2218 675 a(R) p Fm 1 w(\)) p Fk 2334 633 a(n) p Fe(\000) p Fi(1) p Fg 0 891 a(where) p Fl 34 w(n) p Fm 28 w(=) p Fl 28 w(n) p Fi 541 906 a(1) p Fm 607 891 a(+) p Fl 23 w(n) p Fi 767 906 a(2) p Fg 811 891 a(.) p Fn 0 1050 a(Pro) s(of.) p Fm 34 w(W) -8 b(e) 34 b(b) s(egin) f(b) m(y) h(considering) g(the) p Fl 33 w(\030) p Fm 38 w(comp) s(onen) m(t) f(of) p Fl 33 w(X) p Fi 2402 1060 a(\024) p Fk 2401 1078 a(h) p Fm 2453 1050 a(,) f(one) i(has) p Fl 1505 1334 a(X) p Fk 1596 1286 a(\030) p Fi 1589 1359 a(\024) p Fk 1588 1377 a(h) p Fm 1667 1334 a(=) p Fl 28 w(i) 1818 1266 y(n) p Fi 1878 1281 a(2) p Fl 1923 1266 a(F) 14 b(\030) p Fk 2050 1230 a(n) p Fb 2099 1240 a(1) p Fl 2142 1266 a(\021) p Fk 2195 1230 a(n) p Fb 2244 1240 a(2) p Fe 2283 1230 a(\000) p Fi(1) p 1819 1311 572 4 v Fm 1930 1422 a(\(2) p Fl(\031) p Fm 4 w(\)) p Fd 2131 1356 a(n) p Fc(\000) p Fb(1) p 2130 1372 132 4 v 2179 1411 a(2) p Fl 2436 1334 a(;) p Fm 0 1623 a(or) 33 b(more) f(precisely) p Fl 1181 1801 a(X) p Fk 1272 1753 a(\030) p Fi 1265 1826 a(\024) p Fk 1264 1845 a(h) p Fm 1315 1801 a(\() p Fl(x) p Fm(\)) 27 b(=) p Fl 29 w(i) 1629 1733 y(n) p Fi 1689 1748 a(2) p Fl 1733 1733 a(F) p Fm 14 w(\() p Fl(x) p Fm(\)[) p Fl(\030) p Fm 5 w(\() p Fl(x) p Fm(\)]) p Fk 2186 1697 a(n) p Fb 2235 1707 a(1) p Fm 2277 1733 a([) p Fl(\021) p Fm 4 w(\() p Fl(x) p Fm(\)]) p Fk 2521 1697 a(n) p Fb 2570 1707 a(2) p Fe 2608 1697 a(\000) p Fi(1) p 1629 1778 1086 4 v Fm 1998 1890 a(\(2) p Fl(\031) p Fm 4 w(\)) p Fd 2199 1823 a(n) p Fc(\000) p Fb(1) p 2197 1839 132 4 v 2246 1879 a(2) p Fl 2760 1801 a(;) p Fm 0 2053 a(from) 33 b(whic) m(h,) g(using) h(the) f(inequalit) m(y) p Fh 1253 2259 a(\015) 1253 2319 y(\015) p Fl 1309 2344 a(\030) p Fi 1358 2303 a(2) p Fh 1402 2259 a(\015) 1402 2319 y(\015) p Fk 1457 2384 a(H) p Fd 1527 2364 a(s) p Fj 1599 2344 a(\024) p Fh 1704 2187 a(r) p 1803 2187 85 4 v Fm 1821 2276 a(2) p 1815 2321 61 4 v Fl 1815 2412 a(\031) p Fm 1888 2344 a(2) p Fk 1938 2303 a(s) p Fi(+1) p Fj 2098 2344 a(k) p Fl -1 w(\030) p Fj 5 w(k) p Fk 2245 2374 a(H) p Fd 2315 2354 a(s) p Fj 2376 2344 a(k) p Fl(\030) p Fj 5 w(k) p Fk 2524 2374 a(H) p Fd 2594 2354 a(s) p Fl 2687 2344 a(;) p Fm 1049 w(\(5) p Fl(:) p Fm(6\)) 0 2667 y(and,) f(remarking) e(that) i(\006) p Fk 979 2682 a(s) p Fm 1049 2667 a(=) p Fj 28 w(k) p Fl(F) p Fj 14 w(k) p Fk 1332 2697 a(H) p Fd 1402 2677 a(s) p Fm 1445 2667 a(,) g(one) g(has) p Fh 1862 2552 a(\015) 1862 2612 y(\015) 1862 2672 y(\015) p Fl 1918 2667 a(X) p Fk 2009 2619 a(\030) p Fi 2002 2692 a(\024) p Fk 2001 2711 a(h) p Fh 2052 2552 a(\015) 2052 2612 y(\015) 2052 2672 y(\015) p Fk 2107 2577 a(R) 2107 2737 y(s) p Fj 2200 2667 a(\024) p Fl 28 w(n) p Fi 2365 2682 a(2) p Fm 2410 2667 a(\006) p Fk 2482 2682 a(s) p Fm 2524 2667 a(\() p Fl(\032) p Fk 2615 2682 a(s) p Fl 2657 2667 a(R) p Fm 1 w(\)) p Fk 2773 2631 a(n) p Fe(\000) p Fi(1) p Fm 2929 2667 a(.) 43 b(Considering) 32 b(also) f(the) p Fl 32 w(\021) p Fm 0 2824 a(comp) s(onen) m(t) i(one) h(has) g(the) f(thesis.) p 3891 2750 78 4 v 3891 2820 4 70 v 3965 2820 V 3891 2824 78 4 v Fn 0 2984 a(Prop) s(osition) 177 b(5.3.) p Fg 39 w(Let) p Fl 39 w(h) p Fj 37 w(2) 37 b(M) p Fk 1476 2936 a(\026) 1476 3013 y(R) p Fg 1541 2984 a(;) k(then,) f(for) f(an) m (y) f(p) s(ositiv) m(e) p Fl 38 w(R) p Fi 2673 2947 a(\() p Fk(s) p Fi(\)) p Fg 2816 2984 a(suc) m(h) i(that) p Fl 38 w(\032) p Fk 3319 2999 a(s) p Fl 3361 2984 a(R) p Fi 3438 2947 a(\() p Fk(s) p Fi(\)) p Fj 3579 2984 a(\024) p Fl 37 w(R) p Fg 39 w(and) 0 3103 y(an) m(y) p Fl 35 w(\016) p Fi 238 3067 a(\() p Fk(s) p Fi(\)) p Fl 375 3103 a(<) 32 b(R) p Fi 561 3067 a(\() p Fk(s) p Fi(\)) p Fg 665 3103 a(,) k(the) g(Hamiltonian) d(v) m(ector) i(\014eld) p Fl 36 w(X) p Fk 2080 3118 a(h) p Fg 2167 3103 a(is) g(analytic) g(as) g (a) h(map) e(from) p Fj 35 w(B) p Fk 3387 3118 a(s) p Fm 3430 3103 a(\() p Fl(R) p Fi 3546 3067 a(\() p Fk(s) p Fi(\)) p Fm 3650 3103 a(\)) p Fg 35 w(to) p Fj 35 w(P) p Fk 3925 3067 a(s) p Fg 0 3223 a(and) g(ful\014lls) f(the) h(estimate) p Fj 1260 3368 a(k) p Fl -1 w(X) p Fk 1392 3383 a(h) p Fj 1444 3368 a(k) p Fk 1493 3318 a(R) p Fb 1553 3288 a(\() p Fd(s) p Fb(\)) p Fe 1646 3318 a(\000) p Fk(\016) p Fb 1747 3288 a(\() p Fd(s) p Fb(\)) p Fk 1493 3398 a(s) p Fj 1872 3368 a(\024) p Fm 2055 3301 a(\006) p Fk 2127 3316 a(s) p 1989 3345 247 4 v Fl 1989 3438 a(\032) p Fk 2041 3453 a(s) p Fl 2083 3438 a(\016) p Fi 2131 3409 a(\() p Fk(s) p Fi(\)) p Fj 2265 3368 a(h) o(j) p Fl(h) p Fj(ji) p Fk 2455 3318 a(\026) 2455 3405 y(\032) p Fd 2496 3415 a(s) p Fk 2535 3405 a(R) p Fb 2595 3385 a(\() p Fd(s) p Fb(\)) p Fm 3764 3368 a(\(5) p Fl(:) p Fm(7\)) p Fn 0 3767 a(Pro) s(of.) p Fm 31 w(First) 29 b(w) m(ork) h(on) g (homogenous) g(p) s(olynomials) d(of) j(degree) p Fl 31 w(n) p Fm(.) 43 b(One) 31 b(has) p Fl 30 w(h) p Fm 28 w(=) p Fh 3186 3692 a(P) p Fd 3365 3762 a(n) p Fb 3409 3777 a(1) p Fd 3448 3762 a(;n) p Fb 3515 3777 a(2) p Fd 3303 3822 a(n) p Fb 3347 3837 a(1) 3386 3822 y(+) p Fd(n) p Fb 3481 3837 a(2) 3520 3822 y(=) p Fd(n) p Fl 3649 3767 a(h) p Fk 3706 3730 a(n) p Fb 3755 3740 a(1) p Fk 3794 3730 a(;n) p Fb 3867 3740 a(2) p Fl 3941 3767 a(;) p Fm 0 3916 a(with) p Fl 33 w(h) p Fk 284 3880 a(n) p Fb 333 3890 a(1) p Fk 372 3880 a(;n) p Fb 445 3890 a(2) p Fm 522 3916 a(homogeneous) j(in) p Fl 33 w(\030) 5 b(;) p Fm 32 w(and) 34 b(in) p Fl 33 w(\021) p Fm 4 w(.) 44 b(One) 34 b(has) p Fj 213 4177 a(k) p Fl(X) p Fk 346 4192 a(h) p Fj 397 4177 a(k) p Fk 447 4127 a(R) p Fb 507 4097 a(\() p Fd(s) p Fb(\)) p Fe 599 4127 a(\000) p Fk(\016) p Fb 700 4097 a(\() p Fd(s) p Fb(\)) p Fk 447 4207 a(s) p Fj 825 4177 a(\024) p Fh 958 4082 a(X) p Fk 930 4292 a(n) p Fb 979 4302 a(1) p Fk 1019 4292 a(;n) p Fb 1092 4302 a(2) p Fj 1147 4177 a(k) p Fl(X) p Fk 1280 4192 a(h) p Fd 1327 4170 a(n) p Fb 1371 4185 a(1) p Fd 1409 4170 a(;n) p Fb 1476 4185 a(2) p Fj 1524 4177 a(k) p Fk 1574 4127 a(R) p Fb 1634 4097 a(\() p Fd(s) p Fb(\)) p Fe 1727 4127 a(\000) p Fk(\016) p Fb 1828 4097 a(\() p Fd(s) p Fb(\)) p Fk 1574 4207 a(s) p Fj 1953 4177 a(\024) p Fh 2086 4082 a(X) p Fk 2058 4292 a(n) p Fb 2107 4302 a(1) p Fk 2146 4292 a(;n) p Fb 2219 4302 a(2) p Fj 2274 4177 a(j) p Fl(h) p Fk 2359 4136 a(n) p Fb 2408 4146 a(1) p Fk 2448 4136 a(;n) p Fb 2521 4146 a(2) p Fj 2564 4177 a(j) p Fk 2592 4207 a(\026) p Fl 2662 4177 a(n) p Fm(\006) p Fk 2794 4192 a(s) p Fh 2853 4067 a(\020) p Fl 2913 4177 a(\032) p Fk 2965 4192 a(s) p Fm 3007 4177 a(\() p Fl(R) p Fi 3123 4136 a(\() p Fk(s) p Fi(\)) p Fj 3249 4177 a(\000) p Fl 22 w(\016) p Fi 3396 4136 a(\() p Fk(s) p Fi(\)) p Fm 3501 4177 a(\)) p Fh 3540 4067 a(\021) p Fk 3600 4087 a(n) p Fe(\000) p Fi(1) p Fj 825 4491 a(\024) p Fm 1009 4423 a(\006) p Fk 1081 4438 a(s) p 942 4468 V Fl 942 4561 a(\032) p Fk 994 4576 a(s) p Fl 1036 4561 a(\016) p Fi 1084 4532 a(\() p Fk(s) p Fi(\)) p Fh 1246 4396 a(X) p Fk 1218 4605 a(n) p Fb 1267 4615 a(1) p Fk 1306 4605 a(;n) p Fb 1379 4615 a(2) p Fj 1434 4491 a(j) p Fl(h) p Fk 1519 4449 a(n) p Fb 1568 4459 a(1) p Fk 1608 4449 a(;n) p Fb 1681 4459 a(2) p Fj 1724 4491 a(j) p Fk 1752 4520 a(\026) p Fh 1822 4380 a(h) p Fl 1869 4491 a(\032) p Fk 1921 4506 a(s) p Fl 1963 4491 a(R) p Fi 2040 4449 a(\() p Fk(s) p Fi(\)) p Fh 2145 4380 a(i) p Fk 2192 4401 a(n) p Fm 2273 4491 a(=) 2457 4423 y(\006) p Fk 2529 4438 a(s) p 2391 4468 V Fl 2391 4561 a(\032) p Fk 2443 4576 a(s) p Fl 2485 4561 a(\016) p Fi 2533 4532 a(\() p Fk(s) p Fi(\)) p Fj 2666 4491 a(hj) p Fl(h) p Fj(ji) p Fk 2856 4440 a(\026) 2856 4527 y(\032) p Fd 2897 4537 a(s) p Fk 2936 4527 a(R) p Fb 2996 4507 a(\() p Fd(s) p Fb(\)) p Fm 0 4816 a(where) 45 b(w) m(e) g(used) h(the) f (inequalit) m(y) p Fl 43 w(n) p Fm(\() p Fl(R) p Fj 30 w(\000) p Fl 30 w(d) p Fk 1716 4780 a(n) p Fe(\000) p Fi(1) p Fm 1872 4816 a(\)) p Fj 46 w(\024) p Fl 47 w(R) p Fk 2158 4780 a(n) p Fl 2212 4816 a(d) p Fe 2264 4780 a(\000) p Fi(1) p Fm 2371 4816 a(.) 78 b(Then) 45 b(the) g(result) g (easily) f(extends) h(to) 0 4936 y(analytic) 32 b(functions.) p 3891 4862 78 4 v 3891 4932 4 70 v 3965 4932 V 3891 4936 78 4 v 199 5085 a(Consider) 41 b(no) m(w) e(a) h(homogeneous) g(p) s (olynomial) d(of) j(degree) p Fl 40 w(m) p Fi 2585 5100 a(1) p Fm 2630 5085 a(,) p Fl 41 w(m) p Fi 2786 5100 a(2) p Fm 2831 5085 a(,) p Fl 41 w(m) p Fi 2987 5100 a(3) p Fm 3032 5085 a(,) p Fl 41 w(m) p Fi 3188 5100 a(4) p Fm 3233 5085 a(,) h(in) p Fl 40 w(q) t(;) 17 b(p;) g(Q;) g(P) p Fm 51 w(re-) 0 5205 y(sp) s(ectiv) m(ely) -8 b(,) 32 b(and) i(write) f(it) g(in) g(the) g(form) p Fl 576 5436 a(h) p Fm(\() p Fl(q) t(;) 17 b(p;) g(Q;) g(P) p Fm 14 w(\)) 25 b(=) p Fh 1260 5341 a(X) p Fk 1231 5556 a(k) r(;i;j;l) p Fl 1449 5436 a(h) p Fk 1506 5451 a(k) r(;i;j;l) p Fl 1713 5436 a(q) p Fk 1757 5451 a(k) p Fb 1799 5461 a(1) p Fl 1844 5436 a(:::q) p Fk 1972 5451 a(k) p Fd 2014 5461 a(m) p Fb 2075 5476 a(1) p Fl 2124 5436 a(p) p Fk 2174 5451 a(i) p Fb 2202 5461 a(1) p Fl 2246 5436 a(:::p) p Fk 2380 5451 a(i) p Fd 2408 5461 a(m) p Fb 2469 5476 a(2) p Fl 2517 5436 a(Q) p Fk 2596 5451 a(j) p Fb 2629 5461 a(1) p Fl 2673 5436 a(:::Q) p Fk 2836 5451 a(j) p Fd 2869 5461 a(m) p Fb 2930 5476 a(3) p Fl 2978 5436 a(P) p Fk 3042 5451 a(l) p Fb 3067 5461 a(1) p Fl 3111 5436 a(:::P) p Fk 3259 5451 a(l) p Fd 3284 5461 a(m) p Fb 3345 5476 a(4) p Fm 3764 5480 a(\(5) p Fl(:) p Fm(8\)) 1807 5778 y(20.03.02) p 90 rotate dyy eop %%Page: 13 13 13 12 bop Fg 1222 60 a(Birkho\013) 33 b(normal) f(form) g(for) i(PDE's) p Fm 1222 w(13) 0 299 y(Corresp) s(ondingly) f(w) m(e) h(de\014ne) h (the) e(the) h(functional) 578 539 y(\024) 575 559 y(\024) p Fl 574 585 a(h) p Fm(\() p Fl(p;) 17 b(q) t(;) g(P) s(;) g(Q) p Fm(\)) 25 b(:=) 1407 518 y(1) p 1258 562 349 4 v 1258 674 a(\(2) p Fl(\031) p Fm 4 w(\)) p Fd 1459 608 a(n) p Fc(\000) p Fb(1) p 1457 624 132 4 v 1506 663 a(2) p Fh 1635 450 a(Z) p Fk 1734 474 a(\031) p Fe 1690 676 a(\000) p Fk(\031) p Fl 1823 585 a(F) p Fm 14 w(\() p Fl(x) p Fm(\)) p Fl(q) p Fm 4 w(\() p Fl(x) p Fm(\)) p Fk 2219 544 a(m) p Fb 2290 554 a(1) p Fl 2332 585 a(p) p Fm(\() p Fl(x) p Fm(\)) p Fk 2517 544 a(m) p Fb 2588 554 a(2) p Fl 2631 585 a(Q) p Fm(\() p Fl(x) p Fm(\)) p Fk 2845 544 a(m) p Fb 2916 554 a(3) p Fl 2959 585 a(P) p Fm 14 w(\() p Fl(x) p Fm(\)) p Fk 3172 544 a(m) p Fb 3243 554 a(4) p Fl 3286 585 a(dx) p Fm 0 900 a(with) p Fl 33 w(q) p Fm 4 w(\() p Fl(x) p Fm(\),) p Fl 32 w(p) p Fm(\() p Fl(x) p Fm(\),) p Fl 33 w(Q) p Fm(\() p Fl(x) p Fm(\),) p Fl 32 w(P) p Fm 14 w(\() p Fl(x) p Fm(\)) 32 b(de\014ned) j(in) e(analogy) f(with) h (\(5.5\).) 43 b(F) -8 b(or) 33 b(its) f(v) m(ector) h(\014eld) h(w) m (e) g(will) e(pro) m(v) m(e) 0 1019 y(an) g(estimate) e(stronger) i (than) g(\(5.7\);) f(this) h(is) f(the) h(heart) g(of) g(the) g(momen) m (tum) e(cuto\013.) 44 b(W) -8 b(orking) 31 b(exactly) 0 1139 y(as) i(in) h(the) f(pro) s(of) h(of) f(lemma) e(5.1) i(one) g (has) p Fn 0 1298 a(Lemma) 180 b(5.4.) p Fg 39 w(Let) p Fl 39 w(h) p Fg 40 w(b) s(e) 39 b(homogeneous) g(p) s(olynomial) e(of) i (degree) p Fl 40 w(m) p Fi 2845 1313 a(1) p Fl 2890 1298 a(;) 17 b(m) p Fi 3022 1313 a(2) p Fl 3067 1298 a(;) g(m) p Fi 3199 1313 a(3) p Fl 3243 1298 a(;) g(m) p Fi 3375 1313 a(4) p Fg 3459 1298 a(in) p Fl 39 w(q) t(;) g(p;) g(Q;) g(P) p Fg 0 1418 a(resp) s(ectiv) m(ely) -8 b(,) 33 b(then) h(one) f(has) p Fj 1522 1585 a(k) p Fl(X) p Fk 1655 1600 a(h) p Fj 1706 1585 a(k) p Fk 1756 1535 a(R) 1756 1615 y(s) p Fj 1849 1585 a(\024) 28 b(j) p Fl(h) p Fj(j) p Fk 2067 1600 a(\026) p Fh 2137 1470 a(\015) 2137 1530 y(\015) 2137 1590 y(\015) p Fl 2192 1585 a(X) p Fi 2278 1595 a(\024) 2276 1608 y(\024) p Fk 2275 1627 a(h) p Fh 2326 1470 a(\015) 2326 1530 y(\015) 2326 1590 y(\015) p Fk 2381 1495 a(R) 2381 1655 y(s) p Fm 199 1932 a(T) -8 b(o) 35 b(come) f(to) h(the) g(estimate) f(of) i(the) f (v) m(ector) g(\014eld) h(of) 2229 1886 y(\024) 2226 1906 y(\024) p Fl 2225 1932 a(h) p Fm 36 w(w) m(e) g(need) g(a) f(few) h (lemmas) d(concerning) j(the) 0 2052 y(v) m(ector) d(\014eld) h(of) f (the) h(pro) s(duct) g(of) f(the) h(functions) p Fl 34 w(P) p Fm 14 w(\() p Fl(x) p Fm(\)) f(and) p Fl 34 w(Q) p Fm(\() p Fl(x) p Fm(\).) p Fn 0 2211 a(Lemma) 153 b(5.5.) p Fg 33 w(Let) p Fl 34 w(\036) p Fg 33 w(b) s(e) 34 b(suc) m(h) h(that) p Fl 1415 2488 a(\036) p Fm(\() p Fl(x) p Fm(\)) 27 b(=) 1825 2421 y(1) p 1753 2465 194 4 v Fj 1753 2485 a(p) p 1836 2485 111 4 v Fm 1836 2568 a(2) p Fl(\031) p Fh 2015 2393 a(X) p Fe 1975 2612 a(j) p Fk(k) p Fe 2 w(j) p Fk(>N) p Fl 2216 2488 a(\036) p Fk 2275 2503 a(k) p Fl 2324 2488 a(e) p Fk 2370 2447 a(ik) r(x) p Fl 2526 2488 a(;) p Fg 0 2850 a(then,) 34 b(giv) m(en) f(p) s(ositiv) m(e) f(in) m(tegers) p Fl 34 w(s;) 17 b(j) p Fg 38 w(suc) m(h) 35 b(that) p Fl 33 w(s) 27 b(>) h(j) p Fm 28 w(+) p Fi 2224 2810 a(1) p 2224 2827 40 4 v 2224 2884 a(2) p Fg 2275 2850 a(,) 33 b(one) h(has) p Fh 1265 3030 a(\014) 1265 3090 y(\014) 1265 3150 y(\014) 1265 3209 y(\014) p Fl 1310 3107 a(d) p Fk 1362 3071 a(j) p Fl 1404 3107 a(\036) p 1310 3152 154 4 v 1311 3243 a(dx) p Fk 1420 3214 a(j) p Fm 1475 3175 a(\() p Fl(x) p Fm(\)) p Fh 1610 3030 a(\014) 1610 3090 y(\014) 1610 3150 y(\014) 1610 3209 y(\014) p Fj 1671 3175 a(\024) p Fh 1776 3018 a(r) p 1875 3018 85 4 v Fm 1893 3107 a(2) p 1887 3152 61 4 v Fl 1887 3243 a(\031) p Fm 2123 3107 a(1) p 1972 3152 353 4 v Fl 1972 3257 a(N) p Fk 2063 3222 a(s) p Fe(\000) p Fk(j) p Fe 4 w(\000) p Fb 2274 3196 a(1) p 2274 3207 34 4 v 2274 3246 a(2) p Fj 2353 3175 a(k) p Fl(\036) p Fj(k) p Fk 2512 3204 a(H) p Fd 2582 3184 a(s) p Fl 2676 3175 a(:) p Fn 0 3570 a(Pro) s(of.) p Fm 34 w(One) h(has) p Fj 727 3813 a(p) p 810 3813 111 4 v Fm 810 3900 a(2) p Fl(\031) p Fh 937 3756 a(\014) 937 3815 y(\014) 937 3875 y(\014) 937 3935 y(\014) p Fl 982 3833 a(d) p Fk 1034 3796 a(j) p Fl 1076 3833 a(\036) p 982 3877 154 4 v 983 3968 a(dx) p Fk 1092 3940 a(j) p Fm 1147 3900 a(\() p Fl(x) p Fm(\)) p Fh 1282 3756 a(\014) 1282 3815 y(\014) 1282 3875 y(\014) 1282 3935 y(\014) p Fm 1343 3900 a(=) p Fh 1448 3696 a(\014) 1448 3756 y(\014) 1448 3815 y(\014) 1448 3875 y(\014) 1448 3935 y(\014) 1448 3995 y(\014) 1521 3805 y(X) p Fe 1481 4024 a(j) p Fk(k) p Fe 2 w(j) p Fk(>N) p Fm 1705 3900 a(\() p Fl(ik) p Fm 3 w(\)) p Fk 1872 3859 a(j) p Fl 1914 3900 a(\036) p Fk 1973 3915 a(k) p Fl 2022 3900 a(e) p Fk 2068 3859 a(ik) r(x) p Fh 2191 3696 a(\014) 2191 3756 y(\014) 2191 3815 y(\014) 2191 3875 y(\014) 2191 3935 y(\014) 2191 3995 y(\014) p Fj 2252 3900 a(\024) p Fh 2397 3805 a(X) p Fe 2357 4024 a(j) p Fk(k) p Fe 2 w(j) p Fk(>N) p Fj 2598 3900 a(j) p Fl(\036) p Fk 2685 3915 a(k) p Fj 2734 3900 a(jj) p Fl(k) p Fj 3 w(j) p Fk 2873 3859 a(j) p Fj 1343 4326 a(\024) p Fh 1448 4126 a(2) 1448 4305 y(4) 1554 4232 y(X) p Fe 1514 4450 a(j) p Fk(k) p Fe 2 w(j) p Fk(>N) p Fm 1919 4259 a(1) p 1767 4303 355 4 v Fj 1767 4396 a(j) p Fl(k) p Fj 3 w(j) p Fi 1878 4368 a(2\() p Fk(s) p Fe(\000) p Fk(j) p Fi 4 w(\)) p Fh 2133 4126 a(3) 2133 4305 y(5) p Fi 2200 4147 a(1) p Fk(=) p Fi(2) p Fh 2341 4126 a(2) 2341 4305 y(4) 2448 4232 y(X) p Fe 2408 4450 a(j) p Fk(k) p Fe 2 w(j) p Fk(>N) p Fj 2649 4326 a(j) p Fl(k) p Fj 3 w(j) p Fi 2760 4285 a(2) p Fk(s) p Fj 2841 4326 a(j) p Fl(\036) p Fk 2928 4341 a(k) p Fj 2977 4326 a(j) p Fi 3005 4285 a(2) p Fh 3050 4126 a(3) 3050 4305 y(5) p Fi 3116 4147 a(1) p Fk(=) p Fi(2) p Fj 1343 4693 a(\024) p Fh 1448 4552 a(\032) p Fm 2013 4625 a(2) p 1534 4670 1007 4 v 1534 4763 a([2\() p Fl(s) p Fj 22 w(\000) p Fl 22 w(j) p Fm 6 w(\)) p Fj 22 w(\000) p Fm 22 w(1]) p Fl(N) p Fi 2195 4734 a(2\() p Fk(s) p Fe(\000) p Fk(j) p Fi 4 w(\)) p Fe(\000) p Fi(1) p Fh 2553 4552 a(\033) p Fi 2627 4573 a(1) p Fk(=) p Fi(2) p Fj 2769 4693 a(k) p Fl(\036) p Fj(k) p Fk 2928 4722 a(H) p Fd 2998 4703 a(s) p 3891 4918 78 4 v 3891 4987 4 70 v 3965 4987 V 3891 4991 78 4 v Fn 0 5151 a(Lemma) 153 b(5.6.) p Fg 33 w(Let) p Fl 34 w(\036) p Fg 33 w(and) p Fl 34 w( ) p Fg 37 w(b) s(e) 33 b(suc) m(h) i(that) p Fl 780 5428 a( ) p Fm 4 w(\() p Fl(x) p Fm(\)) 26 b(=) 1199 5361 y(1) p 1127 5405 194 4 v Fj 1127 5425 a(p) p 1210 5425 111 4 v Fm 1210 5507 a(2) p Fl(\031) p Fh 1389 5333 a(X) p Fe 1349 5552 a(j) p Fk(k) p Fe 2 w(j) p Fk(>N) p Fl 1590 5428 a( ) p Fk 1655 5443 a(k) p Fl 1704 5428 a(e) p Fk 1750 5387 a(ik) r(x) p Fl 1906 5428 a(;) 116 b(\036) p Fm(\() p Fl(x) p Fm(\)) 27 b(=) 2460 5361 y(1) p 2388 5405 194 4 v Fj 2388 5425 a(p) p 2471 5425 111 4 v Fm 2471 5507 a(2) p Fl(\031) p Fh 2650 5333 a(X) p Fe 2610 5552 a(j) p Fk(k) p Fe 2 w(j) p Fk(>N) p Fl 2851 5428 a(\036) p Fk 2910 5443 a(k) p Fl 2959 5428 a(e) p Fk 3005 5387 a(ik) r(x) p Fl 3161 5428 a(;) p Fm 1807 5778 a(20.03.02) p 90 rotate dyy eop %%Page: 14 14 14 13 bop Fg 0 60 a(14) 1660 b(D.) 33 b(Bam) m(busi) 0 299 y(then,) h(for) f(an) m(y) g(in) m(teger) p Fl 34 w(s) p Fj 27 w(\025) p Fm 28 w(1) p Fg(,) g(one) h(has) p Fj 1193 575 a(k) p Fl -1 w( ) t(\036) p Fj(k) p Fk 1420 604 a(H) p Fd 1490 585 a(s) p Fj 1562 575 a(\024) p Fh 1667 418 a(r) p 1767 418 85 4 v Fm 1784 507 a(2) p 1779 552 61 4 v Fl 1779 643 a(\031) p Fm 1884 507 a(2) p Fk 1934 471 a(s) p Fi(+1) p 1863 552 236 4 v Fl 1863 643 a(N) p Fk 1954 614 a(s) p Fe(\000) p Fi(1) p Fj 2127 575 a(k) p Fl( ) p Fj 4 w(k) p Fk 2295 604 a(H) p Fd 2365 585 a(s) p Fj 2425 575 a(k) p Fl(\036) p Fj(k) p Fk 2584 604 a(H) p Fd 2654 585 a(s) p Fl 2748 575 a(:) p Fm 988 w(\(5) p Fl(:) p Fm(9\)) p Fn 0 935 a(Pro) s(of.) p Fm 34 w(The) g(main) e(p) s(oin) m(t) h(consists) h(in) f(estimating) f (the) p Fl 34 w(L) p Fi 2268 899 a(2) p Fm 2345 935 a(norm) h(of) p Fl 1348 1144 a(d) p Fk 1400 1108 a(s) p 1320 1188 152 4 v Fl 1320 1279 a(dx) p Fk 1429 1251 a(s) p Fm 1483 1211 a(\() p Fl( ) t(\036) p Fm(\)) 27 b(=) p Fk 1874 1087 a(s) p Fh 1821 1117 a(X) p Fk 1824 1329 a(j) p Fi 4 w(=0) p Fh 1982 1071 a(\022) p Fl 2071 1151 a(s) 2071 1271 y(j) p Fh 2135 1071 a(\023) p Fl 2225 1211 a( ) p Fi 2294 1170 a(\() p Fk(j) p Fi 4 w(\)) p Fl 2397 1211 a(\036) p Fi 2456 1170 a(\() p Fk(s) p Fe(\000) p Fk(j) p Fi 4 w(\)) p Fm 0 1519 a(Denote) 34 b(b) m(y) p Fl 33 w(D) p Fk 562 1534 a(j) p Fm 638 1519 a(the) p Fl 33 w(j) p Fj 28 w(\000) p Fl 23 w(th) p Fm 33 w(addendum) h(at) d(r.h.s.) 44 b(F) -8 b(or) 33 b(1) p Fj 28 w(\024) p Fl 28 w(j) p Fj 33 w(\024) p Fl 28 w(s) p Fj 22 w(\000) p Fm 23 w(1) g(one) g(has) h (\(b) m(y) f(lemma) e(5.5\)) p Fj 1151 1784 a(j) p Fl(D) p Fk 1261 1799 a(j) p Fm 1303 1784 a(\() p Fl(x) p Fm(\)) p Fj(j) c(\024) p Fh 1598 1643 a(\022) p Fl 1688 1724 a(s) 1688 1843 y(j) p Fh 1751 1643 a(\023) p Fm 1859 1716 a(2) p 1853 1761 61 4 v Fl 1853 1852 a(\031) p Fm 2030 1716 a(1) p 1938 1761 236 4 v Fl 1938 1852 a(N) p Fk 2029 1823 a(s) p Fe(\000) p Fi(1) p Fj 2202 1784 a(k) p Fl -1 w(\036) p Fj(k) p Fk 2361 1813 a(H) p Fd 2431 1794 a(s) p Fj 2491 1784 a(k) p Fl( ) p Fj 4 w(k) p Fk 2659 1813 a(H) p Fd 2729 1794 a(s) p Fl 2790 1784 a(;) p Fm 0 2013 a(moreo) m(v) m(er,) p Fj 235 2289 a(j) p Fl -1 w(D) p Fi 344 2304 a(0) p Fm 389 2289 a(\() p Fl(x) p Fm(\)) p Fj(j) g(\024) p Fh 684 2132 a(r) p 784 2132 85 4 v Fm 801 2221 a(2) p 796 2266 61 4 v Fl 796 2357 a(\031) p Fm 1013 2221 a(1) p 880 2266 317 4 v Fl 880 2359 a(N) p Fk 971 2330 a(s) p Fe(\000) p Fi(1) p Fk(=) p Fi(2) p Fj 1225 2289 a(k) p Fl( ) p Fj 4 w(k) p Fk 1393 2319 a(H) p Fd 1463 2299 a(s) p Fh 1523 2174 a(\014) 1523 2234 y(\014) 1523 2294 y(\014) p Fl 1557 2289 a(\036) p Fi 1616 2248 a(\() p Fk(s) p Fi(\)) p Fm 1721 2289 a(\() p Fl(x) p Fm(\)) p Fh 1856 2174 a(\014) 1856 2234 y(\014) 1856 2294 y(\014) p Fl 1938 2289 a(;) p Fj 116 w(j) p Fl(D) p Fk 2192 2304 a(s) p Fm 2235 2289 a(\() p Fl(x) p Fm(\)) p Fj(j) g(\024) p Fh 2530 2132 a(r) p 2629 2132 85 4 v Fm 2647 2221 a(2) p 2641 2266 61 4 v Fl 2641 2357 a(\031) p Fm 2859 2221 a(1) p 2726 2266 317 4 v Fl 2726 2359 a(N) p Fk 2817 2330 a(s) p Fe(\000) p Fi(1) p Fk(=) p Fi(2) p Fj 3070 2289 a(k) p Fl(\036) p Fj(k) p Fk 3229 2319 a(H) p Fd 3299 2299 a(s) p Fh 3360 2174 a(\014) 3360 2234 y(\014) 3360 2294 y(\014) p Fl 3393 2289 a( ) p Fi 3462 2248 a(\() p Fk(s) p Fi(\)) p Fm 3566 2289 a(\() p Fl(x) p Fm(\)) p Fh 3701 2174 a(\014) 3701 2234 y(\014) 3701 2294 y(\014) p Fm 0 2528 a(from) 33 b(whic) m(h) p Fj 671 2785 a(k) p Fl(\036 ) p Fj 4 w(k) p Fk 899 2815 a(L) p Fb 954 2795 a(2) p Fm 1019 2785 a(+) p Fk 1172 2660 a(s) p Fh 1119 2690 a(X) p Fk 1122 2902 a(j) p Fi 4 w(=0) p Fj 1279 2785 a(k) p Fl(D) p Fk 1411 2800 a(j) p Fj 1454 2785 a(k) p Fk 1504 2818 a(L) p Fb 1559 2798 a(2) p Fj 1630 2785 a(\024) p Fm 28 w(2) p Fh 1785 2628 a(r) p 1884 2628 85 4 v Fm 1902 2717 a(2) p 1896 2762 61 4 v Fl 1896 2853 a(\031) p Fm 2073 2717 a(1) p 1981 2762 236 4 v Fl 1981 2853 a(N) p Fk 2072 2824 a(s) p Fe(\000) p Fi(1) p Fj 2245 2785 a(k) p Fl -1 w(\036) p Fj(k) p Fk 2404 2815 a(H) p Fd 2474 2795 a(s) p Fj 2534 2785 a(k) p Fl( ) p Fj 4 w(k) p Fk 2702 2815 a(H) p Fd 2772 2795 a(s) p Fk 2886 2660 a(s) p Fh 2833 2690 a(X) p Fk 2836 2902 a(j) p Fi 4 w(=0) p Fh 2993 2644 a(\022) p Fl 3083 2725 a(s) 3083 2844 y(j) p Fh 3147 2644 a(\023) p Fl 3270 2785 a(;) p Fm 0 3162 a(using) p Fh 261 3087 a(P) p Fk 366 3112 a(s) 366 3192 y(j) p Fi 4 w(=0) p Fh 525 3021 a(\022) p Fl 615 3102 a(s) 615 3222 y(j) p Fh 679 3021 a(\023) p Fm 780 3162 a(=) 28 b(2) p Fk 935 3126 a(s) p Fm 1010 3162 a(the) 34 b(thesis) g(follo) m(ws.) p 3891 3088 78 4 v 3891 3158 4 70 v 3965 3158 V 3891 3162 78 4 v Fn 0 3415 a(Lemma) 160 b(5.7.) p Fg 35 w(Let) p Fm 924 3369 a(\024) 922 3389 y(\024) p Fl 920 3415 a(h) p Fg 36 w(b) s(e) 35 b(of) g(degree) g(at) g (least) f(three) h(in) p Fl 35 w(P) s(;) 17 b(Q) p Fg(,) 34 b(namely) f(with) p Fl 34 w(m) p Fi 3223 3430 a(3) p Fm 3292 3415 a(+) p Fl 23 w(m) p Fi 3479 3430 a(4) p Fj 3555 3415 a(\025) p Fm 30 w(3) p Fg(,) i(then) 0 3535 y(one) f(has) p Fh 1384 3568 a(\015) 1384 3627 y(\015) 1384 3687 y(\015) p Fl 1440 3682 a(X) p Fi 1526 3692 a(\024) 1524 3705 y(\024) p Fk 1523 3724 a(h) p Fh 1574 3568 a(\015) 1574 3627 y(\015) 1574 3687 y(\015) p Fk 1629 3592 a(R) 1629 3752 y(s) p Fj 1722 3682 a(\024) p Fl 28 w(n) p Fm 1960 3615 a(\006) p Fk 2032 3630 a(s) p 1899 3659 236 4 v Fl 1899 3751 a(N) p Fk 1990 3722 a(s) p Fe(\000) p Fi(1) p Fm 2147 3682 a(\() p Fl(\032) p Fk 2238 3697 a(s) p Fl 2279 3682 a(R) p Fm 1 w(\)) p Fk 2395 3641 a(n) p Fe(\000) p Fi(1) p Fm 3714 3682 a(\(5) p Fl(:) p Fm(10\)) p Fg 0 3884 a(where) p Fl 34 w(n) p Fm 28 w(=) p Fl 28 w(m) p Fi 568 3899 a(1) p Fm 635 3884 a(+) p Fl 23 w(m) p Fi 822 3899 a(2) p Fm 889 3884 a(+) p Fl 23 w(m) p Fi 1076 3899 a(3) p Fm 1143 3884 a(+) p Fl 23 w(m) p Fi 1330 3899 a(4) p Fg 1375 3884 a(.) p Fn 0 4043 a(Pro) s(of.) p Fm 33 w(F) -8 b(or) 32 b(simplicit) m(y) e(w) m(e) i(will) f(consider) i (only) e(the) i(case) p Fl 32 w(m) p Fi 2376 4058 a(3) p Fj 2449 4043 a(\025) p Fm 28 w(3,) f(the) g(other) h(cases) g(can) f(b) s(e) g(treated) 0 4163 y(exactly) g(in) h(the) g(same) g(w) m(a) m(y) -8 b(.) 43 b(W) -8 b(e) 34 b(pro) s(ceed) g(as) f(in) g(the) g(pro) s(of) g (of) h(lemma) c(5.2,) i(but) i(consider) g(\014rst) f(the) p Fl 34 w(P) p Fm 0 4283 a(comp) s(onen) m(t) g(of) h(the) f(v) m(ector) g (\014eld.) 45 b(Its) 33 b(norm) g(is) g(estimated) f(b) m(y) 1253 4508 y(\(2) p Fl(\031) p Fm 4 w(\)) p Fd 1454 4435 a(n) p Fc(\000) p Fb(1) p 1453 4452 132 4 v 1502 4491 a(2) p Fh 1618 4394 a(\015) 1618 4454 y(\015) 1618 4513 y(\015) p Fl 1673 4508 a(X) p Fk 1764 4467 a(P) p Fi 1759 4527 a(\024) 1757 4541 y(\024) p Fk 1756 4559 a(h) p Fh 1830 4394 a(\015) 1830 4454 y(\015) 1830 4513 y(\015) p Fk 1885 4578 a(H) p Fd 1955 4558 a(s) p Fm 2027 4508 a(=) p Fl 28 w(m) p Fi 2219 4523 a(3) p Fh 2281 4424 a(\015) 2281 4483 y(\015) p Fl 2336 4508 a(F) f(q) p Fk 2479 4467 a(m) p Fb 2550 4477 a(1) p Fl 2593 4508 a(p) p Fk 2643 4467 a(m) p Fb 2714 4477 a(2) p Fl 2758 4508 a(Q) p Fk 2837 4467 a(m) p Fb 2908 4477 a(3) p Fe 2946 4467 a(\000) p Fi(1) p Fl 3053 4508 a(P) p Fk 3131 4467 a(m) p Fb 3202 4477 a(4) p Fh 3246 4424 a(\015) 3246 4483 y(\015) p Fk 3301 4548 a(H) p Fd 3371 4528 a(s) p Fj 1285 4782 a(\024) p Fl 28 w(m) p Fi 1477 4797 a(3) p Fh 1539 4698 a(\015) 1539 4758 y(\015) p Fl 1594 4782 a(q) p Fk 1642 4741 a(m) p Fb 1713 4751 a(1) p Fl 1757 4782 a(p) p Fk 1807 4741 a(m) p Fb 1878 4751 a(2) p Fl 1922 4782 a(Q) p Fk 2001 4741 a(m) p Fb 2072 4751 a(3) p Fe 2110 4741 a(\000) p Fi(3) p Fl 2217 4782 a(P) p Fk 2295 4741 a(m) p Fb 2366 4751 a(4) p Fl 2409 4782 a(F) p Fh 2487 4698 a(\015) 2487 4758 y(\015) p Fk 2543 4822 a(H) p Fd 2613 4802 a(s) p Fh 2673 4626 a(r) p 2773 4626 85 4 v Fm 2790 4715 a(2) p 2785 4760 61 4 v Fl 2785 4851 a(\031) p Fm 2857 4782 a(2) p Fk 2907 4741 a(s) p Fi(+1) p Fh 3067 4698 a(\015) 3067 4758 y(\015) p Fl 3122 4782 a(Q) p Fi 3201 4741 a(2) p Fh 3246 4698 a(\015) 3246 4758 y(\015) p Fk 3301 4822 a(H) p Fd 3371 4802 a(s) p Fj 554 5096 a(\024) p Fl 28 w(m) p Fi 746 5111 a(3) p Fh 807 5011 a(\015) 807 5071 y(\015) p Fl 863 5096 a(F) 14 b(q) p Fk 989 5054 a(m) p Fb 1060 5064 a(1) p Fl 1103 5096 a(p) p Fk 1153 5054 a(m) p Fb 1224 5064 a(2) p Fl 1268 5096 a(Q) p Fk 1347 5054 a(m) p Fb 1418 5064 a(3) p Fe 1456 5054 a(\000) p Fi(3) p Fl 1563 5096 a(P) p Fk 1641 5054 a(m) p Fb 1712 5064 a(4) p Fh 1756 5011 a(\015) 1756 5071 y(\015) p Fk 1811 5135 a(H) p Fd 1881 5115 a(s) p Fh 1942 4925 a( ) 2020 4939 y(r) p 2120 4939 85 4 v Fm 2137 5028 a(2) p 2132 5073 61 4 v Fl 2132 5164 a(\031) p Fm 2204 5096 a(2) p Fk 2254 5054 a(s) p Fi(+1) p Fh 2398 4925 a(!) p Fi 2476 4946 a(2) p Fm 2643 5028 a(1) p 2550 5073 236 4 v Fl 2550 5164 a(N) p Fk 2641 5135 a(s) p Fe(\000) p Fi(1) p Fj 2814 5096 a(k) p Fl(Q) p Fj(k) p Fk 2992 5125 a(H) p Fd 3062 5105 a(s) p Fj 3123 5096 a(k) p Fl -1 w(Q) p Fj(k) p Fk 3301 5125 a(H) p Fd 3371 5105 a(s) p Fj 1942 5455 a(\024) p Fl 28 w(m) p Fi 2134 5470 a(3) p Fm 2179 5455 a(\006) p Fk 2251 5470 a(s) p Fm 2399 5387 a(1) p 2306 5432 V Fl 2306 5523 a(N) p Fk 2397 5494 a(s) p Fe(\000) p Fi(1) p Fh 2570 5284 a( ) 2649 5298 y(r) p 2748 5298 85 4 v Fm 2766 5387 a(2) p 2760 5432 61 4 v Fl 2760 5523 a(\031) p Fm 2833 5455 a(2) p Fk 2883 5413 a(s) p Fi(+1) p Fl 3026 5455 a(R) p Fh 3103 5284 a(!) p Fk 3181 5305 a(n) p Fe(\000) p Fi(1) p Fl 3387 5455 a(:) p Fm 1807 5778 a(20.03.02) p 90 rotate dyy eop %%Page: 15 15 15 14 bop Fg 1222 60 a(Birkho\013) 33 b(normal) f(form) g(for) i(PDE's) p Fm 1222 w(15) 0 299 y(Considering) g(the) f(other) h(comp) s(onen) m (ts) f(and) h(summing) e(up) i(one) f(has) h(the) g(thesis.) p 3891 225 78 4 v 3891 295 4 70 v 3965 295 V 3891 299 78 4 v 199 448 a(Finally) e(w) m(e) i(ha) m(v) m(e) p Fn 0 608 a(Prop) s(osition) 144 b(5.8.) p Fg 32 w(Let) p Fl 32 w(h) p Fg 31 w(b) s(e) 32 b(a) f(p) s(olynomial) e(of) j(degree) g (less) g(or) f(equal) g(than) p Fl 32 w(r) p Fg 34 w(and) h(of) f (class) p Fj 32 w(M) p Fk 3876 560 a(\026) 3876 637 y(R) p Fg 3941 608 a(,) 0 727 y(assume) e(also) f(that) g(it) g(is) h(at) f (least) h(cubic) g(in) p Fm 29 w(\() p Fl(P) s(;) 17 b(Q) p Fm(\)) p Fg 27 w(namely) 27 b(that) i(it) f(is) g(the) h(sum) g (of) g(p) s(olynomials) d(of) j(the) 0 847 y(form) h(\(5.8\)) g(eac) m (h) i(one) g(ful\014lling) p Fl 31 w(m) p Fi 1347 862 a(3) p Fm 1410 847 a(+) p Fl 18 w(m) p Fi 1592 862 a(4) p Fj 1665 847 a(\025) p Fm 28 w(3) p Fg(,) f(then,) h(for) g(an) m(y) f (p) s(ositiv) m(e) p Fl 30 w(R) p Fi 2912 811 a(\() p Fk(s) p Fi(\)) p Fg 3048 847 a(suc) m(h) h(that) p Fl 31 w(\032) p Fk 3536 862 a(s) p Fl 3578 847 a(R) p Fi 3655 811 a(\() p Fk(s) p Fi(\)) p Fj 3787 847 a(\024) p Fl 28 w(R) p Fg 0 966 a(and) i(an) m(y) p Fl 33 w(\016) p Fi 430 930 a(\() p Fk(s) p Fi(\)) p Fl 562 966 a(<) 29 b(R) p Fi 745 930 a(\() p Fk(s) p Fi(\)) p Fg 849 966 a(,) k(one) g(has) p Fj 1099 1245 a(k) p Fl(X) p Fk 1232 1260 a(h) p Fj 1283 1245 a(k) p Fk 1333 1195 a(R) p Fb 1393 1165 a(\() p Fd(s) p Fb(\)) p Fe 1486 1195 a(\000) p Fk(\016) p Fb 1587 1165 a(\() p Fd(s) p Fb(\)) p Fk 1333 1275 a(s) p Fj 1712 1245 a(\024) p Fm 1855 1178 a(2) p Fk 1905 1141 a(r) p Fi 2 w(+1) p 1829 1222 247 4 v Fl 1829 1315 a(\032) p Fk 1881 1330 a(s) p Fl 1923 1315 a(\016) p Fi 1971 1286 a(\() p Fk(s) p Fi(\)) p Fm 2160 1178 a(\006) p Fk 2232 1193 a(s) p 2100 1222 236 4 v Fl 2100 1313 a(N) p Fk 2191 1284 a(s) p Fe(\000) p Fi(1) p Fj 2364 1245 a(hj) p Fl -1 w(h) p Fj(ji) p Fk 2554 1195 a(\026) 2554 1282 y(\032) p Fd 2595 1292 a(s) p Fk 2634 1282 a(R) p Fb 2694 1262 a(\() p Fd(s) p Fb(\)) p Fl 2841 1245 a(:) p Fm 845 w(\(5) p Fl(:) p Fm(11\)) p Fn 0 1684 a(Pro) s(of.) p Fm 28 w(First) 26 b(consider) h(a) g(p) s (olynomial) p Fl 24 w(h) p Fk 1596 1648 a(n) p Fb 1645 1658 a(1) p Fk 1684 1648 a(;n) p Fb 1757 1658 a(2) p Fm 1827 1684 a(of) g(degree) p Fl 28 w(n) p Fi 2298 1699 a(1) p Fl 2342 1684 a(;) 17 b(n) p Fi 2447 1699 a(2) p Fm 2517 1684 a(in) p Fl 27 w(\030) p Fm 30 w(and) p Fl 27 w(\021) p Fm 31 w(resp) s(ectiv) m(ely) -8 b(.) 41 b(Expand) 27 b(it) 0 1804 y(in) d(T) -8 b(a) m(ylor) 24 b(series) h(in) p Fl 25 w(P) s(;) 17 b(Q) p Fm(.) 39 b(Suc) m(h) 26 b(a) f(T) -8 b(a) m(ylor) 23 b(series) i(con) m(tains) g (at) f(most) g(2) p Fk 2686 1768 a(n) p Fb 2735 1778 a(1) p Fi 2773 1768 a(+) p Fk(n) p Fb 2883 1778 a(2) p Fm 2952 1804 a(hogeneous) i(p) s(olynomials.) 0 1923 y(Denote) 31 b(b) m(y) p Fl 30 w(h) p Fk 531 1887 a(n) p Fb 580 1897 a(1) p Fk 620 1887 a(;n) p Fb 693 1897 a(2) p Fk 732 1887 a(;m) p Fb 827 1897 a(3) p Fk 865 1887 a(;m) p Fb 960 1897 a(4) p Fm 1033 1923 a(the) g(p) s(olynomial) d(of) i (degree) p Fl 31 w(m) p Fi 2218 1938 a(3) p Fl 2264 1923 a(;) 17 b(m) p Fi 2396 1938 a(4) p Fm 2470 1923 a(in) p Fl 30 w(Q;) g(P) p Fm 44 w(resp) s(ectiv) m(ely) -8 b(.) 42 b(Then) 31 b(one) g(has) p Fj 1277 2144 a(hj) p Fl -1 w(h) p Fk 1400 2103 a(n) p Fb 1449 2113 a(1) p Fk 1489 2103 a(;n) p Fb 1562 2113 a(2) p Fk 1601 2103 a(;m) p Fb 1696 2113 a(3) p Fk 1734 2103 a(;m) p Fb 1829 2113 a(4) p Fj 1872 2144 a(ji) p Fk 1939 2094 a(\026) 1939 2174 y(R) p Fj 2048 2144 a(\024) d(hj) p Fl(h) p Fk 2277 2103 a(n) p Fb 2326 2113 a(1) p Fk 2365 2103 a(;n) p Fb 2438 2113 a(2) p Fj 2482 2144 a(ji) p Fk 2549 2094 a(\026) 2549 2174 y(R) p Fl 2664 2144 a(;) p Fm 0 2364 a(and) 34 b(th) m(us,) g(b) m(y) f(lemma) e(5.7,) h(w) m(orking) h(as) g (in) g(the) h(pro) s(of) f(of) h(prop) s(osition) f(5.3) f(one) i(has) p Fj 644 2630 a(k) p Fl(X) p Fk 777 2645 a(h) p Fd 824 2622 a(n) p Fb 868 2637 a(1) p Fd 906 2622 a(;n) p Fb 973 2637 a(2) p Fd 1012 2622 a(;m) p Fb 1096 2637 a(3) p Fd 1134 2622 a(;m) p Fb 1218 2637 a(4) p Fj 1267 2630 a(k) p Fk 1317 2579 a(R) p Fb 1377 2549 a(\() p Fd(s) p Fb(\)) p Fe 1470 2579 a(\000) p Fk(\016) p Fb 1571 2549 a(\() p Fd(s) p Fb(\)) p Fk 1317 2659 a(s) p Fj 1696 2630 a(\024) p Fm 1911 2562 a(1) p 1813 2607 247 4 v Fl 1813 2700 a(\032) p Fk 1865 2715 a(s) p Fl 1907 2700 a(\016) p Fi 1955 2671 a(\() p Fk(s) p Fi(\)) p Fm 2144 2562 a(\006) p Fk 2216 2577 a(s) p 2084 2607 236 4 v Fl 2084 2698 a(N) p Fk 2175 2669 a(s) p Fe(\000) p Fi(1) p Fj 2348 2630 a(hj) p Fl -1 w(h) p Fk 2471 2588 a(n) p Fb 2520 2598 a(1) p Fk 2560 2588 a(;n) p Fb 2633 2598 a(2) p Fk 2672 2588 a(;m) p Fb 2767 2598 a(3) p Fk 2805 2588 a(;m) p Fb 2900 2598 a(4) p Fj 2943 2630 a(ji) p Fk 3009 2579 a(\026) 3009 2666 y(\032) p Fd 3050 2676 a(s) p Fk 3089 2666 a(R) p Fb 3149 2646 a(\() p Fd(s) p Fb(\)) p Fl 3297 2630 a(:) p Fm 389 w(\(5) p Fl(:) p Fm(12\)) 0 2911 y(Summing) 28 b(o) m(v) m(er) i(all) f(the) h (di\013eren) m(t) g(p) s(olynomials) e(one) i(gets) f(at) h(most) e (the) i(factor) g(2) p Fk 3131 2875 a(r) p Fi 2 w(+1) p Fm 3305 2911 a(and) g(the) g(desired) 0 3030 y(result.) p 3891 2957 78 4 v 3891 3026 4 70 v 3965 3026 V 3891 3030 78 4 v Fg 0 3648 a(5.2) j(Lie) g(transform) g(estimates) p Fm 199 3917 a(In) 46 b(this) f(sub{section) h(w) m(e) g(\014rst) g (recall) f(some) f(facts) i(ab) s(out) f(Lie) g(the) h(transform,) h (and) f(then) g(w) m(e) 0 4037 y(estimate) 32 b(the) p Fj 33 w(M) p Fk 690 3989 a(\026) 690 4067 y(R) p Fe(\000) p Fk(d) p Fm 891 4037 a(norm) g(of) h(the) g(Lie) g(transform) f(of) h (an) p Fj 33 w(M) p Fk 2430 3989 a(\026) 2430 4066 y(R) p Fm 2528 4037 a(function.) 45 b(This) 33 b(is) f(giv) m(en) h(b) m(y) g (lemma) 0 4156 y(5.12) f(b) s(elo) m(w,) h(whic) m(h) h(constitutes) g (the) g(main) e(result) h(of) h(this) f(sub{section.) 199 4306 y(Consider) k(a) f(function) p Fl 38 w(\037) p Fm 36 w(and) h(the) g(corresp) s(onding) g(Hamilton) d(equations) 3123 4279 y(_) p Fl 3103 4306 a(\020) p Fm 40 w(=) p Fl 33 w(X) p Fk 3380 4321 a(\037) p Fm 3434 4306 a(\() p Fl(\020) p Fm 7 w(\),) j(the) f(cor-) 0 4425 y(resp) s(onding) k(\015o) m(w) p Fl 38 w(T) p Fk 797 4389 a(t) p Fm 871 4425 a(is) f(a) f(canonical) g (transformation,) h(and) p Fj 39 w(T) p Fm 62 w(:=) p Fl 37 w(T) p Fi 2750 4389 a(1) p Fj 2831 4425 a(\021) p Fl 37 w(T) p Fk 3017 4389 a(t) p Fh 3052 4340 a(\014) 3052 4400 y(\014) p Fk 3085 4465 a(t) p Fi(=1) p Fm 3260 4425 a(is) f(called) h(the) p Ff 39 w(Lie) 0 4545 y(tr) -5 b(ansform) p Fm 36 w(generated) 38 b(b) m(y) p Fl 36 w(\037) p Fm(.) 55 b(Giv) m(en) 36 b(an) h(analytic) e(function) p Fl 38 w(h) p Fm 37 w(one) i(can) g(consider) g(the) g(transformed) 0 4664 y(function) p Fl 34 w(h) p Fj 23 w(\016) 22 b(T) p Fm 25 w(.) 44 b(W) -8 b(e) 34 b(recall) f(that) g(one) g(has) p Fl 1648 4939 a(h) p Fj 22 w(\016) 22 b(T) p Fm 53 w(=) p Fe 2044 4815 a(1) p Fh 2012 4845 a(X) p Fk 2020 5059 a(l) p Fi(=0) p Fl 2172 4939 a(h) p Fk 2229 4954 a(l) p Fl 2293 4939 a(;) p Fm 1393 w(\(5) p Fl(:) p Fm(13\)) 0 5249 y(where) 34 b(the) p Fl 34 w(h) p Fk 517 5264 a(l) p Fm 548 5249 a('s) f(are) g(de\014ned) i(b) m(y) p Fl 1161 5511 a(h) p Fk 1218 5526 a(l) p Fm 1277 5511 a(=) 1394 5443 y(1) p 1394 5488 50 4 v Fl 1403 5579 a(l) p Fj 1473 5511 a(f) p Fl -1 w(\037;) 17 b(h) p Fk 1686 5526 a(l) p Fe(\000) p Fi(1) p Fj 1819 5511 a(g) p Fl 50 w(;) 115 b(l) p Fj 30 w(\025) p Fm 28 w(1) 33 b(;) p Fl 116 w(h) p Fi 2511 5526 a(0) p Fm 2584 5511 a(=) p Fl 28 w(h) g(;) p Fm 907 w(\(5) p Fl(:) p Fm(14\)) 1807 5778 y(20.03.02) p 90 rotate dyy eop %%Page: 16 16 16 15 bop Fm 0 60 a(16) p Fg 1660 w(D.) 33 b(Bam) m(busi) p Fm 0 299 a(as) g(is) g(easily) g(seen) h(b) m(y) g(iterating) e(the) i (equalit) m(y) p Fl 1472 485 a(d) p 1454 530 88 4 v 1454 621 a(dt) p Fm 1554 553 a([) p Fl(h) p Fj 22 w(\016) p Fl 22 w(T) p Fk 1805 512 a(t) p Fm 1840 553 a(]) 27 b(=) p Fj 29 w(f) p Fl -1 w(\037;) 17 b(h) p Fj(g) 22 b(\016) p Fl 22 w(T) p Fk 2430 512 a(t) p Fl 2499 553 a(;) p Fm 0 798 a(and) 33 b(considering) h(the) g(T) -8 b(a) m(ylor) 31 b(expansion) j(in) p Fl 33 w(t) p Fm 33 w(of) p Fl 33 w(h) p Fj 22 w(\016) p Fl 22 w(T) p Fk 2182 762 a(t) p Fm 2217 798 a(.) 44 b(In) 33 b(order) g(to) g(estimate) f(the) h(norm) f (of) p Fl 33 w(h) p Fj 22 w(\016) 22 b(T) p Fm 0 917 a(w) m(e) 36 b(need) h(\014rst) f(to) f(estimate) g(the) h(P) m(oisson) g(brac) m(k) m(et) g(of) g(t) m(w) m(o) p Fj 35 w(M) p Fk 2434 870 a(\026) 2434 947 y(R) p Fm 2535 917 a(functions.) 52 b(W) -8 b(e) 36 b(will) f(do) h(it) f(in) g(a) h(few) 0 1037 y(steps.) p Fn 0 1196 a(Lemma) 190 b(5.9.) p Fg 42 w(Let) p Fl 41 w(h) p Fg 42 w(b) s(e) 42 b(a) f(homogeneous) g(p) s (olynomial) e(of) j(degree) p Fl 42 w(m) p Fg 42 w(and) p Fl 42 w(g) p Fg 44 w(a) f(homogeneous) 0 1316 y(p) s(olynomial) 26 b(of) j(degree) p Fl 29 w(n) p Fg(;) h(assume) e(that) h(b) s(oth) p Fl 28 w(h) p Fg(,) p Fl 30 w(g) p Fg 31 w(are) g(homogenea) f(in) p Fl 28 w(\030) p Fg 33 w(and) p Fl 29 w(\021) p Fg 33 w(indep) s(enden) m(tly) -8 b(,) 30 b(and) 0 1436 y(that) j(they) g(ha) m(v) m(e) h(\014nite) p Fl 33 w(\026) p Fg(-mo) s(dulus,) f(then) h (one) g(has) p Fj 1462 1643 a(jf) p Fl -1 w(h;) 17 b(g) p Fj 4 w(g) o(j) p Fk 1770 1673 a(\026) p Fj 1851 1643 a(\024) p Fl 28 w(mn) p Fj(j) p Fl(h) p Fj(j) p Fk 2216 1658 a(\026) p Fj 2286 1643 a(j) p Fl(g) p Fj 4 w(j) p Fk 2394 1658 a(\026) p Fl 2479 1643 a(;) p Fm 1207 w(\(5) p Fl(:) p Fm(15\)) p Fn 0 2010 a(Pro) s(of.) p Fm 34 w(W) -8 b(rite) p Fl 33 w(h) p Fm 34 w(and) p Fl 34 w(g) p Fm 36 w(explicitely) 32 b(as) h(follo) m(ws) p Fl 658 2232 a(h) p Fm(\() p Fl(\030) 5 b(;) 17 b(\021) p Fm 4 w(\)) 26 b(=) p Fh 1326 2137 a(X) p Fk 1071 2349 a(i) p Fb 1099 2359 a(1) p Fk 1138 2349 a(;:::;i) p Fd 1286 2359 a(n) p Fb 1330 2374 a(1) p Fk 1372 2349 a(;j) p Fb 1429 2359 a(1) p Fk 1468 2349 a(;:::;j) p Fd 1621 2359 a(m) p Fb 1682 2374 a(1) p Fl 1741 2232 a(h) p Fk 1798 2247 a(i) p Fb 1826 2257 a(1) p Fk 1865 2247 a(;:::;i) p Fd 2013 2257 a(n) p Fb 2057 2272 a(1) p Fk 2100 2247 a(;j) p Fb 2157 2257 a(1) p Fk 2195 2247 a(;:::;j) p Fd 2348 2257 a(m) p Fb 2409 2272 a(1) p Fl 2456 2232 a(\030) p Fk 2500 2247 a(i) p Fb 2528 2257 a(1) p Fl 2572 2232 a(:::\030) p Fk 2700 2247 a(i) p Fd 2728 2257 a(n) p Fb 2772 2272 a(1) p Fl 2819 2232 a(\021) p Fk 2868 2247 a(j) p Fb 2901 2257 a(1) p Fl 2946 2232 a(:::\021) p Fk 3079 2247 a(j) p Fd 3112 2257 a(m) p Fb 3173 2272 a(1) p Fl 664 2528 a(g) p Fm 4 w(\() p Fl(\030) 5 b(;) 17 b(\021) p Fm 4 w(\)) 25 b(=) p Fh 1332 2434 a(X) p Fk 1071 2648 a(k) p Fb 1113 2658 a(1) p Fk 1152 2648 a(;:::;k) p Fd 1314 2658 a(n) p Fb 1358 2673 a(2) p Fk 1401 2648 a(;l) p Fb 1450 2658 a(1) p Fk 1488 2648 a(;:::;l) p Fd 1633 2658 a(m) p Fb 1694 2673 a(2) p Fl 1754 2528 a(g) p Fk 1802 2543 a(k) p Fb 1844 2553 a(1) p Fk 1882 2543 a(;:::;k) p Fd 2044 2553 a(n) p Fb 2088 2568 a(2) p Fk 2131 2543 a(;l) p Fb 2180 2553 a(1) p Fk 2219 2543 a(;:::;l) p Fd 2364 2553 a(m) p Fb 2425 2568 a(2) p Fl 2472 2528 a(\030) p Fk 2516 2543 a(k) p Fb 2558 2553 a(1) p Fl 2602 2528 a(:::\030) p Fk 2730 2543 a(k) p Fd 2772 2553 a(n) p Fb 2816 2568 a(2) p Fl 2864 2528 a(\021) p Fk 2913 2543 a(l) p Fb 2938 2553 a(1) p Fl 2982 2528 a(:::\021) p Fk 3115 2543 a(l) p Fd 3140 2553 a(m) p Fb 3201 2568 a(2) p Fl 3283 2528 a(;) p Fm 0 2850 a(One) 34 b(has) p Fl 1010 3005 a(') p Fm 28 w(:=) p Fj 28 w(f) p Fl -1 w(h;) 17 b(g) p Fj 4 w(g) p Fm 27 w(=) p Fl 28 w(i) p Fh 1672 2910 a(X) p Fk 1729 3119 a(t) p Fh 1832 2864 a(\022) p Fl 1928 2937 a(@) 6 b(h) p 1918 2982 138 4 v 1918 3073 a(@) g(\030) p Fk 2021 3088 a(t) p Fl 2126 2937 a(@) g(g) p 2079 2982 206 4 v 2079 3073 a(@) g(\021) p Fe 2187 3088 a(\000) p Fk(t) p Fj 2318 3005 a(\000) p Fl 2443 2937 a(@) g(g) p 2430 2982 138 4 v 2430 3073 a(@) g(\030) p Fk 2533 3088 a(t) p Fl 2635 2937 a(@) g(h) p 2591 2982 206 4 v 2591 3073 a(@) g(\021) p Fe 2699 3088 a(\000) p Fk(t) p Fh 2808 2864 a(\023) p Fl 2931 3005 a(:) p Fm 755 w(\(5) p Fl(:) p Fm(16\)) 0 3264 y(W) -8 b(e) 34 b(\014rst) f(consider) h(the) g(case) g (where) p Fl 34 w(h) p Fm 34 w(dep) s(ends) h(on) p Fl 33 w(\030) p Fm 38 w(only) d(and) p Fl 34 w(g) p Fm 36 w(dep) s(ends) j(on) p Fl 34 w(\021) p Fm 4 w(:) 44 b(W) -8 b(e) 33 b(ha) m(v) m(e) p Fl 767 3486 a(') p Fm(\() p Fl(\030) 5 b(;) 17 b(\021) p Fm 4 w(\)) 26 b(=) p Fh 1404 3391 a(X) p Fk 1188 3603 a(i) p Fb 1216 3613 a(2) p Fk 1255 3603 a(;:::;i) p Fd 1403 3613 a(n) p Fk 1451 3603 a(;j) p Fb 1508 3613 a(2) p Fk 1546 3603 a(;:::;j) p Fd 1699 3613 a(m) p Fl 1780 3486 a(') p Fk 1845 3501 a(i) p Fb 1873 3511 a(2) p Fk 1912 3501 a(;:::;i) p Fd 2060 3511 a(n) p Fk 2108 3501 a(;j) p Fb 2165 3511 a(2) p Fk 2203 3501 a(;:::;j) p Fd 2356 3511 a(m) p Fl 2426 3486 a(\030) p Fk 2470 3501 a(i) p Fb 2498 3511 a(2) p Fl 2541 3486 a(;) 17 b(:::;) g(\030) p Fk 2759 3501 a(i) p Fd 2787 3511 a(n) p Fl 2838 3486 a(\021) p Fk 2887 3501 a(j) p Fb 2920 3511 a(2) p Fl 2965 3486 a(:::\021) p Fk 3098 3501 a(j) p Fd 3131 3511 a(m) p Fm 0 3794 a(with) p Fl 943 3913 a(') p Fk 1008 3928 a(i) p Fb 1036 3938 a(2) p Fk 1075 3928 a(;:::;i) p Fd 1223 3938 a(n) p Fk 1270 3928 a(;j) p Fb 1327 3938 a(2) p Fk 1366 3928 a(;:::;j) p Fd 1519 3938 a(m) p Fm 1616 3913 a(=) p Fl 28 w(mn) p Fh 1885 3819 a(X) p Fk 1921 4031 a(j) p Fb 1954 4041 a(1) p Fl 2045 3913 a(h) p Fe 2102 3928 a(\000) p Fk(j) p Fb 2197 3938 a(1) p Fk 2237 3928 a(;i) p Fb 2289 3938 a(2) p Fk 2327 3928 a(;:::;i) p Fd 2475 3938 a(n) p Fl 2528 3913 a(g) p Fk 2576 3928 a(j) p Fb 2609 3938 a(1) p Fk 2647 3928 a(;j) p Fb 2704 3938 a(2) p Fk 2743 3928 a(;:::;j) p Fd 2896 3938 a(m) p Fl 2998 3913 a(:) p Fm 0 4189 a(In) 34 b(order) f(to) g(estimate) f(the) p Fl 34 w(\026) p Fm({mo) s(dulus) g(of) p Fl 34 w(') p Fm 33 w(w) m(e) i(write) f(it) f(explicitely) g(as) i(follo) m(ws:) 297 4443 y(1) p 248 4488 148 4 v Fl 248 4579 a(nm) p Fj 424 4511 a(j) p Fl -1 w(') p Fj(j) p Fk 544 4541 a(\026) p Fm 625 4511 a(=) p Fh 731 4416 a(X) p Fk 790 4631 a(l) p Fm 1148 4511 a(sup) p Fk 891 4593 a(i) p Fb 919 4603 a(2) p Fk 958 4593 a(;:::;i) p Fd 1106 4603 a(n) p Fk 1153 4593 a(;j) p Fb 1210 4603 a(2) p Fk 1249 4593 a(;:::;j) p Fd 1402 4603 a(m) p Fc(\000) p Fb(1) p Fh 1571 4306 a(\014) 1571 4366 y(\014) 1571 4426 y(\014) 1571 4486 y(\014) 1571 4546 y(\014) 1571 4605 y(\014) 1604 4416 y(X) p Fk 1640 4628 a(j) p Fb 1673 4638 a(1) p Fl 1765 4511 a(g) p Fk 1813 4552 a(j) p Fb 1846 4562 a(1) p Fk 1884 4552 a(;:::;j) p Fd 2037 4562 a(m) p Fc(\000) p Fb(1) p Fk 2189 4552 a(;l) p Fe(\000) p Fh 2300 4485 a(P) p Fd 2406 4510 a(n) 2406 4590 y(a) p Fb(=2) p Fk 2548 4552 a(i) p Fd 2576 4562 a(a) p Fe 2620 4552 a(\000) p Fh 2682 4485 a(P) p Fd 2787 4510 a(m) p Fc(\000) p Fb(1) p Fd 2787 4590 a(b) p Fb(=2) p Fk 2955 4552 a(j) p Fd 2988 4563 a(b) p Fl 3029 4511 a(h) p Fe 3086 4526 a(\000) p Fk(j) p Fb 3181 4536 a(1) p Fk 3221 4526 a(;i) p Fb 3273 4536 a(2) p Fk 3312 4526 a(;:::;i) p Fd 3460 4536 a(n) p Fh 3512 4306 a(\014) 3512 4366 y(\014) 3512 4426 y(\014) 3512 4486 y(\014) 3512 4546 y(\014) 3512 4605 y(\014) p Fm 3562 4511 a(e) p Fk 3606 4470 a(\026) p Fe(j) p Fk(l) p Fe(j) p Fm 0 4843 a(Chainging) f(the) h(v) -6 b(ariable) 32 b(in) i(the) f(sum) g(from) p Fl 33 w(j) p Fi 1797 4858 a(1) p Fm 1875 4843 a(to) p Fl 32 w(l) p Fi 2026 4858 a(1) p Fm 2098 4843 a(:=) p Fj 28 w(\000) p Fl(j) p Fi 2349 4858 a(1) p Fm 2417 4843 a(+) p Fh 2516 4768 a(P) p Fk 2621 4793 a(n) 2621 4873 y(a) p Fi(=2) p Fl 2787 4843 a(i) p Fk 2821 4858 a(a) p Fm 2870 4843 a(,) f(this) i(b) s(ecomes) p Fh 122 5021 a(X) p Fk 181 5235 a(l) p Fm 539 5115 a(sup) p Fk 282 5197 a(i) p Fb 310 5207 a(2) p Fk 349 5197 a(;:::;i) p Fd 497 5207 a(n) p Fk 545 5197 a(;j) p Fb 602 5207 a(2) p Fk 640 5197 a(;:::;j) p Fd 793 5207 a(m) p 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3684 4971 a(\014) 3684 5031 y(\014) 3684 5090 y(\014) 3684 5150 y(\014) p Fm 3734 5115 a(e) p Fk 3778 5074 a(\026) p Fe(j) p Fk(l) p Fe(j) p Fj 64 5439 a(\024) p Fh 169 5345 a(X) p Fk 228 5559 a(l) p Fh 330 5345 a(X) p Fk 370 5559 a(l) p Fb 395 5569 a(1) p Fm 723 5439 a(sup) p Fk 490 5521 a(i) p Fb 518 5531 a(2) p Fk 557 5521 a(;::;i) p Fd 681 5531 a(n) p Fk 729 5521 a(;j) p Fb 786 5531 a(2) p Fk 825 5521 a(;::;j) p Fd 954 5531 a(m) p Fc(\000) p Fb(1) p Fh 1123 5295 a(\014) 1123 5355 y(\014) 1123 5414 y(\014) 1123 5474 y(\014) p Fl 1156 5439 a(g) p Fh 1204 5414 a(P) p Fd 1309 5438 a(n) 1309 5518 y(a) p Fb(=2) p Fk 1451 5481 a(i) p Fd 1479 5491 a(a) p Fe 1523 5481 a(\000) p Fk(l) p Fb 1610 5491 a(1) p Fk 1649 5481 a(;j) p Fb 1706 5491 a(2) p Fk 1745 5481 a(;:::;j) p Fd 1898 5491 a(m) p Fc(\000) p Fb(1) p Fk 2050 5481 a(;l) p Fe(\000) p Fh 2161 5414 a(P) p Fd 2266 5438 a(n) 2266 5518 y(a) p Fb(=2) p Fk 2409 5481 a(i) p Fd 2437 5491 a(a) p Fe 2480 5481 a(\000) p Fh 2542 5414 a(P) p Fd 2648 5438 a(m) p Fc(\000) p Fb(1) p Fd 2648 5518 a(b) p Fb(=2) p Fk 2816 5481 a(j) p Fd 2849 5492 a(b) p Fl 2890 5439 a(h) p Fk 2947 5472 a(l) p Fb 2972 5482 a(1) p Fe 3012 5472 a(\000) p Fh 3074 5405 a(P) p Fd 3179 5430 a(n) 3179 5510 y(a) p Fb(=2) p Fk 3321 5472 a(i) p Fd 3349 5482 a(a) p Fk 3393 5472 a(;i) p Fb 3445 5482 a(2) p Fk 3484 5472 a(;:::;i) p Fd 3632 5482 a(n) p Fh 3684 5295 a(\014) 3684 5355 y(\014) 3684 5414 y(\014) 3684 5474 y(\014) p Fm 3734 5439 a(e) p Fk 3778 5398 a(\026) p Fe(j) p Fk(l) p Fe(j) p Fm 1807 5778 a(20.03.02) p 90 rotate dyy eop %%Page: 17 17 17 16 bop Fg 1222 60 a(Birkho\013) 33 b(normal) f(form) g(for) i(PDE's) 1222 b(17) p Fm 0 299 a(Changing) 32 b(the) f(v) -6 b(ariable) 32 b(of) f(summation) p Fl 30 w(l) p Fm 33 w(to) p Fl 31 w(l) p Fi 1827 314 a(2) p Fm 1899 299 a(=) p Fl 28 w(l) p Fj 21 w(\000) p Fl 19 w(l) p Fi 2181 314 a(1) p Fm 2257 299 a(one) h(can) g(estimate) e(the) i(ab) s(o) m(v) m(e) g(quan) m (tit) m(y) e(b) m(y) p Fh 511 487 a(X) p Fk 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3487 540 a(\026) p Fe(j) p Fk(l) p Fb 3584 550 a(2) p Fe 3624 540 a(j) p Fj 2505 876 a(\002) p Fl 29 w(h) p Fk 2668 909 a(l) p Fb 2693 919 a(1) p Fe 2733 909 a(\000) p Fh 2795 841 a(P) p Fd 2900 866 a(n) 2900 946 y(a) p Fb(=2) p Fk 3043 909 a(i) p Fd 3071 919 a(a) p Fk 3114 909 a(;i) p Fb 3166 919 a(2) p Fk 3205 909 a(;:::;i) p Fd 3353 919 a(n) p Fh 3405 761 a(\014) 3405 821 y(\014) 3405 881 y(\014) p Fm 3455 876 a(e) p Fk 3499 834 a(\026) p Fe(j) p Fk(l) p Fb 3596 844 a(1) p Fe 3636 834 a(j) p Fj 304 1160 a(\024) p Fh 409 990 a(\() 490 1065 y(X) p Fk 530 1280 a(l) p Fb 555 1290 a(2) p Fm 939 1160 a(sup) p Fk 650 1244 a(i) p Fb 678 1254 a(2) p Fk 717 1244 a(;:::;i) p Fd 865 1254 a(n) p Fk 913 1244 a(;j) p Fb 970 1254 a(2) p Fk 1008 1244 a(;::;j) p Fd 1137 1254 a(m) p Fc(\000) p Fb(1) p Fk 1290 1244 a(;l) p Fb 1339 1254 a(1) p Fh 1394 1016 a(\014) 1394 1075 y(\014) 1394 1135 y(\014) 1394 1195 y(\014) p Fl 1427 1160 a(g) p Fh 1475 1134 a(P) p Fd 1580 1159 a(n) 1580 1239 y(a) p Fb(=2) p Fk 1722 1202 a(i) p Fd 1750 1212 a(a) p Fe 1794 1202 a(\000) p Fk(l) p Fb 1881 1212 a(1) p Fk 1920 1202 a(;j) p Fb 1977 1212 a(2) p Fk 2016 1202 a(;:::;j) p Fd 2169 1212 a(m) p Fc(\000) p Fb(1) p Fk 2321 1202 a(;l) p Fb 2370 1212 a(1) p Fi 2409 1202 a(+) p Fk(l) p Fb 2495 1212 a(2) p Fe 2534 1202 a(\000) p Fh 2596 1134 a(P) p Fd 2702 1159 a(n) 2702 1239 y(a) p Fb(=2) p Fk 2844 1202 a(i) p Fd 2872 1212 a(a) p Fe 2916 1202 a(\000) p Fh 2978 1134 a(P) p Fd 3083 1159 a(m) p Fc(\000) p Fb(1) p Fd 3083 1239 a(b) p Fb(=2) p Fk 3251 1202 a(j) p Fd 3284 1213 a(b) p Fh 3325 1016 a(\014) 3325 1075 y(\014) 3325 1135 y(\014) 3325 1195 y(\014) p Fm 3375 1160 a(e) p Fk 3419 1119 a(\026) p Fe(j) p Fk(l) p Fb 3516 1129 a(2) p Fe 3555 1119 a(j) p Fh 3584 990 a(\)) p Fj 1884 1514 a(\002) p Fh 1978 1344 a(\() 2058 1419 y(X) p Fk 2098 1634 a(l) p Fb 2123 1644 a(1) p Fm 2275 1514 a(sup) p Fk 2219 1596 a(i) p Fb 2247 1606 a(2) p Fk 2286 1596 a(;:::;i) p Fd 2434 1606 a(n) p Fh 2498 1399 a(\014) 2498 1459 y(\014) 2498 1519 y(\014) p Fl 2531 1514 a(h) p Fk 2588 1547 a(l) p Fb 2613 1557 a(1) p Fe 2653 1547 a(\000) p Fh 2715 1480 a(P) p Fd 2820 1504 a(n) 2820 1584 y(a) p Fb(=2) p Fk 2962 1547 a(i) p Fd 2990 1557 a(a) p Fk 3034 1547 a(;i) p Fb 3086 1557 a(2) p Fk 3125 1547 a(;:::;i) p Fd 3273 1557 a(n) p Fh 3325 1399 a(\014) 3325 1459 y(\014) 3325 1519 y(\014) p Fm 3375 1514 a(e) p Fk 3419 1473 a(\026) p Fe(j) p Fk(l) p Fb 3516 1483 a(1) p Fe 3555 1473 a(j) p Fh 3584 1344 a(\)) p Fm 0 1842 a(but) f(the) h (\014rst) f(suprem) m(um) g(is) g(tak) m(en) g(o) m(v) m(er) g(all) g (the) g(com) m(bination) f(of) h(indices) h(suc) m(h) h(that) e(their) g (sum) f(is) p Fl 29 w(l) p Fi 3896 1857 a(2) p Fm 3941 1842 a(,) 0 1961 y(and) h(therefore) h(the) f(\014rst) g(curly) f(brac) m(k) m(et) h(is) g(just) g(the) p Fl 29 w(\026) p Fm({mo) s(dulus) f (of) p Fl 29 w(g) p Fm 4 w(,) g(the) h(second) h(is) e(the) p Fl 29 w(\026) p Fm({mo) s(dulus) 0 2081 y(of) p Fl 33 w(h) p Fm(.) 45 b(So) 33 b(the) h(thesis) g(follo) m(ws) f(in) g(the) h (presen) m(t) g(case.) 199 2200 y(The) g(general) f(case) h(follo) m (ws) f(using) p Fj 526 2429 a(j) o(f) p Fl(h;) 17 b(g) p Fj 4 w(g) o(j) p Fk 833 2459 a(\026) p Fj 914 2429 a(\024) p Fm 29 w(\() p Fl(n) p Fi 1119 2444 a(1) p Fl 1163 2429 a(m) p Fi 1250 2444 a(2) p Fm 1317 2429 a(+) p Fl 23 w(n) p Fi 1477 2444 a(2) p Fl 1521 2429 a(m) p Fi 1608 2444 a(1) p Fm 1653 2429 a(\)) p Fj(j) p Fl(h) p Fj(j) p Fk 1805 2444 a(\026) p Fj 1875 2429 a(j) p Fl(g) p Fj 4 w(j) p Fk 1983 2444 a(\026) p Fj 2062 2429 a(\024) p Fm 29 w(\() p Fl(n) p Fi 2267 2444 a(1) p Fm 2333 2429 a(+) p Fl 23 w(m) p Fi 2520 2444 a(1) p Fm 2565 2429 a(\)\() p Fl(m) p Fi 2730 2444 a(2) p Fm 2797 2429 a(+) p Fl 22 w(n) p Fi 2956 2444 a(2) p Fm 3001 2429 a(\)) p Fj(j) p Fl(h) p Fj(j) p Fk 3153 2444 a(\026) p Fj 3222 2429 a(j) p Fl(g) p Fj 4 w(j) p Fk 3330 2444 a(\026) p Fl 3415 2429 a(;) p Fm 0 2668 a(whic) m(h) 34 b(is) f(the) h(thesis.) p 3891 2595 78 4 v 3891 2664 4 70 v 3965 2664 V 3891 2668 78 4 v Fn 0 2828 a(Remark) 158 b(5.10.) p Fg 35 w(Equation) 34 b(\(5.15\)) e(holds) j(also) f(in) h(the) f(case) h(p) s(olynomials) d (whic) m(h) k(are) e(not) g(homo-) 0 2947 y(genea) g(in) p Fl 33 w(\030) 5 b(;) 17 b(\021) p Fg 4 w(,) 31 b(to) i(v) m(erify) g (this) g(p) s(oin) m(t) h(just) g(use) g(the) f(de\014nition) h(of) g (the) p Fl 33 w(\026) p Fg({mo) s(dulus.) p Fn 0 3147 a(Lemma) 174 b(5.11.) p Fg 39 w(Let) p Fl 38 w(h) p Fj 36 w(2) 36 b(M) p Fk 1314 3099 a(\026) 1314 3176 y(R) p Fg 1417 3147 a(and) p Fl 38 w(g) p Fj 39 w(2) g(M) p Fk 1924 3099 a(\026) 1924 3177 y(R) p Fe(\000) p Fk(d) p Fg 2092 3147 a(,) j(then) g(for) f(an) m(y) g(p) s(ositiv) m(e) p Fl 37 w(d) p Fe 3166 3111 a(0) p Fl 3230 3147 a(<) d(R) p Fj 26 w(\000) p Fl 26 w(d) p Fg 38 w(one) j(has) p Fj 0 3266 a(f) p Fl(h;) 17 b(g) p Fj 4 w(g) 26 b(2) i(M) p Fk 494 3218 a(\026) 494 3296 y(R) p Fe(\000) p Fk(d) p Fe(\000) p Fk(d) p Fc 760 3276 a(0) p Fj 1012 3446 a(hjf) p Fl -1 w(h;) 17 b(g) p Fj 4 w(g) o(ji) p Fk 1397 3396 a(\026) 1397 3476 y(R) p Fe(\000) p Fk(d) p Fe(\000) p Fk(d) p Fc 1663 3456 a(0) p Fj 1741 3446 a(\024) p Fm 2039 3379 a(1) p 1859 3423 411 4 v Fl 1859 3515 a(d) p Fe 1911 3486 a(0) p Fm 1938 3515 a(\() p Fl(d) p Fm 22 w(+) p Fl 23 w(d) p Fe 2203 3486 a(0) p Fm 2231 3515 a(\)) p Fj 2298 3446 a(hj) p Fl -1 w(h) p Fj(ji) p Fk 2488 3396 a(\026) 2488 3476 y(R) p Fj 2587 3446 a(h) o(j) p Fl(g) p Fj 4 w(j) o(i) p Fk 2771 3396 a(\026) 2771 3476 y(R) p Fe(\000) p Fk(d) p Fm 3714 3446 a(\(5) p Fl(:) p Fm(17\)) p Fn 0 3790 a(Pro) s(of.) p Fm 34 w(W) -8 b(rite) p Fl 32 w(h) p Fm 28 w(=) p Fh 809 3715 a(P) p Fk 914 3820 a(j) p Fl 973 3790 a(h) p Fk 1030 3805 a(j) p Fm 1105 3790 a(and) p Fl 33 w(g) p Fm 30 w(=) p Fh 1482 3715 a(P) p Fk 1587 3820 a(k) p Fl 1652 3790 a(g) p Fk 1700 3805 a(k) p Fm 1781 3790 a(with) p Fl 32 w(h) p Fk 2064 3805 a(j) p Fm 2139 3790 a(homogeneous) 33 b(of) g(degree) p Fl 33 w(j) p Fm 38 w(and) g(similarly) d(for) p Fl 0 3910 a(g) p Fm 4 w(,) i(w) m(e) i(ha) m(v) m(e) p Fj 1503 4044 a(f) p Fl(h;) 17 b(g) p Fj 4 w(g) p Fm 26 w(=) p Fh 1888 3949 a(X) p Fk 1910 4163 a(j;k) p Fj 2048 4044 a(f) p Fl(h) p Fk 2155 4059 a(j) p Fl 2198 4044 a(;) g(g) p Fk 2291 4059 a(k) p Fj 2338 4044 a(g) p Fl 50 w(:) p Fm 0 4331 a(No) m(w) 33 b(eac) m(h) h(term) e(of) i(the) f(series) h (is) g(estimated) e(b) m(y) p Fj 1213 4559 a(hj) o(f) p Fl(h) p Fk 1386 4574 a(k) p Fl 1436 4559 a(;) 17 b(g) p Fk 1529 4574 a(k) p Fj 1576 4559 a(gji) p Fk 1693 4509 a(\026) 1693 4589 y(R) p Fe(\000) p Fk(d) p Fe(\000) p Fk(d) p Fc 1959 4569 a(0) p Fm 2037 4559 a(=) p Fj 28 w(j) o(f) p Fl(h) p Fk 2276 4574 a(j) p Fl 2319 4559 a(;) g(g) p Fk 2412 4574 a(k) p Fj 2459 4559 a(gj) p Fk 2537 4592 a(\026) p Fm 2607 4559 a(\() p Fl(R) p Fj 22 w(\000) p Fl 23 w(d) p Fj 22 w(\000) p Fl 23 w(d) p Fe 3070 4517 a(0) p Fm 3097 4559 a(\)) p Fk 3136 4517 a(j) p Fi 4 w(+) p Fk(k) p Fe 2 w(\000) p Fi(2) p Fj 1970 4735 a(\024) 28 b(j) p Fl(h) p Fk 2160 4750 a(j) p Fj 2202 4735 a(j) p Fk 2230 4769 a(\026) p Fj 2300 4735 a(j) p Fl(g) p Fk 2376 4750 a(k) p Fj 2424 4735 a(j) p Fk 2452 4750 a(\026) p Fl 2505 4735 a(j) 6 b(k) p Fm 3 w(\() p Fl(R) p Fj 22 w(\000) p Fl 23 w(d) p Fj 22 w(\000) p Fl 23 w(d) p Fe 3070 4694 a(0) p Fm 3097 4735 a(\)) p Fk 3136 4694 a(j) p Fi 4 w(+) p Fk(k) p Fe 2 w(\000) p Fi(2) p Fm 1395 4912 a(=) p Fj 28 w(j) p Fl(h) p Fk 1585 4927 a(j) p Fj 1627 4912 a(j) p Fk 1655 4945 a(\026) p Fj 1725 4912 a(j) p Fl(g) p Fk 1801 4927 a(k) p Fj 1849 4912 a(j) p Fk 1877 4927 a(\026) p Fl 1930 4912 a(j) p Fm 6 w(\() p Fl(R) p Fj 22 w(\000) p Fl 23 w(d) p Fj 22 w(\000) p Fl 23 w(d) p Fe 2440 4870 a(0) p Fm 2467 4912 a(\)) p Fk 2506 4870 a(j) p Fe 4 w(\000) p Fi(1) p Fl 2650 4912 a(k) p Fm 3 w(\() p Fl(R) p Fj 22 w(\000) p Fl 23 w(d) p Fj 22 w(\000) p Fl 23 w(d) p Fe 3168 4870 a(0) p Fm 3196 4912 a(\)) p Fk 3235 4870 a(k) p Fe 2 w(\000) p Fi(1) p Fj 583 5130 a(\024) 28 b(j) p Fl(h) p Fk 773 5145 a(j) p Fj 815 5130 a(j) p Fk 843 5164 a(\026) p Fj 913 5130 a(j) p Fl(g) p Fk 989 5145 a(k) p Fj 1037 5130 a(j) p Fk 1065 5145 a(\026) p Fl 1118 5130 a(R) p Fk 1195 5089 a(j) p Fm 1351 5063 a(1) p 1249 5107 254 4 v Fl 1249 5198 a(d) p Fm 22 w(+) p Fl 22 w(d) p Fe 1474 5170 a(0) p Fm 1541 5063 a(1) p 1526 5107 80 4 v Fl 1526 5198 a(d) p Fe 1578 5170 a(0) p Fm 1618 5130 a(\() p Fl(R) p Fj 22 w(\000) p Fl 23 w(d) p Fm(\)) p Fk 1946 5089 a(k) p Fm 2022 5130 a(=) 2320 5063 y(1) p 2139 5107 411 4 v Fl 2139 5198 a(d) p Fe 2191 5170 a(0) p Fm 2219 5198 a(\() p Fl(d) p Fm 22 w(+) p Fl 22 w(d) p Fe 2483 5170 a(0) p Fm 2511 5198 a(\)) p Fj 2579 5130 a(h) o(j) p Fl(h) p Fk 2702 5145 a(j) p Fj 2744 5130 a(ji) p Fk 2811 5080 a(\026) 2811 5164 y(R) p Fj 2909 5130 a(hj) p Fl(g) p Fk 3024 5145 a(k) p Fj 3072 5130 a(ji) p Fk 3139 5080 a(\026) 3139 5160 y(R) p Fe(\000) p Fk(d) p Fl 3358 5130 a(:) p Fm 3714 4871 a(\(5) p Fl(:) p Fm(18\)) p 3891 5346 78 4 v 3891 5416 4 70 v 3965 5416 V 3891 5420 78 4 v 199 5539 a(W) -8 b(e) 34 b(estimate) e(no) m(w) h (the) h(terms) f(of) g(the) h(series) g(\(5.13\),) d(\(5.14\)) h (de\014ning) i(the) g(Lie) f(transform.) 1807 5778 y(20.03.02) p 90 rotate dyy eop %%Page: 18 18 18 17 bop Fg 0 60 a(18) 1660 b(D.) 33 b(Bam) m(busi) p Fn 0 299 a(Lemma) 173 b(5.12.) p Fg 38 w(Let) p Fl 38 w(h) p Fj 36 w(2) 35 b(M) p Fk 1311 251 a(\026) 1311 328 y(R) p Fg 1414 299 a(and) p Fl 38 w(\037) p Fj 35 w(2) h(M) p Fk 1931 251 a(\026) 1931 328 y(R) p Fg 2034 299 a(b) s(e) i(t) m(w) m(o) f(analytic) g(functions,) j(denote) e(b) m (y) p Fl 38 w(h) p Fk 3738 314 a(n) p Fg 3830 299 a(the) 0 418 y(functions) c(de\014ned) i(recoursiv) m(ely) d(b) m(y) g (\(5.14\);) f(then,) h(for) h(an) m(y) f(p) s(ositiv) m(e) p Fl 32 w(d) 28 b(<) g(R) p Fg 1 w(,) k(one) i(has) p Fl 34 w(h) p Fk 3476 433 a(n) p Fj 3558 418 a(2) 28 b(M) p Fk 3772 371 a(\026) 3772 448 y(R) p Fe(\000) p Fk(d) p Fg 3941 418 a(,) 0 538 y(and) 34 b(the) f(follo) m(wing) g(estimate) f (holds) p Fj 1212 833 a(hj) p Fl -1 w(h) p Fk 1335 848 a(n) p Fj 1390 833 a(ji) p Fk 1457 782 a(\026) 1457 862 y(R) p Fe(\000) p Fk(d) p Fj 1670 833 a(\024) c(hj) p Fl -1 w(h) p Fj(j) q(i) p Fk 1965 782 a(\026) 1965 862 y(R) p Fh 2064 692 a(\022) p Fl 2152 765 a(e) p Fi 2198 729 a(2) p 2149 810 97 4 v Fl 2149 901 a(d) p Fi 2201 872 a(2) p Fj 2274 833 a(hj) p Fl(\037) p Fj(ji) p Fk 2469 782 a(\026) 2469 862 y(R) p Fh 2551 692 a(\023) p Fk 2625 709 a(n) p Fl 2729 833 a(:) p Fm 957 w(\(5) p Fl(:) p Fm(19\)) p Fn 0 1223 a(Pro) s(of.) p Fm 34 w(Fix) p Fl 33 w(n) p Fm(,) k(and) i(denote) 1162 1197 y(~) p Fl 1157 1223 a(\016) p Fm 32 w(:=) p Fl 28 w(d=n) p Fm(,) e(w) m(e) i (lo) s(ok) e(for) h(a) g(sequence) p Fl 35 w(C) p Fi 2677 1171 a(\() p Fk(n) p Fi(\)) p Fk 2670 1253 a(l) p Fm 2827 1223 a(suc) m(h) i(that) p Fj 1342 1478 a(hj) p Fl -1 w(h) p Fk 1465 1493 a(l) p Fj 1496 1478 a(ji) p Fk 1563 1428 a(\026) 1563 1523 y(R) p Fe(\000) p Fi 1689 1505 a(~) p Fk 1685 1523 a(\016) s(l) p Fj 1799 1478 a(\024) p Fl 28 w(C) p Fi 1982 1427 a(\() p Fk(n) p Fi(\)) p Fk 1975 1508 a(l) p Fl 2132 1478 a(;) p Fj 116 w(8) p Fl 10 w(l) p Fj 30 w(\024) p Fl 28 w(n) e(:) p Fm 0 1727 a(By) f(\(5.17\)) g(this) h(sequence) i(can) f(b) s(e) g(de\014ned) h (b) m(y) p Fl 628 2022 a(C) p Fi 706 1970 a(\() p Fk(n) p Fi(\)) 699 2049 y(0) p Fm 851 2022 a(=) p Fj 28 w(hj) p Fl -1 w(h) p Fj(j) q(i) p Fk 1146 1972 a(\026) 1146 2052 y(R) p Fl 1261 2022 a(;) 116 b(C) p Fi 1483 1970 a(\() p Fk(n) p Fi(\)) p Fk 1476 2052 a(l) p Fm 1628 2022 a(=) 1745 1955 y(1) p 1745 1999 50 4 v Fl 1754 2091 a(l) p Fm 1858 1955 a(1) p 1819 1999 128 4 v Fl 1819 2104 a(l) p Fm 1856 2078 a(~) p Fl 1851 2104 a(\016) p Fm 1903 2078 a(~) p Fl 1898 2104 a(\016) 1958 2022 y(C) p Fi 2036 1970 a(\() p Fk(n) p Fi(\)) p Fk 2029 2052 a(l) p Fe(\000) p Fi(1) p Fj 2179 2022 a(hj) p Fl -1 w(\037) p Fj(ji) p Fk 2374 1972 a(\026) 2374 2052 y(R) p Fm 2484 2022 a(=) p Fl 2633 1955 a(n) p Fi 2693 1919 a(2) p 2601 1999 170 4 v Fl 2601 2091 a(l) p Fi 2633 2062 a(2) p Fl 2677 2091 a(\016) p Fi 2725 2062 a(2) p Fl 2782 2022 a(C) p Fi 2860 1970 a(\() p Fk(n) p Fi(\)) p Fk 2853 2052 a(l) p Fe(\000) p Fi(1) p Fj 3002 2022 a(hj) p Fl(\037) p Fj(ji) p Fk 3197 1972 a(\026) 3197 2052 y(R) p Fl 3312 2022 a(:) p Fm 0 2295 a(So) 33 b(one) h(has) p Fl 1212 2460 a(C) p Fi 1290 2419 a(\() p Fk(n) p Fi(\)) p Fk 1283 2484 a(n) p Fm 1435 2460 a(=) 1632 2392 y(1) p 1552 2437 210 4 v 1552 2528 a(\() p Fl(n) p Fm(!\)) p Fi 1718 2499 a(2) p Fh 1790 2319 a(\022) p Fl 1875 2391 a(n) p Fi 1935 2355 a(2) p Fj 1996 2391 a(hj) p Fl(\037) p Fj(ji) p Fk 2192 2341 a(\026) 2192 2421 y(R) p 1875 2437 399 4 v Fl 2028 2528 a(\016) p Fi 2076 2499 a(2) p Fh 2285 2319 a(\023) p Fk 2359 2335 a(n) p Fj 2430 2460 a(h) o(j) p Fl(g) p Fj 4 w(j) o(i) p Fk 2614 2410 a(\026) 2614 2490 y(R) p Fl 2729 2460 a(:) p Fm 0 2700 a(Using) e(the) g(inequalit) m(y) p Fl 30 w(n) p Fk 969 2664 a(n) p Fl 1051 2700 a(<) c(n) p Fm(!) p Fl(e) p Fk 1290 2664 a(n) p Fl 1344 2700 a(;) p Fm 32 w(whic) m(h) k(is) g(easily) f(v) m(eri\014ed) h(b) m(y) g (writing) f(the) h(iterativ) m(e) f(de\014nition) h(of) p Fl 0 2819 a(n) p Fk 60 2783 a(n) p Fl 114 2819 a(=n) p Fm(!) h(one) g(has) h(the) f(thesis.) p 3891 2745 78 4 v 3891 2815 4 70 v 3965 2815 V 3891 2819 78 4 v Fn 0 2979 a(Remark) 199 b(5.13.) p Fg 44 w(Let) p Fl 43 w(\037) p Fg 44 w(b) s(e) 43 b(an) h(analytic) e(function) i(with) f (Hamiltonian) e(v) m(ector) i(\014eld) h(whic) m(h) g(is) 0 3125 y(analytic) j(as) h(a) g(map) f(from) p Fj 47 w(B) p Fk 1180 3140 a(s) p Fm 1223 3125 a(\() p Fl(R) p Fi 1339 3089 a(\() p Fk(s) p Fi(\)) p Fm 1443 3125 a(\)) p Fg 48 w(to) p Fj 48 w(P) p Fk 1736 3140 a(s) p Fg 1779 3125 a(,) k(\014x) p Fl 48 w(\016) p Fi 2062 3089 a(\() p Fk(s) p Fi(\)) p Fl 2219 3125 a(<) i(R) p Fi 2426 3089 a(\() p Fk(s) p Fi(\)) p Fg 2530 3125 a(.) 88 b(Assume) p Fj 48 w(k) p Fl(X) p Fk 3163 3140 a(\037) p Fj 3217 3125 a(k) p Fk 3267 3075 a(R) p Fb 3327 3045 a(\() p Fd(s) p Fb(\)) p Fk 3267 3159 a(s) p Fl 3477 3125 a(<) 53 b(\016) p Fi 3655 3089 a(\() p Fk(s) p Fi(\)) p Fg 3808 3125 a(and) 0 3245 y(consider) 34 b(the) g(time) p Fl 32 w(t) p Fg 33 w(\015o) m(w) p Fl 34 w(T) p Fk 1136 3209 a(t) p Fg 1204 3245 a(of) p Fl 33 w(X) p Fk 1400 3260 a(\037) p Fg 1455 3245 a(.) 44 b(Then,) 33 b(for) p Fj 34 w(j) p Fl(t) p Fj(j) 27 b(\024) p Fm 28 w(1) p Fg(,) 33 b(one) g(has) p Fm 1355 3512 a(sup) p Fe 1146 3606 a(k) p Fk -1 w(\020) p Fe 5 w(k) p Fd 1268 3628 a(s) p Fe 1306 3606 a(\024) p Fk(R) p Fb 1428 3586 a(\() p Fd(s) p Fb(\)) p Fe 1521 3606 a(\000) p Fk(\016) p Fb 1622 3586 a(\() p Fd(s) p Fb(\)) p Fh 1731 3428 a(\015) 1731 3487 y(\015) p Fl 1787 3512 a(T) p Fk 1859 3471 a(t) p Fm 1894 3512 a(\() p Fl(\020) p Fm 7 w(\)) p Fj 21 w(\000) p Fl 23 w(\020) p Fh 2195 3428 a(\015) 2195 3487 y(\015) p Fk 2250 3552 a(s) p Fj 2321 3512 a(\024) 28 b(k) p Fl(\037) p Fj(k) p Fk 2588 3462 a(R) p Fb 2648 3432 a(\() p Fd(s) p Fb(\)) p Fk 2588 3542 a(s) p Fl 2795 3512 a(:) p Fm 891 w(\(5) p Fl(:) p Fm(20\)) p Fn 0 3959 a(Lemma) 175 b(5.14.) p Fg 38 w(Consider) p Fl 39 w(\037) p Fg 38 w(as) 39 b(ab) s(o) m(v) m(e) f(and) g(let) p Fl 38 w(h) p Fm 36 w(:) p Fj 35 w(B) p Fk 2320 3974 a(s) p Fh 2380 3878 a(\000) p Fl 2425 3959 a(R) p Fi 2502 3922 a(\() p Fk(s) p Fi(\)) p Fh 2607 3878 a(\001) p Fj 2688 3959 a(!) p Fn 35 w(C) p Fg 38 w(b) s(e) g(an) h(analytic) e(function) 0 4121 y(with) e(v) m(ector) g (\014eld) h(analytic) e(in) p Fj 36 w(B) p Fk 1313 4136 a(s) p Fh 1372 4040 a(\000) p Fl 1418 4121 a(R) p Fi 1495 4085 a(\() p Fk(s) p Fi(\)) p Fh 1599 4040 a(\001) p Fg 1645 4121 a(,) h(\014x) p Fm 36 w(0) p Fl 31 w(<) c(\016) p Fi 2089 4085 a(\() p Fk(s) p Fi(\)) p Fl 2225 4121 a(<) h(R) p Fi 2411 4085 a(\() p Fk(s) p Fi(\)) p Fg 2550 4121 a(assume) p Fj 36 w(k) p Fl(X) p Fk 3030 4136 a(\037) p Fj 3084 4121 a(k) p Fk 3134 4071 a(R) p Fb 3194 4041 a(\() p Fd(s) p Fb(\)) p Fk 3134 4154 a(s) p Fj 3322 4121 a(\024) p Fl 32 w(\016) p Fi 3479 4085 a(\() p Fk(s) p Fi(\)) p Fl 3584 4121 a(=) p Fm(3) p Fg(,) j(then,) 0 4241 y(for) p Fj 33 w(j) p Fl(t) p Fj(j) 27 b(\024) p Fm 29 w(1) p Fg(,) 32 b(one) i(has) p Fj 902 4531 a(k) p Fl(X) p Fk 1035 4547 a(h) p Fe(\016) p Fk(T) p Fd 1181 4528 a(t) p Fj 1218 4531 a(k) p Fk 1268 4481 a(R) p Fb 1328 4451 a(\() p Fd(s) p Fb(\)) p Fe 1421 4481 a(\000) p Fk(\016) p Fb 1522 4451 a(\() p Fd(s) p Fb(\)) p Fk 1268 4561 a(s) p Fj 1647 4531 a(\024) p Fh 1752 4391 a(\022) p Fm 1825 4531 a(1) 22 b(+) 2060 4464 y(3) p 2009 4508 153 4 v Fl 2009 4601 a(\016) p Fi 2057 4573 a(\() p Fk(s) p Fi(\)) p Fj 2190 4531 a(k) p Fl(X) p Fk 2323 4546 a(\037) p Fj 2377 4531 a(k) p Fk 2427 4481 a(R) p Fb 2487 4451 a(\() p Fd(s) p Fb(\)) p Fk 2427 4565 a(s) p Fh 2585 4391 a(\023) p Fj 2675 4531 a(k) p Fl -1 w(X) p Fk 2807 4546 a(h) p Fj 2859 4531 a(k) p Fk 2908 4481 a(R) p Fb 2968 4451 a(\() p Fd(s) p Fb(\)) p Fk 2908 4561 a(s) p Fm 0 4922 a(F) -8 b(or) 33 b(the) g(pro) s(of) h(see) g([2]) e(pro) s(of) i(of) f (lemma) e(8.2.) p Fg 0 5539 a(5.3) i(Iterativ) m(e) f(lemma) p Fm 1807 5778 a(20.03.02) p 90 rotate dyy eop %%Page: 19 19 19 18 bop Fg 1222 60 a(Birkho\013) 33 b(normal) f(form) g(for) i(PDE's) 1222 b(19) p Fm 199 299 a(In) 28 b(the) g(statemen) m(t) e(of) i(the) g (forthcoming) e(iterativ) m(e) g(lemma) g(w) m(e) h(will) g(use) h(the) g(follo) m(wing) e(notations:) 0 418 y(F) -8 b(or) 39 b(an) m(y) g(p) s(ositiv) m(e) p Fl 38 w(R) p Fm 1 w(,) h(de\014ne) p Fl 41 w(\016) p Fm 42 w(:=) p Fl 37 w(R) q(=) p Fm(2) p Fl(r) p Fm 41 w(and) p Fl 40 w(R) p Fk 1959 433 a(k) p Fm 2045 418 a(:=) p Fl 38 w(R) p Fj 27 w(\000) p Fl 26 w(k) s(\016) p Fm 4 w(.) 62 b(Moreo) m(v) m(er,) 41 b(for) f(an) m(y) f (p) s(ositiv) m(e) p Fl 38 w(R) p Fi 3864 382 a(\() p Fk(s) p Fi(\)) p Fm 0 557 a(de\014ne) p Fl 35 w(\016) p Fi 336 521 a(\() p Fk(s) p Fi(\)) p Fm 468 557 a(:=) p Fl 28 w(R) p Fi 678 521 a(\() p Fk(s) p Fi(\)) p Fl 782 557 a(=) p Fm(2) p Fl(r) p Fm 36 w(and) p Fl 34 w(R) p Fi 1234 506 a(\() p Fk(s) p Fi(\)) p Fk 1233 587 a(k) p Fm 1366 557 a(:=) p Fl 28 w(R) p Fi 1576 521 a(\() p Fk(s) p Fi(\)) p Fj 1702 557 a(\000) p Fl 23 w(k) s(\016) p Fi 1905 521 a(\() p Fk(s) p Fi(\)) p Fm 2009 557 a(.) p Fn 0 717 a(Prop) s(osition) 153 b(5.15.) 34 b(Iterativ) m(e) j(Lemma.) p Fg 96 w(Fix) p Fl 32 w(r) p Fj 31 w(\025) p Fm 28 w(4) p Fg 33 w(and) d(consider) g(the) g(Hamiltonian) d(\(4.1\);) 0 836 y(de\014ne) p Fl 35 w(R) p Fe 364 851 a(\003) p Fg 442 836 a(b) m(y) p Fm 1585 947 a(1) p 1549 991 122 4 v Fl 1549 1082 a(R) p Fe 1625 1097 a(\003) p Fm 1710 1014 a(:=) 1855 947 y(16e) p Fi 1999 911 a(2) p Fl 2044 947 a(AN) p Fk 2210 911 a(\013) p Fl 2266 947 a(r) p Fi 2314 911 a(2) p 1855 991 504 4 v Fl 2078 1082 a(\015) 2404 1014 y(;) p Fg 0 1230 a(and) j(assume) p Fl 1525 1375 a(R) 28 b(<) p Fm 1755 1350 a(\026) p Fl 1734 1375 a(R) p Fi 1810 1390 a(1) p Fl 1887 1375 a(;) 2066 1308 y(R) p 2043 1353 122 4 v 2043 1444 a(R) p Fe 2119 1459 a(\003) p Fj 2204 1375 a(\024) p Fm 2321 1308 a(1) p 2321 1353 50 4 v 2321 1444 a(2) 2416 1375 y(;) 1270 b(\(5) p Fl(:) p Fm(21\)) p Fg 0 1606 a(then,) 32 b(for) g(an) m(y) p Fl 32 w(k) p Fj 31 w(\024) p Fl 28 w(r) p Fj 25 w(\000) p Fm 23 w(4) p Fg 31 w(there) g(exists) g(a) g(canonical) f (transformation) p Fj 31 w(T) p Fi 2832 1570 a(\() p Fk(k) p Fi 2 w(\)) p Fg 2975 1606 a(whic) m(h) h(puts) h(\(4.1\)) d(in) i(the) 0 1726 y(form) p Fl 799 1845 a(H) p Fi 890 1804 a(\() p Fk(k) p Fi 2 w(\)) p Fm 1029 1845 a(:=) p Fl 28 w(H) p Fj 30 w(\016) 22 b(T) p Fi 1427 1804 a(\() p Fk(k) p Fi 2 w(\)) p Fm 1566 1845 a(=) p Fl 28 w(h) p Fi 1728 1860 a(0) p Fm 1795 1845 a(+) p Fl 23 w(Z) p Fi 1970 1804 a(\() p Fk(k) p Fi 2 w(\)) p Fm 2103 1845 a(+) p Fl 23 w(f) p Fi 2263 1804 a(\() p Fk(k) p Fi 2 w(\)) p Fm 2396 1845 a(+) p Fj 22 w(R) p Fi 2579 1794 a(\() p Fk(k) p Fi 2 w(\)) p Fk 2579 1875 a(N) p Fm 2713 1845 a(+) p Fj 23 w(R) p Fi 2897 1794 a(\() p Fk(k) p Fi 2 w(\)) p Fk 2897 1875 a(T) p Fl 3042 1845 a(;) p Fm 644 w(\(5) p Fl(:) p Fm(22\)) p Fg 0 2014 a(with) p Fl 33 w(Z) p Fi 302 1978 a(\() p Fk(k) p Fi 2 w(\)) p Fg 447 2014 a(in) 33 b(Birkho\013) f(normal) g(form;) g(moreo) m(v) m (er) 61 2134 y(1\)) 49 b(the) 34 b(follo) m(wing) e(estimates) h(hold) p Fh 1419 2314 a(D) 1480 2310 y(\014) 1480 2370 y(\014) 1480 2429 y(\014) p Fl 1513 2424 a(Z) p Fi 1588 2383 a(\() p Fk(k) p Fi 2 w(\)) p Fh 1699 2310 a(\014) 1699 2370 y(\014) 1699 2429 y(\014) 1732 2314 y(E) p Fk 1793 2334 a(\026) 1793 2494 y(R) p Fd 1853 2505 a(k) p Fj 1946 2424 a(\024) p Fl 28 w(AR) p Fi 2203 2383 a(3) p Fk 2264 2300 a(k) p Fe 2 w(\000) p Fi(1) p Fh 2265 2330 a(X) p Fk 2274 2544 a(l) p Fi(=0) p Fh 2426 2284 a(\022) p Fl 2534 2357 a(R) p 2512 2401 122 4 v 2512 2493 a(R) p Fe 2588 2508 a(\003) p Fh 2645 2284 a(\023) p Fk 2718 2305 a(l) p Fh 1434 2652 a(D) 1495 2648 y(\014) 1495 2707 y(\014) 1495 2767 y(\014) p Fl 1528 2762 a(f) p Fi 1588 2721 a(\() p Fk(k) p Fi 2 w(\)) p Fh 1699 2648 a(\014) 1699 2707 y(\014) 1699 2767 y(\014) 1732 2652 y(E) p Fk 1793 2672 a(\026) 1793 2832 y(R) p Fd 1853 2843 a(k) p Fj 1946 2762 a(\024) p Fl 28 w(AR) p Fi 2203 2721 a(3) p Fh 2264 2622 a(\022) p Fl 2372 2695 a(R) p 2349 2739 V 2349 2831 a(R) p Fe 2425 2846 a(\003) p Fh 2482 2622 a(\023) p Fk 2556 2642 a(k) p Fm 3714 2574 a(\(5) p Fl(:) p Fm(23\)) p Fg 61 3015 a(2\)) 49 b(for) 34 b(an) m(y) p Fl 33 w(s) p Fj 27 w(\025) p Fm 28 w(1) p Fg 33 w(de\014ne) p Fl 1840 3185 a(R) p Fi 1917 3133 a(\() p Fk(s) p Fi(\)) p Fe 1916 3199 a(\003) p Fm 2049 3185 a(:=) p Fl 2194 3118 a(R) p Fe 2270 3133 a(\003) p 2194 3162 V Fl 2208 3253 a(\032) p Fk 2260 3268 a(s) p Fg 199 3420 a(then,) g(for) f(an) m(y) p Fl 34 w(R) p Fi 872 3384 a(\() p Fk(s) p Fi(\)) p Fg 1009 3420 a(suc) m(h) i(that) p Fl 1586 3611 a(R) p Fi 1663 3575 a(\() p Fk(s) p Fi(\)) p 1586 3655 182 4 v Fl 1586 3772 a(R) p Fi 1663 3720 a(\() p Fk(s) p Fi(\)) p Fe 1662 3786 a(\003) p Fj 1806 3678 a(\024) p Fm 29 w(min) p Fh 2094 3538 a(\032) p Fm 2181 3611 a(1) p 2181 3655 50 4 v 2181 3747 a(2) p Fl 2243 3678 a(;) p Fm 2334 3611 a(2e) p Fi 2428 3575 a(2) p 2299 3655 209 4 v Fl 2299 3747 a(\032) p Fk 2351 3762 a(s) p Fm 2393 3747 a(\006) p Fk 2465 3762 a(s) p Fh 2519 3538 a(\033) p Fm 3714 3678 a(\(5) p Fl(:) p Fm(24\)) p Fg 199 3957 a(the) f(transformation) p Fj 32 w(T) p Fi 1130 3921 a(\() p Fk(k) p Fi 2 w(\)) p Fg 1274 3957 a(is) f(analytic) f(as) i(a) f(map) p Fj 1384 4174 a(T) p Fi 1463 4133 a(\() p Fk(k) p Fi 2 w(\)) p Fm 1602 4174 a(:) p Fj 28 w(B) p Fk 1723 4189 a(s) p Fh 1782 4063 a(\020) p Fl 1842 4174 a(R) p Fi 1919 4122 a(\() p Fk(s) p Fi(\)) p Fk 1918 4204 a(k) p Fi 2 w(+1) p Fh 2067 4063 a(\021) p Fj 2154 4174 a(!) 28 b(B) p Fk 2347 4189 a(s) p Fh 2406 4063 a(\020) p Fl 2466 4174 a(R) p Fi 2543 4122 a(\() p Fk(s) p Fi(\)) 2542 4201 y(1) p Fh 2647 4063 a(\021) p Fl 2756 4174 a(;) p Fm 930 w(\(5) p Fl(:) p Fm(25\)) p Fg 199 4391 a(and) 34 b(the) g(follo) m(wing) e(estimates) h(hold) p Fm 1111 4676 a(sup) p Fe 980 4786 a(k) p Fk(\020) p Fe 5 w(k) p Fd 1103 4808 a(s) p Fe 1141 4786 a(\024) p Fk(R) p Fb 1263 4750 a(\() p Fd(s) p Fb(\)) p Fd 1263 4813 a(k) p Fb 1 w(+1) p Fh 1408 4562 a(\015) 1408 4621 y(\015) 1408 4681 y(\015) p Fl 1464 4676 a(\020) p Fj 29 w(\000) 23 b(T) p Fi 1716 4635 a(\() p Fk(k) p Fi 2 w(\)) p Fm 1827 4676 a(\() p Fl(\020) p Fm 7 w(\)) p Fh 1956 4562 a(\015) 1956 4621 y(\015) 1956 4681 y(\015) p Fk 2011 4746 a(s) p Fj 2081 4676 a(\024) p Fl 29 w(R) p Fi 2264 4635 a(\() p Fk(s) p Fi(\)) p Fl 2394 4609 a(\032) p Fk 2446 4624 a(s) p Fm 2488 4609 a(\006) p Fk 2560 4624 a(s) p 2380 4653 237 4 v Fm 2380 4745 a(16e) p Fi 2524 4716 a(2) p Fl 2568 4745 a(r) p Fk 2695 4552 a(k) p Fh 2645 4582 a(X) p Fk 2653 4796 a(l) p Fi(=1) p Fh 2805 4536 a(\022) p Fl 2890 4609 a(R) p Fi 2967 4573 a(\() p Fk(s) p Fi(\)) p 2890 4653 182 4 v Fl 2890 4769 a(R) p Fi 2967 4718 a(\() p Fk(s) p Fi(\)) p Fe 2966 4783 a(\003) p Fh 3084 4536 a(\023) p Fk 3157 4545 a(l) p Fm 3714 4676 a(\(5) p Fl(:) p Fm(26\)) p Fh 436 4982 a(\015) 436 5042 y(\015) 436 5102 y(\015) p Fl 491 5097 a(X) p Fe 574 5136 a(R) p Fb 642 5100 a(\() p Fd(k) p Fb 1 w(\)) p Fd 642 5162 a(N) p Fh 745 4982 a(\015) 745 5042 y(\015) 745 5102 y(\015) p Fk 800 5007 a(R) p Fb 860 4970 a(\() p Fd(s) p Fb(\)) p Fd 860 5033 a(k) p Fb 1 w(+1) p Fk 800 5167 a(s) p Fj 1021 5097 a(\024) p Fm 1159 5029 a(2) p Fk 1209 4993 a(r) p Fi 2 w(+1) p 1138 5074 236 4 v Fl 1138 5165 a(N) p Fk 1229 5136 a(s) p Fe(\000) p Fi(1) p Fm 1386 5097 a(2) p Fl(r) s(A) p Fm(\006) p Fk 1631 5112 a(s) p Fm 1673 5097 a(\() p Fl(\032) p Fk 1764 5112 a(s) p Fl 1806 5097 a(R) p Fi 1883 5056 a(\() p Fk(s) p Fi(\)) p Fm 1987 5097 a(\)) p Fi 2026 5056 a(2) p Fh 2087 4926 a(") p Fk 2145 4972 a(k) p Fe 2 w(\000) p Fi(1) p Fh 2154 5002 a(Y) p Fk 2155 5217 a(l) p Fi(=0) p Fh 2308 4926 a( ) p Fm 2386 5097 a(1) f(+) 2570 5029 y(3) p Fl(\032) p Fk 2672 5044 a(s) p Fm 2714 5029 a(\006) p Fk 2786 5044 a(s) p 2570 5074 259 4 v Fm 2605 5165 a(16e) p Fi 2749 5136 a(2) p Fh 2857 4956 a(\022) p Fl 2942 5029 a(R) p Fi 3019 4993 a(\() p Fk(s) p Fi(\)) p 2942 5074 182 4 v Fl 2942 5190 a(R) p Fi 3019 5138 a(\() p Fk(s) p Fi(\)) p Fe 3018 5204 a(\003) p Fh 3135 4956 a(\023) p Fk 3209 4966 a(l) p Fi(+1) p Fh 3340 4926 a(!#) p Fj 1016 5449 a(\002) p Fk 1115 5325 a(k) p Fe 2 w(\000) p Fi(1) p Fh 1117 5355 a(X) p Fk 1125 5569 a(l) p Fi(=0) p Fh 1278 5309 a(\022) p Fl 1363 5382 a(R) p Fi 1440 5346 a(\() p Fk(s) p Fi(\)) p 1363 5426 V Fl 1363 5542 a(R) p Fi 1440 5491 a(\() p Fk(s) p Fi(\)) p Fe 1439 5556 a(\003) p Fh 1557 5309 a(\023) p Fk 1630 5318 a(l) p Fm 3714 5271 a(\(5) p Fl(:) p Fm(27\)) 1807 5778 y(20.03.02) p 90 rotate dyy eop %%Page: 20 20 20 19 bop Fm 0 60 a(20) p Fg 1660 w(D.) 33 b(Bam) m(busi) p Fh 373 268 a(\015) 373 327 y(\015) 373 387 y(\015) p Fl 429 382 a(X) p Fe 512 422 a(R) p Fb 580 385 a(\() p Fd(k) p Fb 1 w(\)) p Fd 580 448 a(T) p Fh 682 268 a(\015) 682 327 y(\015) 682 387 y(\015) p Fk 737 292 a(R) p Fb 797 256 a(\() p Fd(s) p Fb(\)) p Fd 797 319 a(k) p Fb 1 w(+1) p Fk 737 453 a(s) p Fj 959 382 a(\024) p Fh 1064 242 a(\022) p Fl 1149 315 a(R) p Fi 1226 279 a(\() p Fk(s) p Fi(\)) p 1149 359 182 4 v Fl 1149 475 a(R) p Fi 1226 424 a(\() p Fk(s) p Fi(\)) p Fe 1225 489 a(\003) p Fh 1342 242 a(\023) p Fk 1416 251 a(r) p Fe 2 w(\000) p Fi(1) p Fm 1578 382 a(\006) p Fk 1650 397 a(s) p Fm 1693 382 a(2) p Fk 1743 341 a(r) p Fe 2 w(\000) p Fi(2) p Fl 1889 382 a(r) s(AR) p Fi 2089 341 a(2) p Fe 2088 407 a(\003) p Fh 2149 212 a(") p Fk 2208 258 a(k) p Fe 2 w(\000) p Fi(1) p Fh 2217 288 a(Y) p Fk 2217 502 a(l) p Fi(=0) p Fh 2370 212 a( ) p Fm 2449 382 a(1) 22 b(+) 2633 315 y(3) p Fl(\032) p Fk 2735 330 a(s) p Fm 2776 315 a(\006) p Fk 2848 330 a(s) p 2633 359 259 4 v Fm 2667 451 a(16e) p Fi 2811 422 a(2) p Fh 2920 242 a(\022) p Fl 3005 315 a(R) p Fi 3082 279 a(\() p Fk(s) p Fi(\)) p 3005 359 182 4 v Fl 3005 475 a(R) p Fi 3082 424 a(\() p Fk(s) p Fi(\)) p Fe 3081 489 a(\003) p Fh 3198 242 a(\023) p Fk 3271 251 a(l) p Fi(+1) p Fh 3403 212 a(!#) p Fj 953 735 a(\002) p Fk 1053 610 a(k) p Fe 2 w(\000) p Fi(1) p Fh 1054 640 a(X) p Fk 1063 855 a(l) p Fi(=0) p Fm 1243 667 a(1) p 1227 712 81 4 v 1227 803 a(2) p Fk 1277 774 a(l) p Fl 1353 735 a(:) p Fm 3714 557 a(\(5) p Fl(:) p Fm(28\)) p Fg 0 1008 a(furthermore) 36 b(if) f(the) h(v) m(ector) f(\014elds) h(of) p Fl 35 w(h) p Fi 1551 1023 a(0) p Fg 1632 1008 a(and) f(of) p Fl 36 w(f) p Fg 46 w(lea) m(v) m(e) g(in) m(v) -6 b(arian) m(t) p Fj 35 w(A) p Fg 35 w(and/or) p Fj 35 w(R) p Fl(e) p Fg(,) 37 b(then) e(the) h(same) 0 1128 y(is) d(true) h(for) p Fj 33 w(T) p Fi 543 1092 a(\() p Fk(k) p Fi 2 w(\)) p Fg 687 1128 a(and) g(for) g(the) f(Hamiltonian) e(v) m(ector) i (\014elds) h(of) g(all) e(the) i(terms) e(of) i(\(5.22\),) p Fn 0 1536 a(Remark) 233 b(5.16.) p Fg 51 w(In) 51 b(particular) p Fl 50 w(X) p Fe 1618 1576 a(R) p Fb 1686 1540 a(\() p Fd(k) p Fb 1 w(\)) p Fd 1686 1602 a(N) p Fg 1839 1536 a(is) g(of) g(order) p Fm 51 w([) p Fl(R) p Fi 2471 1500 a(\() p Fk(s) p Fi(\)) p Fm 2575 1536 a(]) p Fi 2603 1500 a(2) p Fl 2647 1536 a(=) -6 b(N) p Fk 2782 1500 a(s) p Fe(\000) p Fi(1) p Fg 2977 1536 a(and) p Fl 52 w(X) p Fe 3272 1576 a(R) p Fb 3340 1540 a(\() p Fd(k) p Fb 1 w(\)) p Fd 3340 1602 a(T) p Fg 3493 1536 a(is) 50 b(of) h(order) p Fm 0 1716 a(\() p Fl(R) p Fi 116 1680 a(\() p Fk(s) p Fi(\)) p Fl 220 1716 a(=R) p Fi 347 1664 a(\() p Fk(s) p Fi(\)) p Fe 346 1730 a(\003) p Fm 451 1716 a(\)) p Fk 490 1680 a(r) p Fe 2 w(\000) p Fi(1) p Fg 636 1716 a(.) p Fn 0 1875 a(Pro) s(of.) p Fm 35 w(W) -8 b(e) 35 b(pro) s(ceed) g(b) m(y) f(induction.) 47 b(First) 34 b(w) m(e) g(split) p Fl 34 w(f) p Fi 2183 1839 a(\() p Fk(k) p Fi 2 w(\)) p Fm 2328 1875 a(\() p Fl(f) p Fm 44 w(in) g(the) h(case) p Fl 35 w(k) p Fm 32 w(=) 29 b(0\)) 34 b(in) m(to) g(an) g(e\013ectiv) m(e) 0 1995 y(part) f(and) h(a) f(remainder.) 44 b(W) -8 b(rite) p Fl 1601 2128 a(f) p Fi 1661 2087 a(\() p Fk(k) p Fi 2 w(\)) p Fm 1799 2128 a(=) p Fl 28 w(f) p Fi 1964 2077 a(\() p Fk(k) p Fi 2 w(\)) 1953 2155 y(0) p Fm 2097 2128 a(+) p Fl 23 w(f) p Fi 2257 2077 a(\() p Fk(k) p Fi 2 w(\)) p Fk 2246 2158 a(T) p Fm 0 2331 a(where) p Fl 43 w(f) p Fi 357 2279 a(\() p Fk(k) p Fi 2 w(\)) 346 2357 y(0) p Fm 511 2331 a(is) 42 b(the) h(the) g(T) -8 b(a) m(ylor) 42 b(p) s(olynomial) e(of) p Fl 43 w(f) p Fi 2012 2295 a(\() p Fk(k) p Fi 2 w(\)) p Fm 2165 2331 a(truncated) k(at) e(order) p Fl 43 w(r) p Fj 31 w(\000) p Fm 29 w(1) g(and) p Fl 43 w(f) p Fi 3567 2279 a(\() p Fk(k) p Fi 2 w(\)) p Fk 3556 2360 a(T) p Fm 3721 2331 a(is) g(the) 0 2474 y(remainder) 24 b(of) h(the) f(T) -8 b(a) m(ylor) 23 b(series) i(\(whic) m(h) g(b) s (egins) g(with) f(terms) f(of) i(order) p Fl 25 w(r) p Fm 3 w(\).) 40 b(Since) p Fl 25 w(f) p Fi 3214 2422 a(\() p Fk(k) p Fi 2 w(\)) 3203 2501 y(0) p Fm 3349 2474 a(is) 24 b(a) g(truncation) 0 2594 y(of) p Fl 33 w(f) p Fi 173 2558 a(\() p Fk(k) p Fi 2 w(\)) p Fm 284 2594 a(one) 34 b(has) p Fh 1403 2632 a(D) 1464 2628 y(\014) 1464 2688 y(\014) 1464 2748 y(\014) p Fl 1497 2743 a(f) p Fi 1557 2691 a(\() p Fk(k) p Fi 2 w(\)) 1546 2770 y(0) p Fh 1668 2628 a(\014) 1668 2688 y(\014) 1668 2748 y(\014) 1701 2632 y(E) p Fk 1762 2653 a(\026) 1762 2813 y(R) p Fd 1822 2824 a(k) p Fj 1915 2743 a(\024) p Fh 2020 2632 a(D) 2081 2628 y(\014) 2081 2688 y(\014) 2081 2748 y(\014) p Fl 2114 2743 a(f) p Fi 2174 2702 a(\() p Fk(k) p Fi 2 w(\)) p Fh 2285 2628 a(\014) 2285 2688 y(\014) 2285 2748 y(\014) 2318 2632 y(E) p Fk 2379 2653 a(\026) 2379 2813 y(R) p Fd 2439 2824 a(k) p Fl 2538 2743 a(:) p Fm 199 2997 a(Consider) p Fl 34 w(f) p Fi 675 2945 a(\() p Fk(k) p Fi 2 w(\)) 664 3023 y(0) p Fm 786 2997 a(,) f(and) h(its) f(T) -8 b(a) m(ylor) 31 b(series) p Ff 35 w(in) p Fm 35 w(\() p Fl(P) s(;) 17 b(Q) p Fm(\)) p Ff 34 w(only) p Fm(,) 33 b(w) m(e) h(write) p Fl 1601 3233 a(f) p Fi 1661 3181 a(\() p Fk(k) p Fi 2 w(\)) 1650 3260 y(0) p Fm 1799 3233 a(=) 1926 3207 y(~) p Fl 1904 3233 a(f) p Fi 1964 3192 a(\() p Fk(k) p Fi 2 w(\)) p Fm 2097 3233 a(+) p Fl 23 w(f) p Fi 2257 3181 a(\() p Fk(k) p Fi 2 w(\)) p Fk 2246 3262 a(N) p Fm 0 3461 a(where) 321 3435 y(~) p Fl 300 3461 a(f) p Fi 360 3425 a(\() p Fk(k) p Fi 2 w(\)) p Fm 515 3461 a(is) 45 b(the) g(truncation) g(of) g(suc) m(h) i(a) d (series) i(at) e(second) i(order) g(\(it) e(con) m(tains) h(at) f(most) g(terms) 0 3600 y(quadratic) 33 b(in) g(\() p Fl(P) s(;) 17 b(Q) p Fm(\)\)) 31 b(and) p Fl 34 w(f) p Fi 1155 3548 a(\() p Fk(k) p Fi 2 w(\)) p Fk 1144 3629 a(N) p Fm 1299 3600 a(is) i(the) h(remainder) f(of) g(the) h(expansion.) 44 b(One) 35 b(has) p Fh 781 3744 a(D) 842 3740 y(\014) 842 3800 y(\014) 842 3860 y(\014) p Fl 875 3855 a(f) p Fi 935 3803 a(\() p Fk(k) p Fi 2 w(\)) p Fk 924 3884 a(N) p Fh 1046 3740 a(\014) 1046 3800 y(\014) 1046 3860 y(\014) 1079 3744 y(E) p Fk 1140 3765 a(\026) 1140 3925 y(R) p Fd 1200 3936 a(k) p Fj 1293 3855 a(\024) p Fh 1398 3744 a(D) 1459 3740 y(\014) 1459 3800 y(\014) 1459 3860 y(\014) p Fl 1492 3855 a(f) p Fi 1552 3814 a(\() p Fk(k) p Fi 2 w(\)) p Fh 1663 3740 a(\014) 1663 3800 y(\014) 1663 3860 y(\014) 1696 3744 y(E) p Fk 1757 3765 a(\026) 1757 3925 y(R) p Fd 1817 3936 a(k) p Fl 1915 3855 a(;) p Fh 2059 3744 a(D) 2120 3740 y(\014) 2120 3800 y(\014) 2120 3860 y(\014) p Fm 2175 3829 a(~) p Fl 2153 3855 a(f) p Fi 2213 3814 a(\() p Fk(k) p Fi 2 w(\)) p Fh 2324 3740 a(\014) 2324 3800 y(\014) 2324 3860 y(\014) 2357 3744 y(E) p Fk 2418 3765 a(\026) 2418 3925 y(R) p Fd 2478 3936 a(k) p Fj 2571 3855 a(\024) p Fh 2676 3744 a(D) 2737 3740 y(\014) 2737 3800 y(\014) 2737 3860 y(\014) p Fl 2770 3855 a(f) p Fi 2830 3814 a(\() p Fk(k) p Fi 2 w(\)) p Fh 2941 3740 a(\014) 2941 3800 y(\014) 2941 3860 y(\014) 2974 3744 y(E) p Fk 3035 3765 a(\026) 3035 3925 y(R) p Fd 3095 3936 a(k) p Fl 3160 3855 a(:) p Fm 199 4109 a(Rewrite) 1160 4204 y(~) p Fl 1134 4229 a(H) p Fi 1225 4188 a(\() p Fk(k) p Fi 2 w(\)) p Fm 1364 4229 a(:=) p Fl 28 w(h) p Fi 1554 4244 a(0) p Fm 1621 4229 a(+) p Fl 23 w(Z) p Fi 1796 4188 a(\() p Fk(k) p Fi 2 w(\)) p Fm 1929 4229 a(+) 2050 4202 y(~) p Fl 2029 4229 a(f) p Fi 2089 4188 a(\() p Fk(k) p Fi 2 w(\)) p Fm 2222 4229 a(+) 2347 4204 y(~) p Fj 2321 4229 a(R) p Fi 2405 4177 a(\() p Fk(k) p Fi 2 w(\)) p Fk 2405 4258 a(N) p Fm 2539 4229 a(+) 2664 4204 y(~) p Fj 2639 4229 a(R) p Fi 2723 4177 a(\() p Fk(k) p Fi 2 w(\)) p Fk 2723 4258 a(T) p Fm 0 4408 a(where) 1046 4502 y(~) p Fj 1021 4527 a(R) p Fi 1105 4475 a(\() p Fk(k) p Fi 2 w(\)) p Fk 1105 4556 a(N) p Fm 1244 4527 a(:=) p Fj 28 w(R) p Fi 1461 4475 a(\() p Fk(k) p Fi 2 w(\)) p Fk 1461 4556 a(N) p Fm 1595 4527 a(+) p Fl 22 w(f) p Fi 1754 4475 a(\() p Fk(k) p Fi 2 w(\)) p Fk 1743 4556 a(N) p Fl 1898 4527 a(;) p Fm 2068 4502 a(~) p Fj 2042 4527 a(R) p Fi 2126 4475 a(\() p Fk(k) p Fi 2 w(\)) p Fk 2126 4556 a(T) p Fm 2266 4527 a(:=) p Fj 28 w(R) p Fi 2483 4475 a(\() p Fk(k) p Fi 2 w(\)) p Fk 2483 4556 a(T) p Fm 2616 4527 a(+) p Fl 23 w(f) p Fi 2776 4475 a(\() p Fk(k) p Fi 2 w(\)) p Fk 2765 4556 a(T) p Fl 2920 4527 a(:) p Fm 199 4805 a(Consider) i(the) f(Lie) g(transform) p Fj 35 w(T) p Fk 1476 4820 a(k) p Fm 1561 4805 a(generated) h(b) m(y) e (a) h(function) p Fl 37 w(\037) p Fk 2699 4820 a(k) p Fm 2748 4805 a(,) g(and) g(use) h(it) e(to) g(tranform) g(the) 0 4925 y(main) d(part) h(of) h(the) f(Hamiltonian,) e(namely) p Fl 32 w(h) p Fi 1754 4940 a(0) p Fm 1821 4925 a(+) p Fl 23 w(Z) p Fi 1996 4889 a(\() p Fk(k) p Fi 2 w(\)) p Fm 2129 4925 a(+) 2250 4899 y(~) p Fl 2229 4925 a(f) p Fi 2289 4889 a(\() p Fk(k) p Fi 2 w(\)) p Fm 2400 4925 a(;) h(b) m(y) i(form) m(ulae) f(\(5.13\),) e(\(5.14\)) h(one) i(has) p Fh 783 5060 a(h) p Fl 830 5170 a(h) p Fi 887 5185 a(0) p Fm 954 5170 a(+) p Fl 23 w(Z) p Fi 1129 5129 a(\() p Fk(k) p Fi 2 w(\)) p Fm 1262 5170 a(+) 1383 5144 y(~) p Fl 1362 5170 a(f) p Fi 1422 5129 a(\() p Fk(k) p Fi 2 w(\)) p Fh 1533 5060 a(i) p Fj 1602 5170 a(\016) 22 b(T) p Fk 1728 5185 a(k) p Fm 1805 5170 a(=) p Fl 28 w(h) p Fi 1967 5185 a(0) p Fm 2034 5170 a(+) p Fl 23 w(Z) p Fi 2209 5129 a(\() p Fk(k) p Fi 2 w(\)) p Fm 2342 5170 a(+) p Fh 2442 5060 a(h) p Fj 2489 5170 a(f) p Fl(\037) p Fk 2601 5185 a(k) p Fl 2650 5170 a(;) 17 b(h) p Fi 2752 5185 a(0) p Fj 2796 5170 a(g) p Fm 22 w(+) 2989 5144 y(~) p Fl 2968 5170 a(f) p Fi 3028 5129 a(\() p Fk(k) p Fi 2 w(\)) p Fh 3139 5060 a(i) p Fm 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a(and) p Fl 38 w(j) p Fm 39 w(=) p Fj 34 w(f) p Fl(j) p Fk 1289 1971 a(i) p Fj 1322 1956 a(g) p Fk 1372 1971 a(i) p Fe(2) p Fa(Z) p Fm 1551 1956 a(are) 37 b(sequences) h(of) f (nonnegativ) m(e) g(in) m(tegers,) i(and) e(w) m(e) g(used) h(the) 0 2075 y(v) m(ector) 33 b(notation) p Fl 33 w(\030) p Fk 747 2039 a(j) p Fm 815 2075 a(=) p Fh 921 2001 a(Q) p Fk 1015 2105 a(i) p Fe(2) p Fa(Z) p Fl 1173 2075 a(\030) p Fk 1222 2028 a(j) p Fd 1255 2038 a(i) p Fk 1217 2103 a(i) p Fm 1291 2075 a(.) 44 b(Denote) 34 b(b) m(y) p Fj 33 w(RE) p Fm 37 w(:=) p Fj 27 w(f) p Fm(\() p Fl(l) r(;) 17 b(j) p Fm 6 w(\)) 43 b(:) p Fl 44 w(!) p Fj 25 w(\001) p Fm 22 w(\() p Fl(j) p Fj 28 w(\000) p Fl 22 w(l) p Fm 2 w(\)) 28 b(=) g(0) p Fj(g) p Fm -1 w(,) 33 b(and) h(de\014ne) p Fl 536 2433 a(Z) p Fk 604 2448 a(k) p Fm 653 2433 a(\() p Fl(\030) 5 b(;) 17 b(\021) p Fm 4 w(\)) 25 b(:=) p Fh 1124 2338 a(X) p Fi 1036 2556 a(\() p Fk(l;j) p Fi 4 w(\)) p Fe(2RE) p Fl 1372 2433 a(f) p Fi 1432 2381 a(\() p Fk(k) p Fi 2 w(\)) p Fk 1421 2463 a(lj) p Fl 1543 2433 a(\030) p Fk 1592 2391 a(l) p Fl 1622 2433 a(\021) p Fk 1675 2391 a(j) p Fl 1750 2433 a(;) 116 b(\037) p Fk 1956 2448 a(k) p Fm 2005 2433 a(\() p Fl(\030) 5 b(;) 17 b(\021) p Fm 4 w(\)) 26 b(:=) p Fh 2477 2338 a(X) p Fi 2389 2556 a(\() p Fk(l;j) p Fi 4 w(\)) p Fe(62RE) p Fl 2876 2350 a(f) p Fi 2936 2298 a(\() p Fk(k) p Fi 2 w(\)) p Fk 2925 2380 a(lj) p 2737 2410 450 4 v Fl 2737 2501 a(i!) p Fj 26 w(\001) p Fm 22 w(\() p Fl(j) p Fj 27 w(\000) p Fl 23 w(l) p Fm 2 w(\)) p Fl 3198 2433 a(\030) p Fk 3247 2391 a(j) p Fl 3288 2433 a(\021) p Fk 3341 2391 a(l) p Fl 3405 2433 a(;) p Fm 0 2788 a(then) p Fl 34 w(Z) p Fk 295 2803 a(k) p Fm 377 2788 a(is) 33 b(in) h(normal) d(form.) 44 b(The) 34 b(norms) e(of) i(these) g(functions) g(are) g(estimates) e(b) m(y) p Fj 868 3079 a(hj) p Fl(Z) p Fk 1003 3094 a(k) p Fj 1052 3079 a(ji) p Fk 1118 3028 a(\026) 1118 3108 y(R) p Fj 1228 3079 a(\024) p Fh 1333 2968 a(D) 1394 2964 y(\014) 1394 3024 y(\014) 1394 3084 y(\014) p Fm 1449 3052 a(~) p Fl 1427 3079 a(f) p Fi 1487 3037 a(\() p Fk(k) p Fi 2 w(\)) p Fh 1598 2964 a(\014) 1598 3024 y(\014) 1598 3084 y(\014) 1631 2968 y(E) p Fk 1692 2989 a(\026) 1692 3148 y(R) p Fl 1807 3079 a(;) p Fj 116 w(hj) p Fl(\037) p Fk 2080 3094 a(k) p Fj 2129 3079 a(ji) p Fk 2195 3028 a(\026) 2195 3108 y(R) p Fj 2305 3079 a(\024) p Fl 2422 3011 a(N) p Fk 2513 2975 a(\013) p 2422 3056 148 4 v Fl 2467 3147 a(\015) p Fh 2598 2968 a(D) 2659 2964 y(\014) 2659 3024 y(\014) 2659 3084 y(\014) p Fm 2714 3052 a(~) p Fl 2692 3079 a(f) p Fi 2752 3037 a(\() p Fk(k) p Fi 2 w(\)) p Fh 2863 2964 a(\014) 2863 3024 y(\014) 2863 3084 y(\014) 2896 2968 y(E) p Fk 2957 2989 a(\026) 2957 3148 y(R) p Fl 3072 3079 a(;) p Fm 0 3395 a(as) c(one) g(can) h(easily) e (see) i(b) m(y) f(remarking) e(that) p Fl 28 w(Z) p Fk 1777 3410 a(k) p Fm 1854 3395 a(is) i(a) g(truncation) g(of) 2638 3368 y(~) p Fl 2617 3395 a(f) p Fi 2677 3358 a(\() p Fk(k) p Fi 2 w(\)) p Fm 2816 3395 a(and) g(that) g(the) g(denominators) 0 3514 y(in) 33 b(the) h(de\014nition) g(of) p Fl 33 w(\037) p Fm 34 w(are) f(larger) g(than) p Fl 33 w(\015) 6 b(=) -6 b(N) p Fk 1814 3478 a(\013) p Fm 1870 3514 a(.) 44 b(Then,) 34 b(using) g(the) f(inductiv) m(e) h(h) m(yp) s(othesis,) g (one) f(has) p Fj 721 3843 a(hj) p Fl -1 w(Z) p Fk 855 3858 a(k) p Fj 904 3843 a(ji) p Fk 971 3793 a(\026) 971 3873 y(R) p Fd 1031 3884 a(k) p Fj 1124 3843 a(\024) p Fl 28 w(AR) p Fi 1381 3802 a(3) p Fh 1441 3702 a(\022) p Fl 1549 3775 a(R) p 1527 3820 122 4 v 1527 3911 a(R) p Fe 1603 3926 a(\003) p Fh 1660 3702 a(\023) p Fk 1733 3723 a(k) p Fl 1832 3843 a(;) p Fj 116 w(hj) p Fl -1 w(\037) p Fk 2104 3858 a(k) p Fj 2154 3843 a(j) o(i) p Fk 2220 3793 a(\026) 2220 3873 y(R) p Fd 2280 3884 a(k) p Fj 2373 3843 a(\024) p Fl 2490 3775 a(AN) p Fk 2656 3739 a(\013) p 2490 3820 223 4 v Fl 2573 3911 a(\015) 2725 3843 y(R) p Fi 2802 3802 a(3) p Fh 2862 3702 a(\022) p Fl 2970 3775 a(R) p 2948 3820 122 4 v 2948 3911 a(R) p Fe 3024 3926 a(\003) p Fh 3081 3702 a(\023) p Fk 3154 3723 a(k) p Fl 3220 3843 a(:) p Fm 466 w(\(5) p Fl(:) p Fm(31\)) 199 4163 y(De\014ne) p Fj 41 w(T) p Fi 593 4127 a(\() p Fk(k) p Fi 2 w(+1\)) p Fm 843 4163 a(:=) p Fj 37 w(T) p Fi 1065 4127 a(\() p Fk(k) p Fi 2 w(\)) p Fj 1202 4163 a(\016) 26 b(T) p Fk 1332 4178 a(k) p Fm 1420 4163 a(and) 40 b(remark) e(that,) i(b) m(y) f(the) g(standard) h (theory) f(of) g(systems) g(with) 0 4283 y(symmetry) -8 b(,) 49 b(if) p Fj 49 w(T) p Fi 691 4247 a(\() p Fk(k) p Fi 2 w(\)) p Fm 851 4283 a(and) g(the) g(Hamiltonian) d(v) m(ector) i (\014eld) i(of) e(the) h(di\013eren) m(t) h(parts) f(of) 3531 4257 y(~) p Fl 3505 4283 a(H) p Fi 3596 4247 a(\() p Fk(k) p Fi 2 w(\)) p Fm 3755 4283 a(lea) m(v) m(e) 0 4402 y(in) m(v) -6 b(arian) m(t) 33 b(the) h(manifold) p Fj 32 w(A) p Fm 33 w(and/or) p Fj 33 w(R) p Fl(e) p Fm(,) g(then) g (the) f(same) g(is) g(true) h(for) p Fj 33 w(T) p Fi 2830 4366 a(\() p Fk(k) p Fi 2 w(+1\)) p Fm 3042 4402 a(.) 199 4524 y(Then) f(de\014ne) p Fl 34 w(Z) p Fi 820 4488 a(\() p Fk(k) p Fi 2 w(+1\)) p Fm 1060 4524 a(:=) p Fl 28 w(Z) p Fi 1268 4488 a(\() p Fk(k) p Fi 2 w(\)) p Fm 1400 4524 a(+) p Fl 20 w(Z) p Fk 1565 4539 a(k) p Fm 1614 4524 a(,) f(and) p Fl 33 w(f) p Fi 1927 4488 a(\() p Fk(k) p Fi 2 w(+1\)) p Fm 2171 4524 a(as) g(the) h(curly) f (brac) m(k) m(et) h(of) f(\(5.29\).) 42 b(F) -8 b(or) p Fl 32 w(Z) p Fi 3756 4488 a(\() p Fk(k) p Fi 2 w(+1\)) p Fm 0 4643 a(clearly) 38 b(\(5.23\)) f(holds) h(with) p Fl 38 w(k) p Fm 29 w(+) 26 b(1) 38 b(in) g(place) h(of) p Fl 39 w(k) p Fm 3 w(.) 59 b(W) -8 b(e) 39 b(no) m(w) f(estimate) p Fl 37 w(f) p Fi 2881 4607 a(\() p Fk(k) p Fi 2 w(+1\)) p Fm 3093 4643 a(.) 59 b(T) -8 b(o) 38 b(this) h(end) g(let) f(us) 0 4763 y(denote) p Fl 1627 4927 a(\017) p Fm 28 w(:=) 1841 4860 y(e) p Fi 1885 4824 a(2) p 1839 4904 93 4 v Fl 1839 4996 a(\016) p Fi 1887 4967 a(2) p Fj 1961 4927 a(h) o(j) p Fl(\037) p Fj(ji) p Fk 2156 4877 a(\026) 2156 4957 y(R) p Fd 2216 4968 a(k) p Fl 2314 4927 a(;) p Fm 0 5156 a(and) c(notice) f (that,) g(b) m(y) g(\(5.31\),) e(and) j(the) g(de\014nition) g(of) p Fl 33 w(R) p Fe 2182 5171 a(\003) p Fm 2227 5156 a(,) f(w) m(e) h(ha) m (v) m(e) p Fl 1342 5484 a(\017) p Fj 28 w(\024) p Fm 1527 5417 a(1) p 1527 5462 50 4 v 1527 5553 a(4) p Fh 1605 5344 a(\022) p Fl 1713 5417 a(R) p 1691 5462 122 4 v 1691 5553 a(R) p Fe 1767 5568 a(\003) p Fh 1824 5344 a(\023) p Fk 1897 5365 a(k) p Fi 2 w(+1) p Fl 2075 5484 a(<) p Fm 28 w(1) p Fl(=) p Fm(2) p Fl 33 w(;) p Fj 116 w(8) p Fl 10 w(k) p Fm 1807 5778 a(20.03.02) p 90 rotate dyy eop %%Page: 22 22 22 21 bop Fm 0 60 a(22) p Fg 1660 w(D.) 33 b(Bam) m(busi) p Fm 0 299 a(So,) g(using) h(the) f(estimate) f(\(5.19\),) g(from) g (lemma) f(5.12,) h(w) m(e) i(ha) m(v) m(e) p Fh 709 477 a(*) 790 473 y(\014) 790 533 y(\014) 790 592 y(\014) 790 652 y(\014) 790 712 y(\014) p Fe 855 523 a(1) p Fh 823 552 a(X) p Fk 831 767 a(l) p Fi(=1) p Fl 983 647 a(Z) p Fi 1058 595 a(\() p Fk(k) p Fi 2 w(\)) p Fk 1051 677 a(l) p Fh 1170 473 a(\014) 1170 533 y(\014) 1170 592 y(\014) 1170 652 y(\014) 1170 712 y(\014) 1203 477 y(+) p Fk 1283 497 a(\026) 1283 782 y(R) p Fd 1343 793 a(k) p Fb 1 w(+1) p Fj 1521 647 a(\024) p Fe 1658 523 a(1) p Fh 1626 552 a(X) p Fk 1635 767 a(l) p Fi(=1) p Fl 1787 647 a(\017) p Fk 1827 606 a(l) p Fh 1874 536 a(D) 1935 533 y(\014) 1935 592 y(\014) 1935 652 y(\014) p Fl 1968 647 a(Z) p Fi 2043 606 a(\() p Fk(k) p Fi 2 w(\)) p Fh 2155 533 a(\014) 2155 592 y(\014) 2155 652 y(\014) 2188 536 y(E) p Fk 2249 557 a(\026) 2249 717 y(R) p Fd 2309 728 a(k) p Fm 2402 647 a(=) p Fl 2605 580 a(\017) p 2519 624 213 4 v Fm 2519 715 a(1) p Fj 22 w(\000) p Fl 22 w(\017) p Fh 2759 536 a(D) 2820 533 y(\014) 2820 592 y(\014) 2820 652 y(\014) p Fl 2854 647 a(Z) p Fi 2929 606 a(\() p Fk(k) p Fi 2 w(\)) p Fh 3040 533 a(\014) 3040 592 y(\014) 3040 652 y(\014) 3073 536 y(E) p Fk 3134 557 a(\026) 3134 717 y(R) p Fd 3194 728 a(k) p Fj 1521 927 a(\024) p Fm 28 w(4) p Fl(\017AR) p Fi 1868 886 a(1+1) p Fm 0 1174 a(and) g(similarly) p Fh 1173 1227 a(*) 1253 1223 y(\014) 1253 1283 y(\014) 1253 1343 y(\014) 1253 1402 y(\014) 1253 1462 y(\014) p Fe 1319 1273 a(1) p Fh 1286 1303 a(X) p Fk 1295 1517 a(l) p Fi(=1) p Fm 1468 1371 a(~) p Fl 1447 1397 a(f) p Fi 1507 1346 a(\() p Fk(k) p Fi 2 w(\)) p Fk 1496 1427 a(l) p Fh 1618 1223 a(\014) 1618 1283 y(\014) 1618 1343 y(\014) 1618 1402 y(\014) 1618 1462 y(\014) 1651 1227 y(+) p Fk 1731 1248 a(\026) 1731 1532 y(R) p Fd 1791 1543 a(k) p Fb 1 w(+1) p Fj 1969 1397 a(\024) p Fm 28 w(2) p Fl(AR) p Fi 2276 1356 a(2) p Fl 2320 1397 a(\017) p Fh 2377 1257 a(\022) p Fl 2485 1330 a(R) p 2462 1374 122 4 v 2462 1466 a(R) p Fe 2538 1481 a(\003) p Fh 2596 1257 a(\023) p Fk 2669 1278 a(k) p Fl 2768 1397 a(:) p Fm 0 1723 a(Finally) -8 b(,) 38 b(ev) m(en) i(if) p Fl 39 w(h) p Fi 752 1738 a(0) p Fm 835 1723 a(is) f(not) g(of) g(class) p Fj 39 w(M) p Fk 1602 1675 a(\026) 1602 1752 y(R) p Fd 1662 1763 a(k) p Fm 1710 1723 a(,) h(all) e(the) h(terms) f(of) h(the) g (sequence) p Fl 40 w(h) p Fi 3153 1738 a(0) p Fk(l) p Fm 3263 1723 a(generated) h(b) m(y) p Fl 38 w(h) p Fi 3923 1738 a(0) p Fm 0 1842 a(turn) 34 b(out) f(to) g(b) s(e) g(of) h (class) p Fj 33 w(M) p Fk 1123 1794 a(\026) 1123 1871 y(R) p Fd 1183 1882 a(k) p Fb 1 w(+1) p Fm 1316 1842 a(.) 44 b(Indeed,) 35 b(remarking) d(that) p Fl 1222 2127 a(h) p Fi 1279 2142 a(01) p Fm 1392 2127 a(=) p Fj 28 w(f) p Fl(\037) p Fk 1609 2142 a(k) p Fl 1658 2127 a(;) 17 b(h) p Fi 1760 2142 a(0) p Fj 1804 2127 a(g) p Fm 28 w(=) 2008 2101 y(~) p Fl 1987 2127 a(f) p Fi 2047 2086 a(\() p Fk(k) p Fi 2 w(\)) p Fj 2180 2127 a(\000) p Fl 22 w(Z) p Fk 2347 2142 a(k) p Fj 2424 2127 a(2) 28 b(M) p Fk 2638 2079 a(\026) 2638 2157 y(R) p Fd 2698 2168 a(k) p Fm 0 2380 a(and) 34 b(pro) s(ceeding) g(as) f(in) g(the) h (pro) s(of) f(of) h(lemma) d(5.12) h(one) i(gets) p Fj 1110 2663 a(hj) p Fl(h) p Fi 1234 2678 a(0) p Fk 5 w(l) p Fj 1309 2663 a(ji) p Fk 1376 2613 a(\026) 1376 2693 y(R) p Fd 1436 2704 a(k) p Fb 1 w(+1) p Fj 1614 2663 a(\024) p Fl 28 w(\017) p Fk 1759 2622 a(r) p Fe 2 w(\000) p Fi(1) p Fh 1922 2553 a(D) 1983 2549 y(\014) 1983 2609 y(\014) 1983 2668 y(\014) p Fm 2037 2637 a(~) p Fl 2016 2663 a(f) p Fi 2076 2622 a(\() p Fk(k) p Fi 2 w(\)) p Fh 2187 2549 a(\014) 2187 2609 y(\014) 2187 2668 y(\014) 2220 2553 y(E) p Fk 2281 2573 a(\026) 2281 2733 y(R) p Fd 2341 2744 a(k) p Fl 2439 2663 a(;) 116 b(l) p Fj 29 w(\025) p Fm 29 w(1) p Fl 33 w(;) p Fm 0 2956 a(and) 34 b(therefore) p Fh 1182 2990 a(*) 1262 2986 y(\014) 1262 3046 y(\014) 1262 3105 y(\014) 1262 3165 y(\014) 1262 3225 y(\014) p Fe 1328 3036 a(1) p Fh 1296 3065 a(X) p Fk 1304 3280 a(l) p Fi(=2) p Fl 1456 3160 a(h) p Fi 1513 3175 a(0) p Fk 5 w(k) p Fh 1607 2986 a(\014) 1607 3046 y(\014) 1607 3105 y(\014) 1607 3165 y(\014) 1607 3225 y(\014) 1640 2990 y(+) p Fk 1721 3010 a(\026) 1721 3295 y(R) p Fd 1781 3306 a(k) p Fb 1 w(+1) p Fj 1959 3160 a(\024) p Fm 28 w(2) p Fl(AR) p Fi 2266 3119 a(1+1) p Fl 2410 3160 a(\017) p Fh 2467 3020 a(\022) p Fl 2575 3093 a(R) p 2553 3137 V 2553 3228 a(R) p Fe 2629 3243 a(\003) p Fh 2686 3020 a(\023) p Fk 2759 3040 a(k) p Fm 2858 3160 a(;) 0 3496 y(Summing) j(up) i(the) g(di\013eren) m(t) h(con) m(tributions) f(to) p Fl 38 w(f) p Fi 1972 3460 a(\() p Fk(k) p Fi 2 w(+1\)) p Fm 2223 3496 a(and) g(using) g(the) g(de\014nition) g(of) p Fl 39 w(\017) p Fm(,) h(w) m(e) f(obtain) 0 3615 y(the) 34 b(second) g(of) g(\(5.23\)) d(with) p Fl 33 w(k) p Fm 25 w(+) 23 b(1) 33 b(in) g(place) h(of) p Fl 33 w(k) p Fm 3 w(.) 199 3740 y(W) -8 b(e) 34 b(come) e(to) h(item) f(2\).) 44 b(First) 33 b(w) m(e) g(de\014ne) p Fj 1046 4012 a(R) p Fi 1130 3960 a(\() p Fk(k) p Fi 2 w(+1\)) p Fk 1130 4042 a(N) p Fm 1370 4012 a(:=) 1528 3987 y(~) p Fj 1503 4012 a(R) p Fi 1587 3960 a(\() p Fk(k) p Fi 2 w(\)) p Fk 1587 4042 a(N) p Fj 1720 4012 a(\016) 22 b(T) p Fk 1846 4027 a(k) p Fl 1929 4012 a(;) p Fj 116 w(R) p Fi 2157 3960 a(\() p Fk(k) p Fi 2 w(+1\)) p Fk 2157 4042 a(T) p Fm 2397 4012 a(:=) 2556 3987 y(~) p Fj 2530 4012 a(R) p Fi 2614 3960 a(\() p Fk(k) p Fi 2 w(\)) p Fk 2614 4042 a(T) p Fj 2748 4012 a(\016) g(T) p Fk 2874 4027 a(k) p Fm 0 4265 a(and) 34 b(remark) e(that,) g(b) m(y) i(prop) s(osition) e (5.3,) h(one) g(has) p Fj 355 4608 a(k) p Fl(X) p Fk 488 4623 a(\037) p Fd 538 4634 a(k) p Fj 586 4608 a(k) p Fk 635 4558 a(R) p Fb 695 4521 a(\() p Fd(s) p Fb(\)) p Fd 695 4584 a(k) p Fb 1 w(+1) p Fk 635 4641 a(s) p Fj 857 4608 a(\024) p Fm 1040 4541 a(\006) p Fk 1112 4556 a(s) p 974 4585 247 4 v Fl 974 4678 a(\032) p Fk 1026 4693 a(s) p Fl 1068 4678 a(\016) p Fi 1116 4649 a(\() p Fk(s) p Fi(\)) p Fl 1245 4541 a(AN) p Fk 1411 4504 a(\013) p 1245 4585 223 4 v Fl 1327 4676 a(\015) p Fm 1479 4608 a(\() p Fl(\032) p Fk 1570 4623 a(s) p Fl 1612 4608 a(R) p Fi 1689 4567 a(\() p Fk(s) p Fi(\)) p Fm 1793 4608 a(\)) p Fi 1832 4567 a(3) p Fh 1893 4467 a(\022) p Fl 1978 4541 a(R) p Fi 2055 4504 a(\() p Fk(s) p Fi(\)) p 1978 4585 182 4 v Fl 1978 4701 a(R) p Fi 2055 4649 a(\() p Fk(s) p Fi(\)) p Fe 2054 4715 a(\003) p Fh 2172 4467 a(\023) p Fk 2245 4477 a(k) p Fm 2322 4608 a(=) p Fl 2439 4541 a(R) p Fi 2516 4504 a(\() p Fk(s) p Fi(\)) p Fl 2620 4541 a(\032) p Fk 2672 4556 a(s) p Fm 2714 4541 a(\006) p Fk 2786 4556 a(s) p 2439 4585 390 4 v Fm 2516 4676 a(16e) p Fi 2660 4647 a(2) p Fl 2704 4676 a(r) p Fh 2857 4467 a(\022) p Fl 2943 4541 a(R) p Fi 3020 4504 a(\() p Fk(s) p Fi(\)) p 2943 4585 182 4 v Fl 2943 4701 a(R) p Fi 3020 4649 a(\() p Fk(s) p Fi(\)) p Fe 3019 4715 a(\003) p Fh 3136 4467 a(\023) p Fk 3209 4477 a(k) p Fi 2 w(+1) p Fm 3714 4608 a(\(5) p Fl(:) p Fm(32\)) 0 4933 y(where) 42 b(w) m(e) f(used) h(the) f(de\014nition) g(of) p Fl 41 w(R) p Fe 1511 4948 a(\003) p Fm 1556 4933 a(;) j(The) d(ab) s(o) m(v) m (e) g(expression,) i(b) m(y) e(lemma) e(5.13) g(and) i(b) m(y) g (\(5.24\),) 0 5053 y(ensures) 35 b(that) p Fj 1306 5186 a(T) p Fk 1360 5201 a(k) p Fm 1437 5186 a(:) p Fj 27 w(B) p Fk 1557 5201 a(s) p Fh 1617 5076 a(\020) p Fl 1676 5186 a(R) p Fi 1753 5135 a(\() p Fk(s) p Fi(\)) p Fk 1752 5217 a(k) p Fi 2 w(+2) p Fh 1902 5076 a(\021) p Fj 1989 5186 a(!) 27 b(B) p Fk 2181 5201 a(s) p Fh 2241 5076 a(\020) p Fl 2300 5186 a(R) p Fi 2377 5135 a(\() p Fk(s) p Fi(\)) p Fk 2376 5217 a(k) p Fi 2 w(+1) p Fh 2526 5076 a(\021) p Fl 2635 5186 a(;) p Fm 0 5420 a(whic) m(h) 40 b(together) g(with) f(the) h(inductiv) m(e) g(assumptions) g(ensures) h (the) f(v) -6 b(alidit) m(y) 39 b(of) h(the) f(estimate) g(\(5.26\)) 0 5539 y(with) p Fl 33 w(k) p Fm 25 w(+) 23 b(1) 33 b(in) g(place) h(of) p Fl 33 w(k) p Fm 3 w(.) 1807 5778 y(20.03.02) p 90 rotate dyy eop %%Page: 23 23 23 22 bop Fg 1222 60 a(Birkho\013) 33 b(normal) f(form) g(for) i(PDE's) 1222 b(23) p Fm 199 303 a(Estimate) 32 b(of) p Fj 33 w(R) p Fi 819 252 a(\() p Fk(k) p Fi 2 w(+1\)) p Fk 819 333 a(N) p Fm 1032 303 a(:) 44 b(T) -8 b(o) 33 b(b) s(egin) g(with) g (remark) f(that) h(one) g(has) p Fh 901 519 a(\015) 901 579 y(\015) 901 639 y(\015) p Fl 957 634 a(X) p Fi 1056 655 a(~) p Fk 1040 673 a(f) p Fb 1087 637 a(\() p Fd(k) p Fb 1 w(\)) p Fd 1079 699 a(N) p Fh 1188 519 a(\015) 1188 579 y(\015) 1188 639 y(\015) p Fk 1244 544 a(R) p Fb 1304 507 a(\() p Fd(s) p Fb(\)) p Fd 1304 570 a(k) p Fb 1 w(+1) p Fk 1244 704 a(s) p Fj 1465 634 a(\024) p Fl 1582 566 a(A) p Fm(\006) p Fk 1729 581 a(s) p Fm 1772 566 a(\() p Fl(\032) p Fk 1863 581 a(s) p Fl 1904 566 a(R) p Fi 1981 530 a(\() p Fk(s) p Fi(\)) p Fm 2086 566 a(\)) p Fi 2125 530 a(3) p 1582 611 587 4 v Fl 1752 704 a(\032) p Fk 1804 719 a(s) p Fl 1846 704 a(\016) p Fi 1894 675 a(\() p Fk(s) p Fi(\)) p Fh 2198 493 a(\022) p Fl 2283 566 a(R) p Fi 2360 530 a(\() p Fk(s) p Fi(\)) p 2283 611 182 4 v Fl 2283 727 a(R) p Fi 2360 675 a(\() p Fk(s) p Fi(\)) p Fe 2359 741 a(\003) p Fh 2476 493 a(\023) p Fk 2549 503 a(k) p Fm 2647 566 a(2) p Fk 2697 530 a(r) p Fi 2 w(+1) p 2627 611 236 4 v Fl 2627 702 a(N) p Fk 2718 673 a(s) p Fe(\000) p Fi(1) p Fm 1465 961 a(=) 28 b(2) p Fl(r) s(A) p Fm(\006) p Fk 1815 976 a(s) p Fh 1874 850 a(\020) p Fl 1933 961 a(\032) p Fk 1985 976 a(s) p Fl 2027 961 a(R) p Fi 2104 919 a(\() p Fk(s) p Fi(\)) p Fh 2209 850 a(\021) p Fi 2268 871 a(2) p Fh 2329 820 a(\022) p Fl 2415 893 a(R) p Fi 2492 857 a(\() p Fk(s) p Fi(\)) p 2415 938 182 4 v Fl 2415 1054 a(R) p Fi 2492 1002 a(\() p Fk(s) p Fi(\)) p Fe 2491 1068 a(\003) p Fh 2608 820 a(\023) p Fk 2681 829 a(k) p Fm 2779 893 a(2) p Fk 2829 857 a(r) p Fi 2 w(+1) p 2759 938 236 4 v Fl 2759 1029 a(N) p Fk 2850 1000 a(s) p Fe(\000) p Fi(1) p Fl 3039 961 a(;) p Fm 0 1259 a(and,) 33 b(from) g(lemma) e (5.14) h(one) i(has) p Fh 263 1465 a(\015) 263 1524 y(\015) 263 1584 y(\015) p Fl 319 1579 a(X) p Fk 402 1619 a(f) p Fb 449 1582 a(\() p Fd(k) p Fb 1 w(\)) p Fd 441 1645 a(N) p Fe 545 1619 a(\016T) p Fd 631 1630 a(k) p Fh 680 1465 a(\015) 680 1524 y(\015) 680 1584 y(\015) p Fk 735 1489 a(R) p Fb 795 1453 a(\() p Fd(s) p Fb(\)) p Fd 795 1516 a(k) p Fb 1 w(+2) p Fk 735 1650 a(s) p Fj 957 1579 a(\024) p Fh 1062 1409 a( ) p Fm 1141 1579 a(1) 22 b(+) 1324 1512 y(3) p Fl(\032) p Fk 1426 1527 a(s) p Fm 1468 1512 a(\006) p Fk 1540 1527 a(s) p 1324 1556 259 4 v Fm 1359 1648 a(16e) p Fi 1503 1619 a(2) p Fh 1611 1439 a(\022) p Fl 1696 1512 a(R) p Fi 1773 1476 a(\() p Fk(s) p Fi(\)) p 1696 1556 182 4 v Fl 1696 1672 a(R) p Fi 1773 1621 a(\() p Fk(s) p Fi(\)) p Fe 1772 1686 a(\003) p Fh 1890 1439 a(\023) p Fk 1963 1448 a(k) p Fi 2 w(+1) p Fh 2113 1409 a(!) p Fm 2208 1579 a(2) p Fl(r) s(A) p Fm(\006) p Fk 2453 1594 a(s) p Fh 2512 1469 a(\020) p Fl 2571 1579 a(\032) p Fk 2623 1594 a(s) p Fl 2665 1579 a(R) p Fi 2742 1538 a(\() p Fk(s) p Fi(\)) p Fh 2847 1469 a(\021) p Fi 2906 1489 a(2) p Fh 2967 1439 a(\022) p Fl 3053 1512 a(R) p Fi 3130 1476 a(\() p Fk(s) p Fi(\)) p 3053 1556 V Fl 3053 1672 a(R) p Fi 3130 1621 a(\() p Fk(s) p Fi(\)) p Fe 3129 1686 a(\003) p Fh 3246 1439 a(\023) p Fk 3319 1448 a(k) p Fm 3417 1512 a(2) p Fk 3467 1476 a(r) p Fi 2 w(+1) p 3397 1556 236 4 v Fl 3397 1648 a(N) p Fk 3488 1619 a(s) p Fe(\000) p Fi(1) p Fl 3677 1579 a(;) p Fm 0 1895 a(and) 34 b(also) p Fh 397 2081 a(\015) 397 2141 y(\015) 397 2201 y(\015) p Fl 452 2196 a(X) p Fe 535 2235 a(R) p Fb 603 2199 a(\() p Fd(k) p Fb 1 w(\)) p Fd 603 2261 a(N) p Fe 700 2235 a(\016T) p Fd 786 2246 a(k) p Fh 835 2081 a(\015) 835 2141 y(\015) 835 2201 y(\015) p Fk 890 2106 a(R) p Fb 950 2070 a(\() p Fd(s) p Fb(\)) p Fd 950 2133 a(k) p Fb 1 w(+2) p Fk 890 2266 a(s) p Fj 1112 2196 a(\024) p Fh 1217 2026 a( ) p Fm 1296 2196 a(1) 22 b(+) 1479 2129 y(3) p Fl(\032) p Fk 1581 2144 a(s) p Fm 1623 2129 a(\006) p Fk 1695 2144 a(s) p 1479 2173 259 4 v Fm 1514 2264 a(16e) p Fi 1658 2235 a(2) p Fh 1766 2055 a(\022) p Fl 1851 2129 a(R) p Fi 1928 2092 a(\() p Fk(s) p Fi(\)) p 1851 2173 182 4 v Fl 1851 2289 a(R) p Fi 1928 2237 a(\() p Fk(s) p Fi(\)) p Fe 1927 2303 a(\003) p Fh 2045 2055 a(\023) p Fk 2118 2065 a(k) p Fi 2 w(+1) p Fh 2268 2026 a(!) p Fm 2396 2129 a(2) p Fk 2446 2092 a(r) p Fi 2 w(+1) p 2375 2173 236 4 v Fl 2375 2264 a(N) p Fk 2466 2235 a(s) p Fe(\000) p Fi(1) p Fm 2623 2196 a(2) p Fl(r) s(A) p Fm(\006) p Fk 2868 2211 a(s) p Fj 1106 2543 a(\002) p Fm 23 w(\() p Fl(\032) p Fk 1297 2558 a(s) p Fl 1339 2543 a(R) p Fi 1416 2502 a(\() p Fk(s) p Fi(\)) p Fm 1520 2543 a(\)) p Fi 1559 2502 a(2) p Fh 1620 2373 a(") p Fk 1678 2419 a(k) p Fe 2 w(\000) p Fi(1) p Fh 1687 2449 a(Y) p Fk 1688 2663 a(l) p Fi(=0) p Fh 1841 2373 a( ) p Fm 1919 2543 a(1) g(+) 2103 2476 y(3) p Fl(\032) p Fk 2205 2491 a(s) p Fm 2247 2476 a(\006) p Fk 2319 2491 a(s) p 2103 2520 259 4 v Fm 2138 2612 a(16e) p Fi 2282 2583 a(2) p Fh 2390 2403 a(\022) p Fl 2475 2476 a(R) p Fi 2552 2440 a(\() p Fk(s) p Fi(\)) p 2475 2520 182 4 v Fl 2475 2636 a(R) p Fi 2552 2585 a(\() p Fk(s) p Fi(\)) p Fe 2551 2650 a(\003) p Fh 2668 2403 a(\023) p Fk 2742 2412 a(l) p Fi(+1) p Fh 2873 2373 a(!#) p Fk 3027 2419 a(k) p Fe 2 w(\000) p Fi(1) p Fh 3028 2449 a(X) p Fk 3037 2663 a(l) p Fi(=0) p Fh 3189 2403 a(\022) p Fl 3275 2476 a(R) p Fi 3352 2440 a(\() p Fk(s) p Fi(\)) p 3275 2520 V Fl 3275 2636 a(R) p Fi 3352 2585 a(\() p Fk(s) p Fi(\)) p Fe 3351 2650 a(\003) p Fh 3468 2403 a(\023) p Fk 3541 2412 a(l) p Fm 0 2869 a(whic) m(h) 34 b(sho) m(ws) g(that) f(the) h (estimate) e(\(5.27\)) f(holds) j(with) p Fl 33 w(k) p Fm 25 w(+) 23 b(1) 33 b(in) g(place) h(of) p Fl 33 w(k) p Fm 3 w(.) 199 3008 y(Estimate) f(of) h(\(5.28\):) 44 b(First) 34 b(w) m(e) h(estimate) e(the) p Fj 34 w(M) p Fk 2149 2960 a(\026) 2149 3038 y(R) p Fd 2209 3049 a(k) p Fm 2291 3008 a(norm) h(of) p Fl 34 w(f) p Fi 2727 2957 a(\() p Fk(k) p Fi 2 w(\)) p Fk 2716 3038 a(T) p Fm 2838 3008 a(.) 46 b(T) -8 b(o) 34 b(this) g(end) h(remark) e(that) p Fh 0 3071 a(\012) 47 3067 y(\014) 47 3127 y(\014) p Fl 80 3152 a(f) p Fi 140 3116 a(\() p Fk(k) p Fi 2 w(\)) p Fh 251 3067 a(\014) 251 3127 y(\014) 284 3071 y(\013) p Fk 331 3088 a(\026) 331 3192 y(R) p Fd 391 3203 a(k) p Fm 494 3152 a(is) 38 b(a) f(function) i(of) p Fl 38 w(R) p Fm 38 w(whic) m(h) f(is) g(analytic) e(and) i(has) g(all) f(the) h(T) -8 b(a) m(ylor) 37 b(co) s(e\016cien) m(ts) h(that) f(are) 0 3332 y(p) s(ositiv) m(e.) 44 b(Moreo) m(v) m(er,) 34 b(the) g(quan) m(tit) m(y) p Fh 1444 3222 a(D) 1505 3218 y(\014) 1505 3278 y(\014) 1505 3337 y(\014) p Fl 1538 3332 a(f) p Fi 1598 3281 a(\() p Fk(k) p Fi 2 w(\)) p Fk 1587 3362 a(T) p Fh 1709 3218 a(\014) 1709 3278 y(\014) 1709 3337 y(\014) 1742 3222 y(E) p Fk 1803 3242 a(\026) 1803 3402 y(R) p Fd 1863 3413 a(k) p Fm 1962 3332 a(is) f(the) h(sum) f(of) h (the) g(terms) f(of) g(order) h(higher) g(than) p Fl 34 w(r) p Fm 0 3492 a(in) f(suc) m(h) i(a) e(series.) 45 b(F) -8 b(rom) 32 b(this) h(one) g(easily) g(sees) h(that) p Fh 657 3672 a(D) 718 3668 y(\014) 718 3728 y(\014) 718 3788 y(\014) p Fl 751 3783 a(f) p Fi 811 3731 a(\() p Fk(k) p Fi 2 w(\)) p Fk 800 3812 a(T) p Fh 922 3668 a(\014) 922 3728 y(\014) 922 3788 y(\014) 955 3672 y(E) p Fk 1016 3693 a(\026) 1016 3852 y(R) p Fd 1076 3863 a(k) p Fj 1169 3783 a(\024) p Fh 1274 3642 a(\022) p Fm 1360 3715 a(2) p Fl(R) p 1360 3760 127 4 v 1362 3851 a(R) p Fe 1438 3866 a(\003) p Fh 1498 3642 a(\023) p Fk 1571 3663 a(r) p Fh 1632 3672 a(D) 1693 3668 y(\014) 1693 3728 y(\014) 1693 3788 y(\014) p Fl 1726 3783 a(f) p Fi 1786 3742 a(\() p Fk(k) p Fi 2 w(\)) p Fh 1897 3668 a(\014) 1897 3728 y(\014) 1897 3788 y(\014) 1930 3672 y(E) p Fk 1991 3693 a(\026) 1991 3852 y(R) p Fc 2051 3863 a(\003) p Fd(k) p Fk 2131 3852 a(=) p Fi(2) p Fj 2261 3783 a(\024) p Fm 28 w(2) p Fk 2416 3742 a(r) p Fe 2 w(\000) p Fi(3) p Fl 2562 3783 a(AR) p Fi 2714 3742 a(3) p Fe 2713 3807 a(\003) p Fh 2775 3642 a(\022) p Fl 2883 3715 a(R) p 2860 3760 122 4 v 2860 3851 a(R) p Fe 2936 3866 a(\003) p Fh 2993 3642 a(\023) p Fk 3067 3663 a(r) p Fm 3164 3715 a(1) p 3139 3760 99 4 v 3139 3851 a(2) p Fk 3189 3822 a(k) p Fl 3300 3780 a(:) p Fm 386 w(\(5) p Fl(:) p Fm(33\)) 0 4068 y(Then,) g(b) m(y) f(prop) s(osition) g(5.3) g(one) g(has) p Fh 971 4258 a(\015) 971 4318 y(\015) 971 4378 y(\015) p Fl 1026 4373 a(X) p Fk 1109 4413 a(f) p Fb 1156 4376 a(\() p Fd(k) p Fb 1 w(\)) p Fd 1148 4438 a(T) p Fh 1258 4258 a(\015) 1258 4318 y(\015) 1258 4378 y(\015) p Fk 1313 4283 a(R) p Fb 1373 4247 a(\() p Fd(s) p Fb(\)) p Fd 1373 4310 a(k) p Fb 1 w(+1) p Fk 1313 4443 a(s) p Fj 1535 4373 a(\024) p Fm 28 w(\006) p Fk 1712 4388 a(s) p Fm 1855 4306 a(2) p Fl(r) p 1766 4350 276 4 v 1766 4443 a(\032) p Fk 1818 4458 a(s) p Fl 1860 4443 a(R) p Fi 1937 4414 a(\() p Fk(s) p Fi(\)) p Fm 2054 4373 a(2) p Fk 2104 4332 a(r) p Fe 2 w(\000) p Fi(3) p Fl 2250 4373 a(AR) p Fi 2402 4332 a(3) p Fe 2401 4398 a(\003) p Fh 2462 4233 a(\022) p Fl 2548 4306 a(R) p Fi 2625 4269 a(\() p Fk(s) p Fi(\)) p 2548 4350 182 4 v Fl 2548 4466 a(R) p Fi 2625 4414 a(\() p Fk(s) p Fi(\)) p Fe 2624 4480 a(\003) p Fh 2741 4233 a(\023) p Fk 2814 4242 a(r) p Fm 2911 4306 a(1) p 2887 4350 99 4 v 2887 4441 a(2) p Fk 2937 4413 a(k) p Fm 1535 4696 a(=) 28 b(\006) p Fk 1712 4711 a(s) p Fm 1754 4696 a(2) p Fk 1804 4655 a(r) p Fe 2 w(\000) p Fi(2) p Fl 1950 4696 a(r) s(AR) p Fi 2150 4655 a(2) p Fe 2149 4721 a(\003) p Fh 2211 4556 a(\022) p Fl 2296 4629 a(R) p Fi 2373 4593 a(\() p Fk(s) p Fi(\)) p 2296 4673 182 4 v Fl 2296 4789 a(R) p Fi 2373 4738 a(\() p Fk(s) p Fi(\)) p Fe 2372 4803 a(\003) p Fh 2489 4556 a(\023) p Fk 2563 4565 a(r) p Fe 2 w(\000) p Fi(1) p Fm 2762 4629 a(1) p 2737 4673 99 4 v 2737 4765 a(2) p Fk 2787 4736 a(k) p Fl 2881 4696 a(:) p Fm 0 5029 a(Finally) -8 b(,) 31 b(pro) s(ceeding) k(as) e(in) g(the) h(estimate) e(of) p Fj 33 w(R) p Fi 1866 4977 a(\() p Fk(k) p Fi 2 w(\)) p Fk 1866 5058 a(N) p Fm 2011 5029 a(one) i(obtains) f(the) h(thesis.) p 3891 4955 78 4 v 3891 5025 4 70 v 3965 5025 V 3891 5029 78 4 v Fn 0 5188 a(Corollary) 177 b(5.17.) p Fg 39 w(Consider) 39 b(the) f(Hamiltonian) e(\(4.1\);) k(There) f(exists) f(a) g(constan) m (t) p Fl 39 w(C) p Fg 46 w(suc) m(h) i(that) 0 5308 y(for) 33 b(an) m(y) p Fl 34 w(s) p Fj 27 w(\025) p Fm 28 w(1) p Fg 33 w(and) h(an) m(y) p Fl 33 w(R) p Fi 1062 5272 a(\() p Fk(s) p Fi(\)) p Fg 1200 5308 a(suc) m(h) g(that) p Fl 1511 5539 a(R) p Fi 1588 5498 a(\() p Fk(s) p Fi(\)) p Fl 1720 5539 a(<) p Fm 29 w(\() p Fl(C) p Fm 7 w(2) p Fk 1993 5498 a(s) p Fl 2035 5539 a(N) p Fk 2126 5498 a(\013) p Fm 2183 5539 a(\006) p Fk 2255 5498 a(\013) 2255 5564 y(s) p Fm 2311 5539 a(\)) p Fe 2350 5498 a(\000) p Fi(1) p Fm 1807 5778 a(20.03.02) p 90 rotate dyy eop %%Page: 24 24 24 23 bop Fm 0 60 a(24) p Fg 1660 w(D.) 33 b(Bam) m(busi) 0 299 y(the) 26 b(follo) m(wing) f(holds) h(true:) 40 b(There) 26 b(exists) g(a) f(canonical) h(transformation) p Fj 24 w(T) p Fk 2853 314 a(r) p Fm 2925 299 a(:) p Fj 27 w(B) p Fk 3045 314 a(s) p Fm 3088 299 a(\() p Fl(R) p Fi 3204 263 a(\() p Fk(s) p Fi(\)) p Fl 3308 299 a(=) p Fm(3\)) p Fj 27 w(!) i(B) p Fk 3667 314 a(s) p Fm 3710 299 a(\() p Fl(R) p Fi 3826 263 a(\() p Fk(s) p Fi(\)) p Fm 3930 299 a(\)) p Fg 0 418 a(ful\014lling) p Fj 1142 538 a(k) p Fl(\020) p Fj 29 w(\000) 22 b(T) p Fk 1418 553 a(r) p Fm 1463 538 a(\() p Fl(\020) p Fm 7 w(\)) p Fj(k) p Fk 1641 568 a(s) p Fj 1711 538 a(\024) p Fl 28 w(C) p Fm 7 w(2) p Fk 1944 497 a(s) p Fm 1987 538 a(\006) p Fk 2059 553 a(s) p Fm 2102 538 a(\() p Fl(R) p Fi 2218 497 a(\() p Fk(s) p Fi(\)) p Fm 2322 538 a(2) p Fk 2372 497 a(s) p Fl 2414 538 a(N) p Fk 2505 497 a(\013) p Fm 2562 538 a(\)) p Fi 2601 497 a(2) p Fl 2645 538 a(R) p Fi 2722 497 a(\() p Fk(s) p Fi(\)) p Fg 0 721 a(suc) m(h) 35 b(that) e(the) g(transformed) h(Hamiltonian) c(has) k(the) g(form) p Fl 1351 947 a(H) p Fj 30 w(\016) 22 b(T) p Fk 1590 962 a(r) p Fm 1662 947 a(=) p Fl 28 w(h) p Fi 1824 962 a(0) p Fm 1892 947 a(+) p Fl 22 w(Z) p Fm 30 w(+) p Fj 22 w(R) p Fk 2272 962 a(T) p Fm 2358 947 a(+) p Fj 22 w(R) p Fk 2541 962 a(N) p Fg 0 1173 a(with) p Fl 33 w(Z) p Fg 40 w(in) 34 b(normal) d(form,) i(and) g(the) h(follo) m(wing) f(estimates) f(hold) p Fj 1186 1444 a(k) p Fl(X) p Fe 1319 1459 a(R) p Fd 1387 1469 a(N) p Fj 1456 1444 a(k) p Fk 1505 1394 a(R) p Fb 1565 1364 a(\() p Fd(s) p Fb(\)) p Fk 1658 1394 a(=) p Fi(3) p Fk 1505 1474 a(s) p Fj 1771 1444 a(\024) p Fm 1981 1377 a(1) p 1888 1421 236 4 v Fl 1888 1513 a(N) p Fk 1979 1484 a(s) p Fe(\000) p Fi(1) p Fl 2136 1444 a(C) p Fm 7 w(\006) p Fk 2286 1459 a(s) p Fh 2345 1334 a(\020) p Fm 2405 1444 a(2) p Fk 2455 1403 a(s) p Fl 2497 1444 a(R) p Fi 2574 1403 a(\() p Fk(s) p Fi(\)) p Fh 2678 1334 a(\021) p Fi 2738 1354 a(2) p Fj 1195 1687 a(k) p Fl(X) p Fe 1328 1702 a(R) p Fd 1396 1712 a(T) p Fj 1456 1687 a(k) p Fk 1505 1637 a(R) p Fb 1565 1607 a(\() p Fd(s) p Fb(\)) p Fk 1658 1637 a(=) p Fi(3) p Fk 1505 1717 a(s) p Fj 1771 1687 a(\024) p Fl 28 w(C) p Fm 7 w(\006) p Fk 2026 1702 a(s) p Fh 2086 1577 a(\020) p Fl 2145 1687 a(R) p Fi 2222 1646 a(\() p Fk(s) p Fi(\)) p Fm 2327 1687 a(2) p Fk 2377 1646 a(s) p Fl 2419 1687 a(N) p Fk 2510 1646 a(\013) p Fh 2567 1577 a(\021) p Fk 2626 1597 a(r) p Fe 2 w(\000) p Fi(1) p Fm 3714 1556 a(\(5) p Fl(:) p Fm(34\)) p Fn 0 2088 a(Pro) s(of.) p Fm 31 w(Apply) f(prop) s(osition) f(5.15) f(with) p Fl 30 w(k) p Fm 31 w(=) p Fl 28 w(k) p Fk 1829 2103 a(r) p Fm 1901 2088 a(=) f(\() p Fl(r) p Fj 19 w(\000) p Fm 17 w(3\),) i(then) h(estimate) e(the) i(v) m(ector) g(\014eld) g(of) p Fl 30 w(f) p Fi 3819 2051 a(\() p Fk(k) p Fd 3892 2061 a(r) p Fi 3932 2051 a(\)) p Fm 0 2227 a(b) m(y) h(using) h(\(5.7\),) d (and) j(de\014ne) p Fj 33 w(R) p Fk 1224 2242 a(N) p Fm 1328 2227 a(:=) p Fj 28 w(R) p Fi 1545 2175 a(\() p Fk(k) p Fd 1618 2185 a(r) p Fi 1659 2175 a(\)) p Fk 1545 2256 a(N) p Fm 1727 2227 a(and) p Fj 32 w(R) p Fk 2003 2242 a(T) p Fm 2094 2227 a(:=) p Fj 28 w(R) p Fi 2311 2175 a(\() p Fk(k) p Fd 2384 2185 a(r) p Fi 2425 2175 a(\)) p Fk 2311 2256 a(T) p Fm 2481 2227 a(+) p Fl 20 w(f) p Fi 2638 2190 a(\() p Fk(k) p Fd 2711 2200 a(r) p Fi 2751 2190 a(\)) p Fm 2787 2227 a(.) 44 b(Inserting) 32 b(the) h(de\014nition) f(of) p Fl 0 2370 a(R) p Fi 77 2318 a(\() p Fk(s) p Fi(\)) p Fe 76 2384 a(\003) p Fm 214 2370 a(and) i(inserting) g(all) e(irrelev) -6 b(an) m(t) 33 b(constan) m(ts) h(in) p Fl 34 w(C) p Fm 40 w(w) m(e) g(get) f(the) g (result.) p 3891 2296 78 4 v 3891 2366 4 70 v 3965 2366 V 3891 2370 78 4 v 199 2489 a(W) -8 b(e) 34 b(are) f(no) m(w) h(ready) f (for) g(the) p Fn 0 2609 a(Pro) s(of) 41 b(of) f(Theorem) f(4.1.) p Fm 48 w(In) c(corollary) e(5.17) h(w) m(e) h(ha) m(v) m(e) p Fl 35 w(N) p Fm 45 w(and) p Fl 35 w(s) p Fm 34 w(at) f(our) h(disp) s (osal.) 48 b(W) -8 b(e) 35 b(b) s(egin) f(b) m(y) 0 2728 y(c) m(ho) s(osing) p Fl 29 w(N) p Fm 38 w(:=) p Fl 28 w(R) p Fe 728 2692 a(\000) p Fi(1) p Fk(=) p Fi(2) p Fk(\013) p Fm 995 2728 a(so) 29 b(that) f(the) g(brac) m(k) m(et) h(in) f(the) h(second) h(of) e(\(5.34\)) f(b) s(ecomes) i(of) f(order) h([) p Fl(R) p Fi 3684 2692 a(\() p Fk(s) p Fi(\)) p Fm 3788 2728 a(]) p Fi 3816 2692 a(1) p Fk(=) p Fi(2) p Fm 3941 2728 a(.) 0 2848 y(Inserting) 34 b(in) f(\(5.34\)) f(w) m(e) h(get) p Fj 576 3144 a(k) p Fl(X) p Fe 709 3159 a(R) p Fd 777 3169 a(N) p Fm 868 3144 a(+) p Fl 22 w(X) p Fe 1050 3159 a(R) p Fd 1118 3169 a(T) p Fj 1178 3144 a(k) p Fk 1228 3094 a(R) p Fb 1288 3064 a(\() p Fd(s) p Fb(\)) p Fk 1381 3094 a(=) p Fi(3) p Fk 1228 3174 a(s) p Fj 1494 3144 a(\024) p Fl 28 w(C) p Fm 7 w(\006) p Fk 1749 3159 a(s) p Fm 1792 3144 a(\() p Fl(R) p Fi 1908 3103 a(\() p Fk(s) p Fi(\)) p Fm 2012 3144 a(\)) p Fb 2086 3076 a(1) p 2063 3087 80 4 v 2063 3127 a(2) p Fd(\013) p Fi 2154 3103 a(\() p Fk(s) p Fe(\000) p Fi(1\)) p Fm 2383 3144 a(+) p Fl 23 w(C) p Fm 7 w(\006) p Fk 2633 3159 a(s) p Fh 2692 3034 a(\020) p Fm 2752 3144 a(4) p Fk 2802 3103 a(s) p Fl 2844 3144 a(C) 7 b(R) p Fi 2999 3103 a(\() p Fk(s) p Fi(\)) p Fh 3104 3034 a(\021) p Fd 3175 3023 a(r) p Fc 2 w(\000) p Fb(1) p 3175 3039 124 4 v 3220 3078 a(2) p Fl 3365 3144 a(:) p Fm 0 3400 a(Finally) 38 b(c) m(ho) s(osing) p Fl 40 w(s) p Fj 37 w(\025) p Fl 38 w(s) p Fe 1002 3415 a(\003) p Fm 1085 3400 a(:=) g(4) p Fl(\013M) p Fm 50 w(with) p Fl 38 w(M) p Fm 49 w(=) g(\() p Fl(r) p Fj 28 w(\000) p Fm 27 w(1\)) p Fl(=) p Fm(2) g(one) i(has) g (that) e(the) i(\014rst) f(addendum) i(is) 0 3520 y(smaller) 32 b(than) h(the) h(second.) 45 b(T) -8 b(o) 33 b(obtain) g(the) h (statemen) m(t) e(use) i(also) f(eq.) 44 b(\(5.1\).) p 3891 3446 78 4 v 3891 3516 4 70 v 3965 3516 V 3891 3520 78 4 v Fn 0 4117 a(6.) 115 b(Application) 35 b(to) j(the) f(nonlinear) g (w) m(a) m(v) m(e) g(equation) p Fm 199 4386 a(W) -8 b(e) 33 b(split) f(this) h(section) f(in) h(t) m(w) m(o) f(sub) s (ections:) 45 b(in) 32 b(the) h(\014rst) g(w) m(e) g(will) e(sho) m(w) i (ho) m(w) g(to) f(apply) g(theorem) 0 4506 y(4.1) k(in) h(order) g(to) f (obtain) h(theorem) f(2.2,) h(and) g(in) g(the) g(second) h(w) m(e) f (will) f(sho) m(w) h(that) g(the) g(frequencies) h(of) 0 4626 y(the) c(system) e(w) m(e) i(study) f(are) h(strongly) e(non) m(v) -6 b(anishing) 34 b(for) g(most) e(v) -6 b(alues) 34 b(of) f(the) h(parameter) p Fl 32 w(m) p Fm(.) p Fg 0 4895 a(6.1) f(Application) f(of) i(theorem) e(4.1) h(to) g(equation) f (\(0.1\)) p Fm 199 5164 a(Consider) j(the) f(eigenfunctions) p Fl 35 w(') p Fk 1504 5179 a(j) p Fm 1580 5164 a(of) g(the) h (Sturm{Liouville) d(problem) h(\(2.1\)) f(and) j(their) f(F) -8 b(ourier) 0 5283 y(expansion) p Fl 1425 5444 a(') p Fk 1490 5459 a(j) p Fm 1532 5444 a(\() p Fl(x) p Fm(\)) 27 b(=) 1883 5377 y(1) p 1811 5421 194 4 v Fj 1811 5441 a(p) p 1894 5441 111 4 v Fm 1894 5523 a(2) p Fl(\031) p Fh 2037 5349 a(X) p Fk 2033 5564 a(k) p Fe 2 w(2) p Fa(Z) p Fl 2202 5444 a(') p Fk 2267 5396 a(j) 2267 5474 y(k) p Fm 2316 5444 a(e) p Fk 2360 5403 a(ik) r(x) p Fl 2516 5444 a(;) p Fm 1807 5778 a(20.03.02) p 90 rotate dyy eop %%Page: 25 25 25 24 bop Fg 1222 60 a(Birkho\013) 33 b(normal) f(form) g(for) i(PDE's) p Fm 1222 w(25) 0 299 y(it) 39 b(is) g(kno) m(wn) g([11,9]) e(that) i (one) h(has) p Fj 39 w(j) p Fl(') p Fk 1491 251 a(j) 1491 329 y(k) p Fj 1540 299 a(j) d(\024) p Fl 38 w(C) p Fm 7 w(e) p Fe 1842 263 a(\000) p Fk(\033) p Fe 3 w(jj) p Fk(k) p Fe 2 w(j\000) p Fk(j) p Fe 4 w(j) p Fm 2236 299 a(with) i(p) s(ositiv) m(e) p Fl 38 w(\033) i(<) d(\033) p Fi 3113 314 a(1) p Fm 3197 299 a(and) h(a) g(suitable) p Fl 39 w(C) p Fm 7 w(.) 0 433 y(Remark) 28 b(also) g(that) p Fl 28 w(') p Fk 842 385 a(j) p Fi 842 459 a(0) p Fm 914 433 a(=) g(0.) 43 b(W) -8 b(e) 28 b(w) m(an) m(t) h(to) f(in) m(tro) s (duce) i(the) e(analogue) h(for) f(our) h(problem) f(of) h(the) g(F) -8 b(ourier) 0 552 y(basis) 34 b(of) f(the) h(exp) s(onen) m(tials.) 44 b(So,) 33 b(w) m(e) h(de\014ne) p Fl 552 833 a( ) p Fk 617 848 a(j) p Fm 659 833 a(\() p Fl(x) p Fm(\)) 27 b(:=) 1037 766 y(1) p 966 810 194 4 v Fj 966 830 a(p) p 1049 830 111 4 v Fm 1049 913 a(2) p Fl(\031) p Fh 1188 739 a(X) p Fk 1187 953 a(k) r(>) p Fi(0) p Fl 1350 833 a(') p Fk 1415 785 a(j) 1415 863 y(k) p Fm 1464 833 a(e) p Fk 1508 792 a(ik) r(x) p Fl 1664 833 a(;) 116 b( ) p Fe 1873 848 a(\000) p Fk(j) p Fm 1977 833 a(\() p Fl(x) p Fm(\)) 27 b(:=) 2356 766 y(1) p 2284 810 194 4 v Fj 2284 830 a(p) p 2367 830 111 4 v Fm 2367 913 a(2) p Fl(\031) p Fh 2507 739 a(X) p Fk 2506 953 a(k) r(<) p Fi(0) p Fl 2668 833 a(') p Fk 2733 785 a(j) 2733 863 y(k) p Fm 2783 833 a(e) p Fk 2827 792 a(ik) r(x) p Fl 2982 833 a(;) 116 b(j) 34 b(>) p Fm 28 w(0) f(;) 0 1167 y(considering) h(the) g(F) -8 b(ourier) 33 b(expansion) g(of) h(the) p Fl 34 w( ) p Fk 1851 1182 a(j) p Fm 1892 1167 a('s,) f(one) h(has) p Fl 1001 1457 a( ) p Fk 1066 1472 a(j) p Fm 1108 1457 a(\() p Fl(x) p Fm(\)) 27 b(=) 1459 1390 y(1) p 1387 1434 194 4 v Fj 1387 1454 a(p) p 1470 1454 111 4 v Fm 1470 1536 a(2) p Fl(\031) p Fh 1613 1362 a(X) p Fk 1609 1586 a(k) p Fe 2 w(2) p Fi 1715 1568 a(\026) p Fa 1707 1586 a(Z) p Fl 1778 1457 a( ) p Fk 1847 1409 a(j) 1843 1487 y(k) p Fm 1892 1457 a(e) p Fk 1936 1416 a(ik) r(x) p Fl 2092 1457 a(;) p Fj 116 w(j) p Fl( ) p Fk 2333 1409 a(j) 2329 1487 y(k) p Fj 2378 1457 a(j) g(\024) p Fl 28 w(C) p Fm 7 w(e) p Fe 2660 1416 a(\000) p Fk(\033) p Fe 3 w(j) p Fk(k) p Fe 2 w(\000) p Fk(j) p Fe 4 w(j) p Fm 3764 1457 a(\(6) p Fl(:) p Fm(1\)) 0 1800 y(whic) m(h) 34 b(is) f(the) h(prop) s(ert) m(y) f(w) m(e) h(need.) 199 1920 y(W) -8 b(e) 35 b(\014x) f(the) h(phase) g(space) g(to) f(b) s(e) p Fj 35 w(F) p Fk 1574 1884 a(p) 1564 1945 y(s) p Fm 1648 1920 a(:=) p Fl 30 w(`) p Fi 1825 1884 a(2) p Fk 1825 1945 a(s) p Fj 1892 1920 a(\010) p Fl 23 w(`) p Fi 2034 1884 a(2) p Fk 2034 1945 a(s) p Fe(\000) p Fi(1) p Fj 2208 1920 a(3) p Fm 30 w(\() 7 b(^) p Fl -57 w(q) s(;) p Fm 25 w(^) p Fl -58 w(p) p Fm(\)) 34 b(endo) m(w) m(ed) i(b) m(y) e (the) h(symplectic) e(form) p Fl 0 2039 a(i) p Fh 51 1965 a(P) p Fk 156 2069 a(k) p Fe 2 w(2) p Fi 262 2052 a(\026) p Fa 254 2069 a(Z) p Fl 330 2039 a(dp) p Fk 432 2054 a(k) p Fj 504 2039 a(^) p Fl 22 w(dq) p Fe 688 2054 a(\000) p Fk(k) p Fm 800 2039 a(;) g(for) g(an) m(y) g(p) s(oin) m(t) g (\() 7 b(^) p Fl -57 w(q) t(;) p Fm 25 w(^) p Fl -58 w(p) p Fm(\)) p Fj 27 w(2) 28 b(F) p Fk 1885 2003 a(p) 1875 2064 y(s) p Fm 1964 2039 a(w) m(e) 34 b(de\014ne) g(the) g(functions) 1008 2309 y(^) p Fl -56 w(u) p Fm(\() p Fl(x) p Fm(\)) 27 b(:=) p Fh 1358 2214 a(X) p Fk 1354 2438 a(k) p Fe 2 w(2) p Fi 1460 2420 a(\026) p Fa 1452 2438 a(Z) p Fm 1531 2309 a(^) p Fl -58 w(q) p Fk 1567 2324 a(k) p Fl 1617 2309 a( ) p Fk 1682 2324 a(k) p Fm 1731 2309 a(\() p Fl(x) p Fm(\)) p Fl 32 w(;) p Fm 120 w(^) p Fl -54 w(v) p Fm 4 w(\() p Fl(x) p Fm(\)) g(:=) p Fh 2393 2214 a(X) p Fk 2389 2438 a(k) p Fe 2 w(2) p Fi 2495 2420 a(\026) p Fa 2487 2438 a(Z) p Fm 2567 2309 a(^) p Fl -59 w(p) p Fk 2608 2324 a(k) p Fl 2657 2309 a( ) p Fk 2722 2324 a(k) p Fm 2771 2309 a(\() p Fl(x) p Fm(\)) 33 b(;) 797 b(\(6) p Fl(:) p Fm(2\)) 0 2658 y(their) 33 b(sk) m(ew{symmetric) f (part) h(will) f(pla) m(y) h(the) g(role) g(of) h(the) f(unc) m(kno) m (wn) i(of) f(equation) e(\(0.1\).) p Fn 0 2817 a(Remark) 156 b(6.1.) p Fg 34 w(The) 35 b(norm) e(of) p Fj 34 w(F) p Fk 1428 2781 a(p) 1418 2842 y(s) p Fg 1507 2817 a(is) h(de\014ned) i (in) e(terms) f(of) h(the) h(co) s(e\016cien) m(ts) f(of) h(the) f (expansion) g(of) p Fm 0 2937 a(\() 6 b(^) p Fl -56 w(u;) p Fm 20 w(^) p Fl -53 w(v) p Fm 3 w(\)) p Fg 28 w(on) 29 b(the) g(basis) g(of) g(the) g(eigenfunction) h(of) f(a) f(Sturm) g (Liouville) g(problem) g(with) g(analytic) f(p) s(oten) m(tial,) 0 3056 y(so) e(it) f(is) h(equiv) -6 b(alen) m(t) 25 b(to) f(the) h (standard) h(Sob) s(olev) e(norm) g(of) h(the) g(corresp) s(onding) h (functions.) 43 b(In) 25 b(particular) 0 3176 y(\(6.2\)) d(establishes) h(an) g(isomorphism) e(of) p Fj 23 w(F) p Fk 1596 3140 a(p) 1586 3201 y(s) p Fg 1665 3176 a(with) h(the) h(Sob) s(olev) g (space) p Fl 23 w(H) p Fk 2749 3140 a(s) p Fm 2792 3176 a(\() 17 b(I) -17 b(R) p Fl(=) p Fm(2) p Fl(\031) p Fn 4 w(Z) p Fm(\)) p Fj 1 w(\010) p Fl 1 w(H) p Fk 3380 3140 a(s) p Fe(\000) p Fi(1) p Fm 3524 3176 a(\() 17 b(I) -17 b(R) p Fl(=) p Fm(2) p Fl(\031) p Fn 4 w(Z) p Fm(\)) p Fg(.) p Fm 199 3336 a(Consider) 34 b(the) g(Hamiltonian) d (system) p Fh 1057 3538 a(X) p Fk 1052 3762 a(k) p Fe 2 w(2) p Fi 1158 3745 a(\026) p Fa 1150 3762 a(Z) p Fm 1242 3566 a(^) p Fl -58 w(p) p Fk 1284 3581 a(k) p Fm 1341 3566 a(^) p Fl -58 w(p) p Fe 1383 3581 a(\000) p Fk(k) p Fm 1516 3566 a(+) p Fl 23 w(!) p Fi 1682 3530 a(2) p Fk 1678 3594 a(k) p Fm 1734 3566 a(^) p Fl -57 w(q) p Fk 1771 3581 a(k) p Fm 1828 3566 a(^) p Fl -58 w(q) p Fe 1864 3581 a(\000) p Fk(k) p 1234 3610 743 4 v Fm 1580 3701 a(2) 2010 3633 y(+) p Fh 2110 3497 a(Z) p Fk 2209 3522 a(\031) p Fe 2165 3724 a(\000) p Fk(\031) p Fl 2298 3633 a(G) p Fm(\() p Fl(x;) p Fm 23 w(^) p Fl -56 w(u) p Fm -1 w(\() p Fl(x) p Fm(\)\)) p Fl(dx) i(;) p Fm 847 w(\(6) p Fl(:) p Fm(3\)) 0 3985 y(with) p Fl 33 w(!) p Fe 289 4000 a(\000) p Fk(k) p Fm 428 3985 a(:=) p Fl 28 w(!) p Fk 623 4000 a(k) p Fj 699 3985 a(\021) 805 3915 y(p) p 888 3915 319 4 v Fl 888 3985 a(\026) p Fk 948 4000 a(k) p Fm 1019 3985 a(+) p Fl 22 w(m) p Fm 1 w(,) g(\() p Fl(k) p Fj 30 w(\025) p Fm 28 w(1\)) g(and) 40 b(^) p Fl -56 w(u) p Fm 33 w(whic) m(h) 34 b(has) g(to) f(b) s(e) g (though) m(t) h(as) f(a) g(function) h(of) 41 b(^) p Fl -57 w(q) p Fm 4 w(.) p Fn 0 4144 a(Remark) 164 b(6.2.) p Fg 36 w(In) 35 b(general) h(the) g(functions) p Fl 37 w( ) p Fk 1918 4159 a(j) p Fg 1995 4144 a(are) g(not) g(eigenfunctions) h(of) f(the) f(Sturm) h(Liouville) 0 4264 y(problem) 26 b(with) g(p) s(oten) m(tial) p Fl 25 w(V) p Fg 49 w(and) h(p) s(erio) s (dic) f(b) s(oundary) h(conditions,) g(therefore) h(the) e(dynamical) f (system) 0 4384 y(\(6.3\)) e(do) s(es) i(not) g(coincide) g(with) f (the) h(system) e(obtained) i(b) m(y) g(considering) g(\(0.1\)) f(with) g(p) s(erio) s(dic) g(b) s(oundary) 0 4503 y(conditions) 34 b(on) p Fm 33 w([) p Fj(\000) p Fl(\031) t(;) 17 b(\031) p Fm 4 w(]) p Fg(.) p Fn 0 4702 a(Remark) 109 b(6.3.) p Fg 24 w(A) 23 b(remark) -6 b(able) 23 b(exception) h(is) f(the) h (Klein) g(Gordon) g(equation) f(\(corresp) s(onding) i(to) p Fl 23 w(V) p Fm 50 w(=) 0 4822 y(0) p Fg(\)) 31 b(where) h(the) f (functions) p Fl 32 w( ) p Fk 1067 4837 a(j) p Fg 1140 4822 a(are) h(just) f(the) h(imaginary) d(exp) s(onen) m(tials) i(whic) m(h) h(are) f(also) g(eigenfunctions) 0 4941 y(of) 46 b(the) g(linerized) g(problem) f(with) g(p) s(erio) s(dic) h(b) s (oundary) g(conditions.) 81 b(It) 45 b(follo) m(ws) g(that) h(in) f (this) h(case) 0 5061 y(the) 37 b(Hamiltonian) e(\(6.3\)) h(is) h(the) h (restriction) f(of) g(the) h(Hamiltonian) c(of) k(the) f(nonlinear) h (Klein{Gordon) 0 5181 y(equation) 26 b(with) g(p) s(erio) s(dic) g(b) s (oundary) g(conditions) h(in) p Fm 26 w([) p Fj(\000) p Fl(\031) t(;) 17 b(\031) p Fm 4 w(]) p Fg 24 w(to) 26 b(the) h(space) g(of) f(the) h(functions) g(with) f(zero) 0 5300 y(a) m(v) m(erage.) 48 b(T) -8 b(o) 35 b(obtain) f(the) h(true) g (nonlinear) g(Klein) g(Gordon) h(equation) e(one) h(has) g(to) f(add) h (the) h(degree) f(of) 0 5420 y(freedom) e(corresp) s(onding) g(to) f (the) h(a) m(v) m(erage) p Fl 32 w(u) p Fg(.) 44 b(This) 33 b(can) f(b) s(e) h(easily) f(done) h(still) f(\014tting) g(the) g (framew) m(ork) 0 5539 y(of) h(the) h(previous) g(section.) p Fm 1807 5778 a(20.03.02) p 90 rotate dyy eop %%Page: 26 26 26 25 bop Fm 0 60 a(26) p Fg 1660 w(D.) 33 b(Bam) m(busi) p Fm 199 299 a(De\014ne) i(the) e(subspace) j(of) d(sk) m(ewsymmetric) f (functions) i(b) m(y) p Fj 1081 524 a(A) p Fm 28 w(:=) p Fj 27 w(f) p Fm(\() 8 b(^) p Fl -58 w(p) p Fk 1460 539 a(k) p Fl 1509 524 a(;) p Fm 24 w(^) p Fl -57 w(q) p Fk 1598 539 a(k) p Fm 1647 524 a(\)) 60 b(:) 68 b(^) p Fl -57 w(q) p Fk 1879 539 a(k) p Fm 1956 524 a(=) 36 b(^) p Fl -58 w(q) p Fe 2105 539 a(\000) p Fk(k) p Fm 2250 524 a(and) 42 b(^) p Fl -58 w(p) p Fk 2494 539 a(k) p Fm 2571 524 a(=) 36 b(^) p Fl -58 w(p) p Fe 2726 539 a(\000) p Fk(k) p Fj 2837 524 a(g) p Fm 0 749 a(then) e(w) m(e) g(ha) m (v) m(e) f(the) h(follo) m(wing) p Fn 0 908 a(Prop) s(osition) 179 b(6.4.) p Fg 39 w(The) 40 b(space) p Fj 40 w(A) p Fg 38 w(is) f(in) m(v) -6 b(arian) m(t) 39 b(for) g(the) h(dynamics) e(of) h(the) h(system) e(\(6.3\);) i(The) 0 1028 y(dynamics) 33 b(of) g(\(6.3\)) f(in) p Fj 33 w(A) p Fg 33 w(coincides) i(with) f(the) h(dynamics) e(of) i(the) f(nonlinear) h(w) m(a) m(v) m(e) f(equation) g (\(0.1\).) p Fn 0 1247 a(Pro) s(of.) p Fm 38 w(Consider) 38 b(\014rst) g(the) f(v) m(ector) g(\014eld) h(of) p Fl 37 w(h) p Fi 1849 1262 a(0) p Fm 1895 1247 a(.) 55 b(It) 37 b(is) g(clear) h(that) e(it) h(lea) m(v) m(es) g(in) m(v) -6 b(arian) m(t) p Fj 38 w(A) p Fm 36 w(and) 38 b(that) 0 1366 y(here) g(its) e(dynamics) h(coincides) h(with) e(the) i(dynamics) e(of) h(the) g(linear) g(part) g(of) g(\(0.1\)) f(b) m(y) h(iden) m (tifying) 44 b(^) p Fl -57 w(q) p Fk 3919 1381 a(k) p Fm 0 1486 a(\() p Fl(k) p Fj 30 w(\025) p Fm 29 w(1\)) 32 b(with) p Fl 33 w(u) p Fk 632 1501 a(k) p Fm 715 1486 a(and) 42 b(^) p Fl -59 w(p) p Fk 958 1501 a(k) p Fm 1041 1486 a(\() p Fl(k) p Fj 30 w(\025) p Fm 28 w(1\)) 33 b(with) p Fl 33 w(v) p Fk 1664 1501 a(k) p Fm 1713 1486 a(.) 199 1606 y(Consider) h(the) g(nonlinear) f(part) g(and) h(remark) e (\014rst) i(that,) e(since) p Fh 1388 1746 a(Z) p Fk 1488 1770 a(\031) p Fe 1443 1972 a(\000) p Fk(\031) p Fl 1576 1881 a( ) p Fk 1641 1896 a(j) p Fm 1683 1881 a(\() p Fl(x) p Fm(\)) p Fl( ) p Fk 1883 1896 a(k) p Fm 1932 1881 a(\() p Fl(x) p Fm(\)) p Fl(dx) p Fm 26 w(=) p Fl 29 w(\016) p Fk 2352 1896 a(j;) p Fe(\000) p Fk(k) p Fl 2553 1881 a(;) p Fm 1183 w(\(6) p Fl(:) p Fm(4\)) 0 2165 y(the) i(symplectic) e(form) g(can) i(b) s(e) g(written) f(in) g(terms) f(of) i(the) g(v) -6 b(ariables) 39 b(^) p Fl -56 w(u;) p Fm 20 w(^) p Fl -53 w(v) p Fm 36 w(as) p Fh 1509 2305 a(Z) p Fk 1609 2330 a(\031) p Fe 1565 2531 a(\000) p Fk(\031) p Fl 1697 2441 a(d) p Fm 4 w(^) p Fl -54 w(v) p Fm 4 w(\() p Fl(x) p Fm(\)) p Fj 22 w(^) p Fl 22 w(d) p Fm 6 w(^) p Fl -56 w(u) p Fm(\() p Fl(x) p Fm(\)) p Fl(dx) 33 b(:) p Fm 1304 w(\(6) p Fl(:) p Fm(5\)) 0 2730 y(Expand) g(the) h(nonlinearit) m(y) p Fl 33 w(f) p Fm 43 w(of) g(\(6.3\)) e(in) h(p) s(o) m(w) m(er) h(series) g(in) p Fl 33 w(u) p Fm(,) f(namely) f(write) p Fl 33 w(f) p Fm 38 w(=) p Fh 3240 2655 a(P) p Fk 3346 2760 a(n) p Fe(\025) p Fi(3) p Fl 3518 2730 a(f) p Fk 3578 2694 a(n) p Fm 3632 2730 a(,) h(with) p Fl 1331 3029 a(f) p Fk 1391 2988 a(n) p Fm 1444 3029 a(\() 6 b(^) p Fl -56 w(u) p Fm(\)) 28 b(=) p Fh 1712 2893 a(Z) p Fk 1811 2918 a(\031) p Fe 1767 3120 a(\000) p Fk(\031) p Fl 1900 3029 a(G) p Fk 1978 2988 a(n) p Fm 2033 3029 a(\() p Fl(x) p Fm(\)[) 6 b(^) p Fl -56 w(u) p Fm -1 w(\() p Fl(x) p Fm(\)]) p Fk 2415 2988 a(n) p Fl 2468 3029 a(dx) 33 b(;) p Fm 1126 w(\(6) p Fl(:) p Fm(6\)) 0 3318 y(and) h(consider) g(the) g (Hamiltonian) c(v) m(ector) k(\014eld) f(of) p Fl 34 w(f) p Fk 2015 3282 a(n) p Fm 2069 3318 a(.) 44 b(By) 32 b(\(6.5\)) g(it) h(is) g(giv) m(en) g(b) m(y) p Fl 1368 3527 a(d) p 1350 3572 88 4 v 1350 3663 a(dt) p Fm 1454 3595 a(^) p Fl -54 w(v) p Fm 32 w(=) p Fl 28 w(nG) p Fk 1773 3554 a(n) p Fm 1834 3595 a(^) p Fl -57 w(u) p Fk 1884 3554 a(n) p Fe(\000) p Fi(1) p Fl 2074 3595 a(;) 2248 3527 y(d) p 2230 3572 V 2230 3663 a(dt) p Fm 2336 3595 a(^) p Fl -57 w(u) p Fm 28 w(=) 28 b(0) 33 b(;) 1134 b(\(6) p Fl(:) p Fm(7\)) 0 3848 y(b) m(y) 31 b(the) h(sk) m(ew) f (symmetry) e(prop) s(ert) m(y) i(of) p Fl 32 w(g) p Fm 34 w(it) f(lea) m(v) m(es) i(in) m(v) -6 b(arian) m(t) p Fj 31 w(A) p Fm(.) 43 b(On) 32 b(this) f(manifold) f(the) i(v) m(ector) f(\014eld) 0 3967 y(\(6.7\)coincides) j(with) g(the) h(one) g(of) f (the) h(single) f(terms) g(of) h(the) f(expansion) h(of) g(the) f (nonlinearit) m(y) g(of) h(\(0.1\)) 0 4087 y(b) m(y) e(iden) m(tifying) h(\(as) f(natural\)) p Fl 33 w(u) p Fm 33 w(and) p Fl 33 w(v) p Fm 38 w(with) f(the) i(restriction) f(of) 40 b(^) p Fl -56 w(u) p Fm 33 w(and) d(^) p Fl -53 w(v) p Fm 37 w(to) 32 b([0) p Fl(;) 17 b(\031) p Fm 4 w(].) p 3891 4013 78 4 v 3891 4083 4 70 v 3965 4083 V 3891 4087 78 4 v 199 4206 a(In) m(tro) s(duce) 35 b(no) m(w) e(complex) f(v) -6 b(ariables) p Fl 34 w(\030) 5 b(;) 17 b(\021) p Fm 35 w(b) m(y) p Fl 639 4485 a(\030) p Fk 683 4500 a(j) p Fm 752 4485 a(:=) 939 4418 y(1) p 897 4463 133 4 v Fj 897 4483 a(p) p 980 4483 50 4 v Fm 980 4565 a(2) p Fh 1059 4345 a(\022) p Fm 1139 4485 a(^) p Fl -57 w(q) p Fk 1176 4500 a(j) p Fj 1218 4423 a(p) p 1301 4423 104 4 v Fl 1301 4486 a(!) p Fk 1363 4501 a(j) p Fj 1428 4485 a(\000) p Fl 22 w(i) p Fm 1629 4418 a(^) p Fl -58 w(p) p Fk 1671 4433 a(j) p 1573 4463 188 4 v Fj 1573 4496 a(p) p 1656 4496 104 4 v Fl 1656 4554 a(!) p Fk 1718 4569 a(j) p Fh 1772 4345 a(\023) p Fl 1896 4485 a(;) 115 b(\021) p Fk 2088 4500 a(j) p Fm 2159 4485 a(:=) 2345 4418 y(1) p 2303 4463 133 4 v Fj 2303 4483 a(p) p 2386 4483 50 4 v Fm 2386 4565 a(2) p Fh 2465 4345 a(\022) p Fm 2545 4485 a(^) p Fl -57 w(q) p Fk 2582 4500 a(j) p Fj 2625 4423 a(p) p 2708 4423 104 4 v Fl 2708 4486 a(!) p Fk 2770 4501 a(j) p Fm 2834 4485 a(+) p Fl 22 w(i) p Fm 3035 4418 a(^) p Fl -58 w(p) p Fk 3077 4433 a(j) p 2980 4463 188 4 v Fj 2980 4496 a(p) p 3063 4496 104 4 v Fl 3063 4554 a(!) p Fk 3125 4569 a(j) p Fh 3179 4345 a(\023) p Fl 3302 4485 a(;) p Fm 3764 4491 a(\(6) p Fl(:) p Fm(8\)) 0 4782 y(then) 34 b(clearly) e(the) i(quadratic) f(part) g(of) g(the) h (hamiltonian) e(tak) m(es) h(the) g(form) g(\(4.2\).) 199 4902 y(Fix) f(a) g(p) s(oten) m(tial) p Fl 32 w(V) p Fm 54 w(as) h(in) f(sect.) 45 b(2,) 32 b(then,) g(concerning) i(the) f (frequencies,) h(w) m(e) f(ha) m(v) m(e) f(the) h(follo) m(wing) 0 5021 y(theorem) g(that) g(will) f(b) s(e) h(pro) m(v) m(ed) h(in) g (the) f(next) g(subsection.) p Fn 0 5181 a(Theorem) 177 b(6.5.) p Fg 39 w(Ha) m(ving) 38 b(\014xed) p Fl 39 w(m) p Fi 1522 5196 a(1) p Fg 1606 5181 a(and) p Fl 39 w(m) p Fi 1892 5196 a(2) p Fg 1976 5181 a(as) h(ab) s(o) m(v) m(e,) h(there) f (exists) g(a) f(subset) p Fj 40 w(J) p Fg 57 w(of) p Fm 39 w([) p Fl(m) p Fi 3720 5196 a(1) p Fl 3764 5181 a(;) 17 b(m) p Fi 3896 5196 a(2) p Fm 3941 5181 a(]) p Fg 0 5300 a(with) 32 b(measure) p Fl 31 w(m) p Fi 699 5315 a(2) p Fj 764 5300 a(\000) p Fl 20 w(m) p Fi 948 5315 a(1) p Fg 1025 5300 a(suc) m(h) h(that) e(for) i(an) m(y) p Fl 31 w(m) p Fj 28 w(2) c(J) p Fg 49 w(the) k(frequencies) p Fl 33 w(!) p Fk 2868 5315 a(j) p Fm 2938 5300 a(:=) p Fj 3071 5235 a(p) p 3154 5235 312 4 v Fl 3154 5300 a(\026) p Fk 3214 5315 a(j) p Fm 3278 5300 a(+) p Fl 22 w(m) p Fg 33 w(\() p Fl(j) p Fj 33 w(\025) p Fm 28 w(1) p Fg(\)) e(are) 0 5420 y(strongly) h(non) i(v) -6 b(anishing.) 44 b(The) 34 b(index) p Fl 34 w(\013) p Fm 27 w(=) p Fl 29 w(\013) p Fm(\() p Fl(r) p Fm 3 w(\)) p Fg 32 w(can) g(b) s(e) f(c) m(hosen) i (equal) e(to) p Fm 33 w(16) p Fl(r) p Fi 3121 5384 a(5) p Fg 3165 5420 a(.) 44 b(The) 33 b(same) g(is) g(true) 0 5539 y(if) g(one) h(adds) g(the) g(frequencies) p Fl 35 w(!) p Fe 1251 5554 a(\000) p Fk(j) p Fm 1383 5539 a(:=) p Fl 27 w(!) p Fk 1577 5554 a(j) p Fg 1619 5539 a(,) p Fl 33 w(j) p Fj 34 w(\025) p Fm 28 w(1) p Fg(.) p Fm 1807 5778 a(20.03.02) p 90 rotate dyy eop %%Page: 27 27 27 26 bop Fg 1222 60 a(Birkho\013) 33 b(normal) f(form) g(for) i(PDE's) p Fm 1222 w(27) 199 299 y(Concerning) g(the) g(nonlinearit) m(y) f(w) m (e) g(ha) m(v) m(e) h(the) g(follo) m(wing) p Fn 0 458 a(Prop) s(osition) 163 b(6.6.) p Fg 36 w(Under) 36 b(the) g(ab) s(o) m (v) m(e) g(assumptions) f(there) h(exist) g(p) s(ositiv) m(e) p Fl 34 w(\026) p Fg 36 w(and) p Fl 36 w(R) p Fg 36 w(suc) m(h) h(that) 0 578 y(the) d(nonlinearit) m(y) e(of) i(\(6.3\)) e(is) h(of) g(class) p Fj 34 w(M) p Fk 1644 530 a(\026) 1644 607 y(R) p Fg 1709 578 a(.) p Fn 0 797 a(Pro) s(of.) p Fm 34 w(De\014ne) i(\014rst) e(the) h(functions) 903 1001 y(\026) p Fl 893 1027 a(\030) p Fm 4 w(\() p Fl(x) p Fm(\)) 27 b(:=) p Fh 1236 933 a(X) p Fk 1286 1147 a(k) p Fl 1459 960 a(\030) p Fk 1503 975 a(k) p 1408 1005 195 4 v Fj 1408 1031 a(p) p 1491 1031 112 4 v Fl 1491 1096 a(!) p Fk 1553 1111 a(k) p Fl 1614 1027 a( ) p Fk 1679 1042 a(k) p Fm 1728 1027 a(\() p Fl(x) p Fm(\)) p Fl 33 w(;) p Fm 123 w(\026) p Fl -57 w(\021) p Fm 4 w(\() p Fl(x) p Fm(\)) g(:=) p Fh 2388 933 a(X) p Fk 2438 1147 a(k) p Fl 2608 960 a(\021) p Fk 2657 975 a(k) p 2560 1005 195 4 v Fj 2560 1031 a(p) p 2643 1031 112 4 v Fl 2643 1096 a(!) p Fk 2705 1111 a(k) p Fl 2766 1027 a( ) p Fk 2831 1042 a(k) p Fm 2880 1027 a(\() p Fl(x) p Fm(\)) p Fl 33 w(;) p Fm 0 1320 a(and,) 33 b(to) g(start) g(with) g(consider) h(a) f(homogeneous) h(p) s (olynomial) p Fl 1144 1574 a(f) p Fm 11 w(\() p Fl(\030) 5 b(;) 17 b(\021) p Fm 4 w(\)) 25 b(:=) p Fh 1587 1439 a(Z) p Fk 1687 1463 a(\031) p Fe 1642 1665 a(\000) p Fk(\031) p Fl 1775 1574 a(G) p Fm(\() p Fl(x) p Fm(\)[) 2026 1548 y(\026) p Fl 2016 1574 a(\030) p Fm 4 w(\() p Fl(x) p Fm(\)]) p Fk 2227 1533 a(n) p Fb 2276 1543 a(1) p Fm 2319 1574 a([) 7 b(\026) p Fl -57 w(\021) p Fm 4 w(\() p Fl(x) p Fm(\)]) p Fk 2563 1533 a(n) p Fb 2612 1543 a(2) p Fl 2655 1574 a(dx) 33 b(;) p Fm 939 w(\(6) p Fl(:) p Fm(9\)) 0 1842 y(with) p Fl 31 w(G) p Fm 31 w(p) s(erio) s(dic) f(and) f (analytic) f(in) h(a) g(strip) h(of) f(width) p Fl 31 w(\024) p Fm 32 w(around) h(the) f(real) g(axis;) g(the) h(expression) g (\(6.9\)) 0 1962 y(coincides) i(with) p Fh 361 2109 a(X) p Fk 394 2323 a(l;i) p Fl 606 2125 a(\030) p Fk 650 2140 a(l) p Fb 675 2150 a(1) p Fl 719 2125 a(:::\030) p Fk 847 2140 a(l) p Fd 872 2150 a(n) p Fb 916 2165 a(1) p Fl 963 2125 a(\021) p Fk 1012 2140 a(i) p Fb 1040 2150 a(1) p Fl 1085 2125 a(:::\021) p Fk 1218 2140 a(i) p Fd 1246 2150 a(n) p Fb 1290 2165 a(2) p 533 2180 878 4 v Fj 533 2220 a(p) p 616 2220 795 4 v Fl 616 2272 a(!) p Fk 678 2287 a(l) p Fb 703 2297 a(1) p Fl 747 2272 a(:::!) p Fk 893 2287 a(l) p Fd 918 2297 a(n) p Fb 962 2312 a(1) p Fl 1010 2272 a(!) p Fk 1072 2287 a(i) p Fb 1100 2297 a(1) p Fl 1144 2272 a(:::!) p Fk 1290 2287 a(i) p Fd 1318 2297 a(n) p Fb 1362 2312 a(2) p Fh 1439 2068 a(Z) p Fk 1539 2092 a(\031) p Fe 1494 2294 a(\000) p Fk(\031) p Fl 1627 2203 a(G) p Fm(\() p Fl(x) p Fm(\)) p Fl( ) p Fk 1905 2218 a(l) p Fb 1930 2228 a(1) p Fm 1974 2203 a(\() p Fl(x) p Fm(\)) p Fl(::: ) p Fk 2258 2218 a(l) p Fd 2283 2228 a(n) p Fb 2327 2243 a(1) p Fm 2374 2203 a(\() p Fl(x) p Fm(\)) p Fl( ) p Fk 2574 2218 a(i) p Fb 2602 2228 a(1) p Fm 2646 2203 a(\() p Fl(x) p Fm(\)) p Fl(::: ) p Fk 2930 2218 a(i) p Fd 2958 2228 a(n) p Fb 3002 2243 a(2) p Fm 3049 2203 a(\() p Fl(x) p Fm(\)) p Fl(dx) e(;) p Fm 361 w(\(6) p Fl(:) p Fm(10\)) 0 2515 y(w) m(e) g(estimate) f(the) h(in) m(tegral.) 43 b(Denote) p Fl 32 w(k) p Fm 31 w(=) 28 b(\() p Fl(k) p Fi 1733 2530 a(1) p Fl 1777 2515 a(:::k) p Fk 1913 2530 a(n) p Fm 1966 2515 a(\)) g(:=) g(\() p Fl(l) r(;) 17 b(i) p Fm(\)) 26 b(=) i(\() p Fl(l) p Fi 2555 2530 a(1) p Fl 2599 2515 a(;) 17 b(:::;) g(l) p Fk 2803 2530 a(n) p Fb 2852 2540 a(1) p Fl 2894 2515 a(;) g(i) p Fi 2973 2530 a(1) p Fl 3017 2515 a(;) g(:::;) g(i) p Fk 3225 2530 a(n) p Fb 3274 2540 a(2) p Fm 3316 2515 a(\),) 32 b(then) g(suc) m(h) h(an) 0 2635 y(in) m(tegral) g(is) g(giv) m(en) g(b) m(y) p Fh 48 2748 a(\014) 48 2808 y(\014) 48 2868 y(\014) 48 2927 y(\014) 48 2987 y(\014) 48 3047 y(\014) 81 2858 y(X) p Fk 134 3070 a(j) p Fl 241 2952 a( ) p Fk 310 2908 a(k) p Fb 352 2918 a(1) p Fk 306 2980 a(j) p Fb 339 2990 a(1) p Fl 396 2952 a(::: ) p Fk 549 2908 a(k) p Fd 591 2918 a(n) p Fk 545 2980 a(j) p Fd 578 2990 a(n) p Fh 660 2817 a(Z) p Fk 760 2841 a(\031) p Fe 716 3043 a(\000) p Fk(\031) p Fl 849 2952 a(G) p Fm(\() p Fl(x) p Fm(\)e) p Fk 1106 2911 a(i) p Fi(\() p Fk(j) p Fb 1198 2921 a(1) p Fi 1237 2911 a(+) p Fk(:::) p Fi(+) p Fk(j) p Fd 1464 2921 a(n) p Fi 1512 2911 a(\)) p Fk(x) p Fl 1593 2952 a(dx) p Fh 1702 2748 a(\014) 1702 2808 y(\014) 1702 2868 y(\014) 1702 2927 y(\014) 1702 2987 y(\014) 1702 3047 y(\014) p Fj 1763 2952 a(\024) p Fh 1868 2858 a(X) p Fk 1922 3070 a(j) p Fl 2029 2952 a(C) p Fk 2107 2911 a(n) p Fm 2161 2952 a(e) p Fe 2205 2911 a(\000) p Fk(\033) p Fi 3 w(\() p Fe(j) p Fk(k) p Fb 2413 2921 a(1) p Fe 2452 2911 a(\000) p Fk(j) p Fb 2547 2921 a(1) p Fe 2587 2911 a(j) p Fi(+) p Fk(:::) p Fi(+) p Fe(j) p Fk(k) p Fd 2871 2921 a(n) p Fe 2918 2911 a(\000) p Fk(j) p Fd 3013 2921 a(n) p Fe 3062 2911 a(j) p Fi(\)) p Fj 3122 2952 a(j) p Fl(G) p Fj(j) p Fk 3256 2967 a(\024) p Fm 3308 2952 a(e) p Fe 3352 2911 a(\000) p Fk(\024) p Fe(j) p Fk(j) p Fb 3518 2921 a(1) p Fi 3556 2911 a(+) p Fk(:::) p Fi(+) p Fk(j) p Fd 3783 2921 a(n) p Fe 3831 2911 a(j) p Fl 3893 2952 a(;) p Fm 0 3275 a(where) p Fj 39 w(j) p Fl(G) p Fj(j) p Fk 427 3290 a(\024) p Fm 516 3275 a(is) 38 b(the) g(suprem) m(um) f(of) p Fl 38 w(G) p Fm 39 w(in) g(the) i(strip) p Fj 37 w(j) p Fm(Im) p Fl 19 w(x) p Fj(j) p Fl 35 w(<) d(\024) p Fm(;) k(here) f(w) m (e) f(used) h(the) f(estimate) f(\(6.1\)) 0 3395 y(of) p Fl 36 w( ) p Fk 185 3359 a(k) 181 3421 y(j) p Fm 233 3395 a(,) f(the) g(fact) f(that) g(the) h(last) f(in) m(tegral) g(is) g (the) h(F) -8 b(ourier) 36 b(co) s(e\016cien) m(t) g(of) p Fl 35 w(G) p Fm 36 w(and) g(the) g(standard) g(deca) m(y) 0 3514 y(estimate) d(of) h(F) -8 b(ourier) 34 b(co) s(e\016cien) m(ts) h (of) f(analytic) f(functions.) 48 b(T) -8 b(o) 34 b(\014x) g(ideas) g (assume) g(that) p Fl 34 w(\024) 29 b(>) h(\033) p Fm 4 w(,) j(then,) 0 3634 y(denoting) h(b) m(y) p Fl 33 w(f) p Fk 597 3649 a(k) p Fj 673 3634 a(\021) p Fl 29 w(f) p Fk 828 3649 a(k) p Fb 870 3659 a(1) p Fk 909 3649 a(;:::;k) p Fd 1071 3659 a(n) p Fm 1156 3634 a(the) g(in) m(tegral) f (in) g(\(6.10\),) e(and) j(c) m(ho) s(osing) g(a) f(p) s(ositiv) m(e) p Fl 32 w(\016) f(<) c(\033) p Fm 37 w(w) m(e) 34 b(ha) m(v) m(e) p Fj 1581 3831 a(j) p Fl(f) p Fk 1658 3846 a(k) p Fj 1706 3831 a(j) p 1565 3875 186 4 v 1565 3967 a(j) p Fl(G) p Fj(j) p Fk 1699 3982 a(\024) p Fj 1789 3898 a(\024) p Fh 1895 3804 a(X) p Fk 1948 4016 a(j) p Fl 2055 3898 a(C) p Fk 2133 3857 a(n) p Fm 2188 3898 a(e) p Fe 2232 3857 a(\000) p Fk(\033) p Fi 3 w(\() p Fe(j) p Fk(k) p Fb 2440 3867 a(1) p Fe 2479 3857 a(\000) p Fk(j) p Fb 2574 3867 a(1) p Fe 2613 3857 a(j) p Fi(+) p Fk(:::) p Fi(+) p Fe(j) p Fk(k) p Fd 2897 3867 a(n) p Fe 2945 3857 a(\000) p Fk(j) p Fd 3040 3867 a(n) p Fe 3089 3857 a(j) p Fi(\)) p Fm 3149 3898 a(e) p Fe 3193 3857 a(\000) p Fk(\024) p Fe(j) p Fk(j) p Fb 3359 3867 a(1) p Fi 3397 3857 a(+) p Fk(:::) p Fi(+) p Fk(j) p Fd 3624 3867 a(n) p Fe 3673 3857 a(j) p Fj 544 4184 a(\024) p Fh 649 4089 a(X) p Fk 703 4301 a(j) p Fl 810 4184 a(C) p Fk 888 4143 a(n) p Fm 942 4184 a(e) p Fe 986 4143 a(\000) p Fk(\016) p Fi 3 w([) p Fe(j) p Fk(k) p Fb 1176 4153 a(1) p Fe 1215 4143 a(\000) p Fk(j) p Fb 1310 4153 a(1) p Fe 1349 4143 a(j) p Fi(+) p Fk(:::) p Fi(+) p Fe(j) p Fk(k) p Fd 1633 4153 a(n) p Fe 1681 4143 a(\000) p Fk(j) p Fd 1776 4153 a(n) p Fe 1825 4143 a(j) p Fi(]) p Fm 1876 4184 a(e) p Fe 1920 4143 a(\000) p Fi(\() p Fk(\033) p Fe 3 w(\000) p Fk(\016) p Fi 3 w(\)[) p Fe(j) p Fk(k) p Fb 2283 4153 a(1) p Fe 2322 4143 a(\000) p Fk(j) p Fb 2417 4153 a(1) p Fe 2456 4143 a(j) p Fi(+) p Fk(:::) p Fi(+) p Fe(j) p Fk(k) p Fd 2740 4153 a(n) p Fe 2788 4143 a(\000) p Fk(j) p Fd 2883 4153 a(n) p Fe 2932 4143 a(j) p Fi(]) p Fm 2983 4184 a(e) p Fe 3027 4143 a(\000) p Fi(\() p Fk(\033) p Fe 3 w(\000) p Fk(\016) p Fi 3 w(\)) p Fe(j) p Fk(j) p Fb 3358 4153 a(1) p Fi 3397 4143 a(+) p Fk(:::) p Fi(+) p Fk(j) p Fd 3624 4153 a(n) p Fe 3673 4143 a(j) p Fj 267 4469 a(\024) p Fl 28 w(C) p Fk 450 4428 a(n) p Fm 505 4469 a(e) p Fe 549 4428 a(\000) p Fi(\() p Fk(\033) p Fe 3 w(\000) p Fk(\016) p Fi 3 w(\)) p Fe(j) p Fk(k) p Fb 889 4438 a(1) p Fi 928 4428 a(+) p Fk(:::) p Fi(+) p Fk(k) p Fd 1164 4438 a(n) p Fe 1213 4428 a(j) p Fh 1258 4375 a(X) p Fk 1312 4587 a(j) p Fm 1419 4469 a(e) p Fe 1463 4428 a(\000) p Fk(\016) p Fe 3 w(j) p Fk(k) p Fb 1630 4438 a(1) p Fe 1669 4428 a(\000) p Fk(j) p Fb 1764 4438 a(1) p Fe 1803 4428 a(j) p Fm 1832 4469 a(e) p Fe 1876 4428 a(\000) p Fk(\016) p Fe 3 w(j) p Fk(k) p Fb 2043 4438 a(2) p Fe 2082 4428 a(\000) p Fk(j) p Fb 2177 4438 a(2) p Fe 2216 4428 a(j) p Fl 2245 4469 a(:::) p Fm(e) p Fe 2373 4428 a(\000) p Fk(\016) p Fe 3 w(j) p Fk(k) p Fd 2540 4438 a(n) p Fe 2588 4428 a(\000) p Fk(j) p Fd 2683 4438 a(n) p Fe 2732 4428 a(j) p Fj 2788 4469 a(\024) p Fl 28 w(C) p Fk 2971 4428 a(n) p Fm 3026 4469 a(e) p Fe 3070 4428 a(\000) p Fi(\() p Fk(\033) p Fe 3 w(\000) p Fk(\016) p Fi 3 w(\)) p Fe(j) p Fk(k) p Fb 3410 4438 a(1) p Fi 3449 4428 a(+) p Fk(:::k) p Fd 3624 4438 a(n) p Fe 3673 4428 a(j) p Fm 0 4773 a(where) g(w) m(e) g (rede\014ned) h(the) f(constan) m(t) p Fl 34 w(C) p Fm 40 w(and) g(used) g(the) g(triangular) e(inequalit) m(y:) p Fj 0 4977 a(j) p Fl(k) p Fi 80 4992 a(1) p Fm 135 4977 a(+) p Fl 10 w(:::) p Fm 10 w(+) p Fl 10 w(k) p Fk 455 4992 a(n) p Fj 509 4977 a(j) c(\024) g(j) p Fl(k) p Fi 750 4992 a(1) p Fj 804 4977 a(\000) p Fl 10 w(j) p Fi 932 4992 a(1) p Fm 988 4977 a(+) p Fl 10 w(:::) p Fm 10 w(+) p Fl 10 w(k) p Fk 1308 4992 a(n) p Fj 1373 4977 a(\000) p Fl 10 w(j) p Fk 1501 4992 a(n) p Fj 1556 4977 a(j) p Fm 10 w(+) p Fj 10 w(j) p Fl(j) p Fi 1750 4992 a(1) p Fm 1806 4977 a(+) p Fl 10 w(:::) p Fm 10 w(+) p Fl 10 w(j) p Fk 2115 4992 a(n) p Fj 2170 4977 a(j) f(\024) h(j) p Fl(k) p Fi 2410 4992 a(1) p Fj 2465 4977 a(\000) p Fl 10 w(j) p Fi 2593 4992 a(1) p Fj 2639 4977 a(j) p Fm 10 w(+) p Fl 10 w(:::) p Fm 10 w(+) p Fj 10 w(j) p Fl(k) p Fk 3025 4992 a(n) p Fj 3089 4977 a(\000) p Fl 10 w(j) p Fk 3217 4992 a(n) p Fj 3272 4977 a(j) p Fm 10 w(+) p Fj 10 w(j) p Fl(j) p Fi 3466 4992 a(1) p Fm 3522 4977 a(+) p Fl 10 w(:::) p Fm 10 w(+) p Fl 10 w(j) p Fk 3831 4992 a(n) p Fj 3886 4977 a(j) p Fl 27 w(:) p Fm 0 5180 a(T) -8 b(ak) m(e) 33 b(no) m(w) p Fl 33 w(\026) p Fm 28 w(=) p Fl 28 w(\033) p Fj 26 w(\000) p Fm 22 w(2) p Fl(\016) p Fm 4 w(,) g(then) h(one) g(has) p Fj 488 5449 a(j) p Fl(f) p Fj 11 w(j) p Fk 604 5464 a(\026) p Fj 683 5449 a(\024) 29 b(j) p Fl(G) p Fj(j) p Fk 923 5464 a(\024) p Fh 991 5355 a(X) p Fk 1050 5569 a(l) p Fm 1151 5449 a(e) p Fk 1195 5408 a(\026) p Fe(j) p Fk(l) p Fe(j) p Fl 1322 5449 a(c) p Fk 1365 5408 a(n) p Fm 1587 5449 a(sup) p Fk 1436 5534 a(k) p Fb 1478 5544 a(1) p Fi 1517 5534 a(+) p Fk(:::) p Fi(+) p Fk(k) p Fd 1753 5544 a(n) p Fi 1801 5534 a(=) p Fk(l) p Fm 1917 5382 a(e) p Fe 1961 5346 a(\000) p Fi(\() p Fk(\033) p Fe 3 w(\000) p Fk(\016) p Fi 3 w(\)) p Fe(j) p Fk(k) p Fb 2301 5356 a(1) p Fi 2340 5346 a(+) p Fk(:::) p Fi(+) p Fk(k) p Fd 2576 5356 a(n) p Fe 2625 5346 a(j) p 1917 5426 737 4 v Fj 2049 5458 a(p) p 2132 5458 390 4 v Fl 2132 5518 a(!) p Fk 2194 5533 a(k) p Fb 2236 5543 a(1) p Fl 2280 5518 a(:::!) p Fk 2426 5533 a(k) p Fd 2468 5543 a(n) p Fm 2693 5449 a(=) p Fl 28 w(c) p Fk 2841 5408 a(n) p Fj 2895 5449 a(j) p Fl(G) p Fj(j) p Fk 3029 5464 a(\024) p Fh 3097 5355 a(X) p Fk 3156 5569 a(l) p Fm 3258 5449 a(e) p Fe 3302 5408 a(\000) p Fk(\016) p Fe 3 w(j) p Fk(l) p Fe(j) p Fm 1807 5778 a(20.03.02) p 90 rotate dyy eop %%Page: 28 28 28 27 bop Fm 0 60 a(28) p Fg 1660 w(D.) 33 b(Bam) m(busi) p Fm 0 299 a(where) 54 b(w) m(e) g(used) h(the) f(fact) f(that) p Fl 53 w(!) p Fk 1429 314 a(k) p Fm 1532 299 a(is) g(strictly) g(p) s (ositiv) m(e) g(for) g(all) p Fl 53 w(k) p Fm 3 w('s.) 104 b(So,) 59 b(one) 54 b(has) f(that,) 58 b(for) p Fl 0 418 a(\026) 46 b(<) p Fm 46 w(min) p Fj -1 w(f) p Fl(\033) n(;) 17 b(\024) p Fj(g) p Fm(,) 45 b(\(6.9\)) e(has) h(\014nite) p Fl 45 w(\026) p Fm({mo) s(dulus) f(smaller) g(than) p Fl 44 w(C) p Fj 7 w(j) p Fl(G) p Fj(j) p Fk 2760 433 a(\024) p Fl 2812 418 a(c) p Fk 2855 382 a(n) p Fm 2953 418 a(where) p Fl 45 w(C) r(;) 17 b(c) p Fm 43 w(are) 44 b(p) s(ositiv) m(e) 0 538 y(constan) m(ts) 37 b(that) e(dep) s(end) i (on) p Fl 36 w(G) p Fm 36 w(only) e(through) p Fl 36 w(\024) p Fm(.) 52 b(T) -8 b(o) 35 b(obtain) g(the) h(statemen) m(t) f (just) i(remark) d(that) h(b) m(y) 0 658 y(analyticit) m(y) c(there) i (exist) f(constan) m(ts) h(suc) m(h) h(that) p Fj 32 w(j) p Fl(G) p Fk 1969 621 a(n) p Fj 2023 658 a(j) p Fk 2051 673 a(\024) p Fj 2130 658 a(\024) p Fl 28 w(C) 7 b(c) p Fk 2356 621 a(n) p Fm 2411 658 a(,) 32 b(and) h(that) p Fl 32 w(G) p Fk 2957 621 a(n) p Fm 3044 658 a(is) f(the) h(sum) f(of) h (at) f(most) 0 777 y(2) p Fk 50 741 a(n) p Fm 137 777 a(p) s(olynomials) f(of) j(the) f(form) g(\(6.9\).) p 3891 703 78 4 v 3891 773 4 70 v 3965 773 V 3891 777 78 4 v 199 897 a(W) -8 b(e) 30 b(ha) m(v) m(e) f(th) m(us) h(v) m(erify) m (ed) g(that) e(the) h(system) g(\(6.3\)) e(ful\014lls) j(the) f (assumptions) g(of) g(theorem) g(4.1) f(that) 0 1016 y(therefore) 40 b(applies.) 60 b(Theorem) 38 b(2.2) g(immediately) d (follo) m(ws) k(as) f(a) g(corollary) g(in) g(virtue) g(of) h(prop) s (osition) 0 1136 y(6.4) 29 b(and) i(of) f(the) h(remark) d(that,) j (since) f(on) p Fj 31 w(A) p Fl 29 w(\030) p Fk 1735 1151 a(k) p Fm 1812 1136 a(=) p Fl 28 w(\030) p Fe 1961 1151 a(\000) p Fk(k) p Fm 2102 1136 a(and) p Fl 30 w(\021) p Fk 2341 1151 a(k) p Fm 2418 1136 a(=) p Fl 29 w(\021) p Fe 2573 1151 a(\000) p Fk(k) p Fm 2684 1136 a(,) h(the) f(restriction) g (of) g(the) h(normal) 0 1255 y(form) i(to) f(this) i(manifold) e(dep) s (ends) j(only) e(on) g(the) h(actions) p Fl 33 w(I) p Fk 2260 1270 a(k) p Fm 2309 1255 a(.) p Fn 0 1375 a(Pro) s(of) 49 b(of) f(Corollary) h(2.3.) p Fm 70 w(The) 42 b(estimates) f(\(2.7\)) f (can) i(b) s(e) g(obtained) g(just) h(estimating) d(the) i(time) 0 1505 y(deriv) -6 b(ativ) m(es) 40 b(of) f(the) h(functions) p Fj 41 w(k) p Fl(\020) p Fj 7 w(k) p Fi 1387 1455 a(2) p Fe 1387 1535 a(F) p Fd 1444 1545 a(s) p Fm 1527 1505 a(and) p Fl 40 w(j) p Fi 1774 1469 a(2) p Fk(s) p Fl 1856 1505 a(I) p Fk 1900 1520 a(j) p Fm 1942 1505 a(\() p Fl(\020) p Fm 7 w(\),) g(whic) m(h) h(are) e(analytic.) 62 b(The) 41 b(estimate) d(\(2.8\)) h(is) 0 1624 y(obtained) 34 b(b) m(y) f(remarking) f(that) h(the) g(distance) p Fl 34 w(d) p Fk 1849 1639 a(s) p Fc 1887 1619 a(0) p Fm 1919 1624 a(\() p Fl(\020) 7 b(;) p Fn 17 w(T) p Fi 2134 1639 a(0) p Fm 2177 1624 a(\)) 33 b(is) g(estimated) f(b) m(y) 569 1998 y([) p Fl(d) p Fk 649 2013 a(s) p Fc 687 1993 a(0) p Fm 718 1998 a(\() p Fl(\020) 7 b(;) p Fn 17 w(T) p Fi 933 2013 a(0) p Fm 976 1998 a(\)]) p Fi 1043 1957 a(2) p Fj 1115 1998 a(\024) p Fh 1220 1798 a(0) 1220 1977 y(@) 1307 1904 y(X) p Fk 1310 2117 a(j) p Fe 4 w(\025) p Fi(1) p Fl 1468 1998 a(j) p Fe 1515 1957 a(\000) p Fi(2\() p Fk(s) p Fe(\000) p Fk(s) p Fc 1786 1927 a(0) p Fi 1812 1957 a(\)) p Fh 1848 1798 a(1) 1848 1977 y(A) p Fi 1936 1819 a(1) p Fk(=) p Fi(2) p Fh 2077 1798 a(2) 2077 1977 y(4) 2144 1904 y(X) p Fk 2146 2117 a(j) p Fe 4 w(\025) p Fi(1) p Fl 2304 1998 a(j) p Fi 2351 1957 a(4) p Fk(s) p Fm 2433 1998 a(\() p Fl(I) p Fk 2516 2013 a(j) p Fm 2558 1998 a(\() p Fl(\020) p Fm 7 w(\)) p Fj 21 w(\000) p Fl 23 w(I) p Fk 2852 2013 a(j) p Fm 2894 1998 a(\() f(\026) p Fl -56 w(u;) p Fm 20 w(\026) p Fl -53 w(v) p Fm 3 w(\)\)) p Fi 3164 1957 a(2) p Fh 3208 1798 a(3) 3208 1977 y(5) p Fi 3274 1819 a(1) p Fk(=) p Fi(2) p Fm 0 2339 a(and) 38 b(b) m(y) h(estimating) d(the) i(time) f(deriv) -6 b(ativ) m(e) 37 b(of) i(the) f(function) h(in) f(square) g(brac) m(k) m(ets) h(whic) m (h) g(is) f(analytic) 0 2459 y(in) p Fj 33 w(F) p Fk 188 2474 a(s) p Fm 230 2459 a(.) p 3891 2385 V 3891 2455 4 70 v 3965 2455 V 3891 2459 78 4 v 199 2578 a(A) 48 b(further) h(corollary) e(concerns) i(the) f(case) h(of) f(the) g (nonlinear) h(Klein) f(Gordon) g(equation) g(with) 0 2698 y(p) s(erio) s(dic) c(b) s(oundary) h(conditions.) 77 b(T) -8 b(o) 43 b(state) h(it) g(denote) h(b) m(y) p Fl 44 w(J) p Fk 2460 2713 a(k) p Fm 2555 2698 a(:=) p Fl 46 w(I) p Fk 2750 2713 a(k) p Fm 2829 2698 a(+) p Fl 30 w(I) p Fe 2980 2713 a(\000) p Fk(k) p Fm 3135 2698 a(the) f(sum) g(of) h(actions) 0 2817 y(corresp) s(onding) 34 b(to) f(mo) s(des) g(with) g(the) g(same) g(frequency) -8 b(.) p Fn 0 2977 a(Corollary) 188 b(6.7.) p Fg 41 w(Consider) 41 b(equation) f(\(0.1\)) g(with) g(p) s(erio) s(dic) h(b) s(oundary) g (conditions) g(on) p Fm 41 w([) p Fj(\000) p Fl(\031) t(;) 17 b(\031) p Fm 4 w(]) p Fg 0 3096 a(in) 43 b(the) g(particular) g(case) p Fl 43 w(V) p Fm 66 w(=) h(0) p Fg(,) h(and) e(\014x) p Fl 43 w(m) p Fg 43 w(in) g(suc) m(h) i(a) d(w) m(a) m(y) h(that) f(the) i(frequencies) g(are) f(strongly) 0 3216 y(non) m(v) -6 b(anishing.) 51 b(Fix) p Fl 34 w(M) p Fj 42 w(\025) p Fm 31 w(4) p Fg 35 w(and) p Fl 35 w(s) p Fj 31 w(\025) p Fl 31 w(s) p Fe 1583 3231 a(\003) p Fg 1629 3216 a(,) 35 b(then) h(there) g(exist) e(p) s(ositiv) m(e) h(constan) m(ts) p Fl 36 w(R) p Fk 3305 3231 a(s) p Fg 3347 3216 a(,) p Fl 35 w(C) p Fk 3481 3231 a(s) p Fg 3559 3216 a(suc) m(h) i(that) 0 3335 y(the) i(follo) m(wing) g(holds) g(true:) 56 b(let) p Fl 39 w(\020) p Fm 7 w(\() p Fl(t) p Fm(\)) p Fj 37 w(\021) p Fm 38 w(\() p Fl(u) p Fm(\() p Fl(t) p Fm(\)) p Fl(;) 17 b(v) p Fm 4 w(\() p Fl(t) p Fm(\)\)) p Fg 37 w(b) s(e) 39 b(the) g(solution) g(of) g(the) h(Cauc) m(h) m(y) g(problem) e(for) 0 3455 y(equation) 33 b(\(0.1\)) f(with) h(initial) f(datum) p Fm 32 w(\() p Fl(u) p Fi 1571 3470 a(0) p Fl 1615 3455 a(;) 17 b(v) p Fi 1708 3470 a(0) p Fm 1753 3455 a(\)) p Fj 27 w(2) p Fl 28 w(H) p Fk 2004 3419 a(s) p Fm 2047 3455 a(\() g(I) -17 b(R) p Fl(=) p Fm(2) p Fl(\031) p Fn 4 w(Z) p Fm(\)) p Fj 21 w(\002) p Fl 22 w(H) p Fk 2676 3419 a(s) p Fe(\000) p Fi(1) p Fm 2821 3455 a(\() 17 b(I) -17 b(R) p Fl(=) p Fm(2) p Fl(\031) p Fn 4 w(Z) p Fm(\)) p Fg 32 w(if) p Fl 1389 3682 a(R) p Fm 28 w(:=) p Fj 28 w(k) p Fm -1 w(\() p Fl(u;) 17 b(v) p Fm 4 w(\)) p Fj(k) p Fk 1956 3712 a(H) p Fd 2026 3692 a(s) p Fe 2065 3712 a(\002) p Fk(H) p Fd 2197 3692 a(s) p Fc(\000) p Fb(1) p Fj 2357 3682 a(\024) p Fl 28 w(R) p Fk 2538 3697 a(s) p Fg 0 3908 a(then) 34 b(for) f(all) g(times) p Fl 32 w(t) p Fg 33 w(with) p Fj 1670 4050 a(j) p Fl(t) p Fj(j) 27 b(\024) p Fm 2071 3983 a(1) p 1906 4027 381 4 v Fl 1906 4118 a(C) p Fk 1977 4133 a(s) p Fl 2020 4118 a(R) p Fk 2097 4090 a(M) p Fe 7 w(\000) p Fi(3) p Fm 3714 4050 a(\(6) p Fl(:) p Fm(11\)) p Fg 0 4277 a(the) 34 b(follo) m(wing) e(estimates) h(hold) p Fj 892 4555 a(k) p Fl -1 w(\020) p Fm 7 w(\() p Fl(t) p Fm(\)) p Fj(k) p Fk 1156 4585 a(H) p Fd 1226 4565 a(s) p Fe 1265 4585 a(\002) p Fk(H) p Fd 1397 4565 a(s) p Fc(\000) p Fb(1) p Fj 1556 4555 a(\024) p Fm 28 w(2) p Fl(C) p Fk 1782 4570 a(s) p Fl 1825 4555 a(R) h(;) p Fj 2090 4488 a(j) p Fl(J) p Fk 2173 4503 a(j) p Fm 2215 4488 a(\() p Fl(t) p Fm(\)) p Fj 22 w(\000) p Fl 23 w(J) p Fk 2506 4503 a(j) p Fm 2548 4488 a(\(0\)) p Fj(j) p 2090 4532 613 4 v Fl 2336 4624 a(R) p Fi 2413 4595 a(2) p Fj 2742 4555 a(\024) p Fl 2867 4488 a(C) p Fk 2938 4503 a(s) p 2860 4532 130 4 v Fl 2860 4624 a(j) p Fi 2907 4595 a(2) p Fk(s) p Fl 3001 4555 a(R) p Fm 637 w(\(6) p Fl(:) p Fm(12\)) p Fg 0 5151 a(6.2) f(Measure) h(estimates) p Fm 199 5420 a(Since) p Fl 37 w(!) p Fe 524 5435 a(\000) p Fk(j) p Fm 659 5420 a(=) p Fl 32 w(!) p Fk 830 5435 a(j) p Fm 908 5420 a(w) m(e) i(will) e (consider) j(only) e(the) h(case) g(of) g(p) s(ositiv) m(e) p Fl 34 w(j) p Fm 6 w('s.) 51 b(W) -8 b(e) 36 b(\014rst) g(in) m(tro) s (duce) g(some) 0 5539 y(notations.) 1807 5778 y(20.03.02) p 90 rotate dyy eop %%Page: 29 29 29 28 bop Fg 1222 60 a(Birkho\013) 33 b(normal) f(form) g(for) i(PDE's) p Fm 1222 w(29) 199 299 y(Giv) m(en) 44 b(in) m(tegers) h(0) p Fl 45 w(<) h(i) p Fi 1130 314 a(1) p Fl 1175 299 a(;) 17 b(:::;) g(i) p Fk 1383 314 a(r) p Fj 1470 299 a(\024) p Fl 46 w(N) p Fm 11 w(,) 46 b(and) e(a) f(c) m(hoice) p Fl 45 w(\033) p Fm 47 w(of) p Fl 44 w(r) p Fm 47 w(signs,) j(de\014ne) g (the) e(v) m(ector) p Fn 44 w(k) p Fm 45 w(:=) p Fh 0 344 a(P) p Fk 105 368 a(r) 105 448 y(l) p Fi(=1) p Fj 253 418 a(\006) p Fl(e) p Fk 376 433 a(i) p Fd 404 444 a(l) p Fm 482 418 a(where) p Fl 45 w(e) p Fk 827 433 a(i) p Fd 855 444 a(l) p Fj 934 418 a(2) p Fn 44 w(Z) p Fe 1114 382 a(1) p Fm 1242 418 a(is) f(the) h(v) m(ector) f(with) g(al) g(comp) s(onen) m(ts) h(equal) f(to) g(zero) g(but) h(the) p Fl 44 w(i) p Fk 3794 433 a(l) p Fm 3825 418 a({th) 0 538 y(whic) m(h) 34 b(is) f(equal) g(to) g(1,) g(so) g(that) p Fn 1354 687 a(k) p Fj 23 w(\001) p Fl 22 w(!) p Fm 31 w(=) p Fj 28 w(\006) p Fl(!) p Fk 1824 702 a(i) p Fb 1852 712 a(1) p Fj 1919 687 a(\006) p Fl 22 w(!) p Fk 2080 702 a(i) p Fb 2108 712 a(2) p Fj 2175 687 a(\006) p Fl 22 w(:::) p Fj 21 w(\006) p Fl 23 w(!) p Fk 2541 702 a(i) p Fd 2569 712 a(r) p Ff 0 837 a(We) i(wil) 5 b(l) 36 b(c) -5 b(onsider) 35 b(only) h(choic) -5 b(es) 34 b(of) p Fl 35 w(\033) p Ff 40 w(such) h(that) p Fn 36 w(k) p Fj 28 w(6) p Fm(=) 28 b(0.) 44 b(Then,) 34 b(for) f(\014xed) p Fl 34 w(\034) 6 b(;) 17 b(\015) p Fi 3107 852 a(1) p Fm 3183 837 a(w) m(e) 34 b(de\014ne) p Fj 714 1041 a(R) p Fk 798 1000 a(\033) 798 1065 y(i) p Fm 852 1041 a(\() p 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3134 a(:::i) p Fi 3085 3092 a(2) p Fk(K) p Fe 5 w(\000) p Fi(1) p Fk 3085 3164 a(K) p Fl 3348 3055 a(:) p Fm 338 w(\(6) p Fl(:) p Fm(14\)) p Fn 0 3535 a(Pro) s(of.) p Fm 34 w(First) 33 b(remark) f(that,) h(b) m(y) g(explicit) g(computation) f(one) h(has) p Fl 1162 3676 a(d) p Fk 1214 3639 a(n) p Fl 1269 3676 a(!) p Fk 1331 3691 a(i) p 1162 3720 202 4 v Fl 1166 3811 a(dm) p Fk 1305 3783 a(n) p Fm 1403 3743 a(=) 1651 3676 y(\(2) p Fl(n) p Fj 21 w(\000) p Fm 23 w(3\)!) p 1520 3720 647 4 v 1520 3811 a(2) p Fk 1570 3783 a(n) p Fe(\000) p Fi(2) p Fm 1726 3811 a(\() p Fl(n) p Fj 22 w(\000) p Fm 23 w(2\)!2) p Fk 2114 3783 a(n) p Fm 2288 3676 a(\() p Fj(\000) p Fm(1\)) p Fk 2493 3639 a(n) p Fi(+1) p 2191 3720 555 4 v Fm 2191 3826 a(\() p Fl(\026) p Fk 2290 3841 a(i) p Fm 2345 3826 a(+) p Fl 23 w(m) p Fm(\)) p Fk 2571 3791 a(n) p Fe(\000) p Fb 2694 3764 a(1) p 2694 3775 34 4 v 2694 3815 a(2) p Fl 2790 3743 a(:) p Fm 896 w(\(6) p Fl(:) p Fm(15\)) 0 3965 y(Substituting) d (\(6.15\)) e(in) i(the) g(l.h.s.) 42 b(of) 30 b(\(6.14\)) e(w) m(e) j (get) e(the) h(determinan) m(t) g(to) f(b) s(e) h(estimated.) 42 b(T) -8 b(o) 29 b(obtain) 0 4085 y(the) 24 b(estimate) f(factorize) i (from) e(the) p Fl 25 w(i) p Fj 4 w(\000) p Fl 4 w(th) p Fm 25 w(column) h(the) g(term) f(\() p Fl(\026) p Fk 2401 4100 a(i) p Fm 2438 4085 a(+) p Fl 4 w(m) p Fm(\)) p Fi 2645 4049 a(1) p Fk(=) p Fi(2) p Fm 2771 4085 a(,) j(and) e(from) g (the) p Fl 24 w(j) p Fj 10 w(\000) p Fl 4 w(th) p Fm 25 w(ro) m(w) g(the) 0 4220 y(term) p Fi 353 4172 a(\(2) p Fk(j) p Fe 4 w(\000) p Fi(3\)!) p 252 4197 466 4 v 252 4255 a(2) p Fd 292 4235 a(j) p Fc 3 w(\000) p Fb(2) p Fi 417 4255 a(\() p Fk(j) p Fe 4 w(\000) p Fi(2\)!2) p Fd 681 4235 a(j) p Fm 730 4220 a(.) 50 b(F) -8 b(orgetting) 34 b(the) i(inessen) m(tial) g(p) s(o) m(w) m(ers) g(of) p Fj 36 w(\000) p Fm(1,) f(w) m(e) h(obtain) f(that) g(the) h(determinan) m(t) 0 4351 y(to) d(b) s(e) g(estimated) g(is) g(giv) m(en) g(b) m(y) p Fh 870 4422 a(") p Fk 955 4468 a(K) p Fh 928 4498 a(Y) p Fk 928 4712 a(l) p Fi(=1) p Fl 1072 4593 a(!) p Fk 1134 4608 a(i) p Fd 1162 4619 a(l) p Fh 1196 4422 a(#) 16 b(") p Fk 1329 4468 a(K) p Fe 5 w(\000) p Fi(1) p Fh 1352 4498 a(Y) p Fk 1340 4709 a(n) p Fi(=1) p Fm 1661 4525 a(\(2) p Fl(n) p Fj 22 w(\000) p Fm 22 w(3\)!) p 1531 4570 647 4 v 1531 4661 a(2) p Fk 1581 4632 a(n) p Fe(\000) p Fi(2) p Fm 1737 4661 a(\() p Fl(n) p Fj 22 w(\000) p Fm 22 w(2\)!2) p Fk 2124 4632 a(n) p Fh 2189 4422 a(#) p Fj 1276 5196 a(\002) p Fh 1376 4782 a(\014) 1376 4842 y(\014) 1376 4902 y(\014) 1376 4961 y(\014) 1376 5021 y(\014) 1376 5081 y(\014) 1376 5141 y(\014) 1376 5201 y(\014) 1376 5260 y(\014) 1376 5320 y(\014) 1376 5380 y(\014) 1376 5440 y(\014) 1376 5499 y(\014) p Fm 1518 4837 a(1) 286 b(1) f(1) p Fl 192 w(:) 100 b(:) f(:) p Fm 192 w(1) p Fl 1479 4956 a(x) p Fk 1536 4971 a(i) p Fb 1564 4981 a(1) p Fl 1814 4956 a(x) p Fk 1871 4971 a(i) p Fb 1899 4981 a(2) p Fl 2149 4956 a(x) p Fk 2206 4971 a(i) p Fb 2234 4981 a(3) p Fl 2431 4956 a(:) h(:) f(:) 140 b(x) p Fk 2911 4971 a(i) p Fd 2939 4981 a(K) p Fl 1479 5076 a(x) p Fi 1536 5040 a(2) p Fk 1536 5102 a(i) p Fb 1564 5112 a(1) p Fl 1814 5076 a(x) p Fi 1871 5040 a(2) p Fk 1871 5102 a(i) p Fb 1899 5112 a(2) p Fl 2149 5076 a(x) p Fi 2206 5040 a(2) p Fk 2206 5102 a(i) p Fb 2234 5112 a(3) p Fl 2431 5076 a(:) 100 b(:) f(:) 140 b(x) p Fi 2911 5040 a(2) p Fk 2911 5102 a(i) p Fd 2939 5112 a(K) p Fl 1529 5196 a(:) 308 b(:) f(:) 203 b(:) 100 b(:) f(:) 203 b(:) 1529 5315 y(:) 308 b(:) f(:) 203 b(:) 100 b(:) f(:) 203 b(:) 1529 5435 y(:) 308 b(:) f(:) 203 b(:) 100 b(:) f(:) 203 b(:) 1425 5554 y(x) p Fk 1482 5512 a(K) p Fe 5 w(\000) p Fi(1) p Fk 1482 5582 a(i) p Fb 1510 5592 a(1) p Fl 1761 5554 a(x) p Fk 1818 5512 a(K) p Fe 5 w(\000) p Fi(1) p Fk 1818 5582 a(i) p Fb 1846 5592 a(2) p Fl 2096 5554 a(x) p Fk 2153 5512 a(K) p Fe 5 w(\000) p Fi(1) p Fk 2153 5582 a(i) p Fb 2181 5592 a(3) p Fl 2431 5554 a(:) 100 b(:) f(:) g(x) p Fk 2870 5512 a(K) p Fe 5 w(\000) p Fi(1) p Fk 2870 5582 a(i) p Fd 2898 5592 a(K) p Fh 3066 4782 a(\014) 3066 4842 y(\014) 3066 4902 y(\014) 3066 4961 y(\014) 3066 5021 y(\014) 3066 5081 y(\014) 3066 5141 y(\014) 3066 5201 y(\014) 3066 5260 y(\014) 3066 5320 y(\014) 3066 5380 y(\014) 3066 5440 y(\014) 3066 5499 y(\014) p Fm 3714 5020 a(\(6) p Fl(:) p Fm(16\)) 1807 5778 y(20.03.02) p 90 rotate dyy eop %%Page: 30 30 30 29 bop Fm 0 60 a(30) p Fg 1660 w(D.) 33 b(Bam) m(busi) p Fm 0 299 a(where) e(w) m(e) f(denoted) i(b) m(y) p Fl 30 w(x) p Fk 994 314 a(i) p Fm 1055 299 a(:=) c(\() p Fl(\026) p Fk 1287 314 a(i) p Fm 1335 299 a(+) p Fl 16 w(m) p Fm(\)) p Fe 1554 263 a(\000) p Fi(1) p Fm 1662 299 a(.) 43 b(The) 30 b(last) g(determinan) m(t) g(is) g(a) g(V) -8 b(andermond) 30 b(determinan) m(t) 0 418 y(whose) k(v) -6 b(alue) 33 b(is) g(giv) m(en) h(b) m(y) p Fh 1692 458 a(Y) p Fk 1620 672 a(l) p Fm 28 w(1) p Fk 1753 4816 a(d) p Fd 1794 4786 a(i) p Fk 1826 4816 a(!) p 1744 4832 V 1744 4890 a(dm) p Fd 1856 4870 a(i) p Fm 2143 4855 a(if) p Fh 2309 4741 a(\015) 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y(\015) 2252 5514 y(\015) p Fk 2326 5469 a(d) p Fd 2367 5449 a(l) p Fk 2396 5469 a(!) p 2318 5486 142 4 v 2318 5545 a(dm) p Fd 2430 5525 a(l) p Fh 2471 5395 a(\015) 2471 5455 y(\015) 2471 5514 y(\015) p Fk 2526 5579 a(`) p Fb 2559 5559 a(1) p Fh 2604 5399 a(o) p Fk 2726 5281 a(K) p Fh 2699 5311 a(Y) p Fk 2699 5526 a(l) p Fi(=1) p Fm 2874 5338 a(1) p 2854 5383 89 4 v Fl 2854 5508 a(i) p Fi 2899 5442 a(~) p Fk 2888 5460 a(\014) 2888 5538 y(l) p Fm 1807 5778 a(20.03.02) p 90 rotate dyy eop %%Page: 31 31 31 30 bop Fg 1222 60 a(Birkho\013) 33 b(normal) f(form) g(for) i(PDE's) p Fm 1222 w(31) 0 299 y(with) 247 273 y(~) p Fl 233 299 a(\014) p Fm 44 w(=) k(2) p Fl(r) p Fj 29 w(\000) p Fm 27 w(1) h(and) p Fl 40 w(!) p Fm 43 w(denotes) i(here) f(the) g(v) m (ector) f(with) h(all) e(comp) s(onen) m(ts) i(equal) f(to) g(zero) h (except) 0 418 y(those) 29 b(with) f(indexes) p Fl 29 w(i) p Fi 859 433 a(1) p Fl 904 418 a(;) 17 b(:::;) g(i) p Fk 1112 433 a(K) p Fm 1215 418 a(whic) m(h) 29 b(are) g(put) g(equal) f (to) p Fl 28 w(!) p Fk 2272 433 a(i) p Fb 2300 443 a(1) p Fl 2344 418 a(:::!) p Fk 2490 433 a(i) p Fd 2518 443 a(K) p Fm 2587 418 a(.) 43 b(Then) 29 b(remark) e(that) h(the) p Fl 29 w(`) p Fi 3669 382 a(1) p Fm 3741 418 a(norm) 0 557 y(of) p Fk 137 517 a(d) p Fd 178 487 a(i) p Fk 210 517 a(!) p 129 535 144 4 v 129 592 a(dm) p Fd 241 572 a(i) p Fm 322 557 a(is) 36 b(b) s(ounded) j(b) m(y) e(a) f(constan) m (t) i(\(indep) s(enden) m(t) h(of) p Fl 37 w(w) p Fm 3 w(\)) d(for) i(all) p Fl 36 w(i) p Fj 34 w(\025) p Fm 34 w(2,) f(while) g(for) p Fl 37 w(i) p Fm 34 w(=) d(1) j(w) m(e) g (ha) m(v) m(e) 0 677 y(that) d(it) g(is) h(giv) m(en) g(b) m(y) f(the) h (sum) g(of) g(at) f(most) p Fl 33 w(r) p Fm 38 w(comp) s(onen) m(ts) h (di\013eren) m(t) g(from) f(zero.) 49 b(Remark) 34 b(also) g(that) 0 797 y(eac) m(h) g(comp) s(onen) m(t) g(of) p Fk 867 756 a(d) 5 b(!) p 859 774 118 4 v 859 831 a(dm) p Fm 1022 797 a(is) 33 b(b) s(ounded) i(uniformly) e(with) g(resp) s(ect) h(to) f (the) h(c) m(hoice) h(of) e(the) h(indexes,) g(and) 0 916 y(therefore) j(also) f(for) p Fl 37 w(i) p Fm 32 w(=) e(1) i(the) g(ab) s(o) m(v) m(e) g(norm) g(is) g(b) s(ounded) i(b) m(y) e(a) g(constan) m(t) h(indep) s(enden) m(t) h(of) p Fl 37 w(w) p Fm 3 w(.) 52 b(Th) m(us) 0 1036 y(w) m(e) 34 b(ha) m(v) m(e) p Fh 1401 1083 a(\014) 1401 1143 y(\014) 1401 1203 y(\014) 1401 1263 y(\014) p Fl 1434 1228 a(w) p Fj 25 w(\001) p Fl 1602 1160 a(d) p Fk 1654 1124 a(i) p Fl 1687 1160 a(!) p 1592 1205 173 4 v 1592 1296 a(dm) p Fk 1731 1267 a(i) p Fh 1776 1083 a(\014) 1776 1143 y(\014) 1776 1203 y(\014) 1776 1263 y(\014) p Fj 1837 1228 a(\025) p Fl 28 w(C) p Fj 2033 1159 a(k) p Fl -1 w(w) p Fj 3 w(k) p Fk 2206 1189 a(`) p Fb 2239 1169 a(1) p 2033 1205 251 4 v Fj 2037 1296 a(k) p Fl(!) p Fj 4 w(k) p Fk 2202 1326 a(`) p Fb 2235 1306 a(1) p Fk 2340 1103 a(K) p Fh 2312 1133 a(Y) p Fk 2312 1348 a(l) p Fi(=1) p Fm 2487 1160 a(1) p 2468 1205 89 4 v Fl 2468 1330 a(i) p Fi 2513 1264 a(~) p Fk 2502 1282 a(\014) 2502 1360 y(l) p Fm 0 1509 a(No) m(w,) f(as) g(one) h(can) f(easily) g(c) m(hec) m(k,) p Fj 34 w(k) p Fl(!) p Fj 4 w(k) p Fm 32 w(is) g(b) s(ounded) i(b) m(y) p Fl 1803 1832 a(C) p Fk 1925 1707 a(K) p Fh 1898 1737 a(Y) p Fk 1898 1952 a(l) p Fi(=1) p Fl 2042 1832 a(!) p Fk 2104 1847 a(i) p Fd 2132 1858 a(l) p Fm 0 2156 a(from) e(whic) m(h,) g(using) h(the) f(asymptotic) f(of) h(the) h(frequencies) h(w) m(e) f (get) f(the) g(thesis.) p 3891 2082 78 4 v 3891 2152 4 70 v 3965 2152 V 3891 2156 78 4 v Fn 0 2315 a(Lemma) 162 b(6.11.) p Fg 36 w(\(Lemma) 34 b(8.4) g(of) i([3]\)) e(Let) p Fl 35 w(g) p Fm 35 w(:) p Fj 30 w(I) 39 b(!) p Fm 48 w(I) -17 b(R) p Fg 35 w(b) s(e) p Fl 36 w(r) p Fg 38 w(times) 34 b(di\013eren) m(tiable,) j(and) e(assume) 0 2435 y(that) 22 2554 y(\(1\)) p Fj 49 w(8) p Fl 10 w(m) p Fj 29 w(2) 28 b(I) p Fg 41 w(there) 33 b(exists) p Fl 34 w(i) p Fj 28 w(\024) p Fl 28 w(r) p Fj 25 w(\000) p Fm 22 w(1) p Fg 33 w(suc) m(h) i(that) p Fl 33 w(g) p Fi 2011 2518 a(\() p Fk(i) p Fi(\)) p Fm 2105 2554 a(\() p Fl(m) p Fm(\)) p Fl 28 w(>) 28 b(d) p Fg 22 2674 a(\(2\)) 49 b(there) 34 b(exists) p Fl 33 w(A) p Fg 33 w(suc) m(h) h(that) p Fj 33 w(j) p Fl(g) p Fi 1358 2638 a(\() p Fk(i) p Fi(\)) p Fm 1452 2674 a(\() p Fl(m) p Fm(\)) p Fj(j) 27 b(\024) p Fl 28 w(A) p Fj 33 w(8) p Fl 10 w(m) p Fj 29 w(2) h(I) p Fg 7 w(,) 33 b(and) p Fj 34 w(8) p Fl 10 w(i) p Fg 34 w(with) p Fm 33 w(1) p Fj 27 w(\024) p Fl 28 w(i) p Fj 28 w(\024) p Fl 29 w(r) p Fg 3 w(.) 199 2793 y(De\014ne) p Fj 1322 2913 a(I) p Fk 1376 2928 a(h) p Fm 1455 2913 a(:=) p Fj 28 w(f) p Fl(m) p Fj 28 w(2) 28 b(I) p Fm 68 w(:) p Fj 61 w(j) p Fl(g) p Fm 4 w(\() p Fl(m) p Fm(\)) p Fj(j) e(\024) p Fl 28 w(h) p Fj(g) p Fl 50 w(;) p Fg 0 3096 a(then) p Fj 1170 3181 a(jI) p Fk 1252 3196 a(h) p Fj 1304 3181 a(j) p 1170 3225 162 4 v 1192 3317 a(jI) 7 b(j) 1371 3248 y(\024) p Fl 1488 3181 a(A) p 1488 3225 75 4 v 1499 3317 a(d) p Fm 1575 3248 a(2\(2) 21 b(+) i(3) f(+) p Fl 23 w(:::) p Fm 21 w(+) p Fl 22 w(r) p Fm 25 w(+) h(2) p Fl(d) p Fe 2483 3207 a(\000) p Fi(1) p Fm 2589 3248 a(\)) p Fl(h) p Fi 2685 3207 a(1) p Fk(=r) p Fm 199 3591 a(F) -8 b(or) 33 b(the) h(pro) s(of) f(see) h([3]) e(and) i([20].) 199 3711 y(By) f(com) m(bining) f(lemma) f(6.11) i(and) g(lemma) e(6.10) i (w) m(e) g(get) g(the) h(follo) m(wing) p Fn 0 3870 a(Lemma) 177 b(6.12.) p Fg 39 w(F) -8 b(or) 38 b(an) m(y) h(c) m(hoice) g(of) p Fl 39 w(i) p Fi 1651 3885 a(1) p Fj 1732 3870 a(\024) p Fl 37 w(i) p Fi 1880 3885 a(2) p Fj 1962 3870 a(\024) p Fl 37 w(:::) p Fj 35 w(\024) p Fl 37 w(i) p Fk 2343 3885 a(r) p Fg 2426 3870 a(and) g(an) m(y) g(acceptable) g(c) m(hoice) p Fl 39 w(\033) p Fg 42 w(of) g(the) 0 3990 y(signs,) 33 b(one) h(has) p Fj 1415 4175 a(jR) p Fk 1527 4134 a(\033) 1527 4199 y(i) p Fm 1581 4175 a(\() p Fl(\015) p Fi 1672 4190 a(1) p Fl 1716 4175 a(;) 17 b(\034) p Fm 11 w(\)) p Fj(j) 26 b(\024) i(jI) 7 b(j) p Fl(C) 2271 4107 y(\015) p Fi 2323 4122 a(1) 2367 4071 y(1) p Fk(=r) p 2221 4152 321 4 v Fh 2221 4172 a(Q) p Fk 2315 4197 a(r) 2315 4277 y(l) p Fi(=1) p Fl 2464 4247 a(i) p Fk 2498 4212 a(\016) 2498 4277 y(l) p Fm 3714 4175 a(\(6) p Fl(:) p Fm(18\)) p Fg 0 4425 a(with) p Fl 33 w(\016) p Fm 32 w(=) p Fj 28 w(\000) p Fm(2) p Fl(\014) p Fm 28 w(+) p Fl 22 w(\034) k(=r) p Fg 3 w(.) 44 b(Moreo) m(v) m(er,) 33 b(for) h(an) m(y) f(c) m(hoice) h (also) f(of) p Fl 33 w(j) h(>) 28 b(N) p Fg 44 w(one) 34 b(has) p Fh 1394 4638 a(\014) 1394 4697 y(\014) p Fj 1427 4722 a(R) p Fk 1511 4681 a(\033) 1511 4747 y(ij) p Fm 1582 4722 a(\() p Fl(\015) p Fi 1673 4737 a(1) p Fl 1717 4722 a(;) 17 b(\034) p Fm 11 w(\)) p Fh 1856 4638 a(\014) 1856 4697 y(\014) p Fj 1915 4722 a(\024) 29 b(jI) 7 b(j) p Fl(C) 2235 4655 y(\015) p Fi 2287 4670 a(1) 2331 4619 y(1) p Fk(=r) p Fl 2455 4655 a(j) p Fk 2502 4619 a(\014) p 2228 4699 335 4 v Fh 2228 4732 a(Q) p Fk 2322 4756 a(r) p Fe 2 w(\000) p Fi(1) p Fk 2322 4836 a(l) p Fi(=1) p Fl 2485 4806 a(i) p Fk 2519 4772 a(\016) 2519 4836 y(l) p Fm 3714 4722 a(\(6) p Fl(:) p Fm(19\)) p Fg 0 5022 a(and,) 33 b(for) h(an) m(y) f(c) m(hoice) h(of) p Fl 34 w(k) p Fj 30 w(\025) p Fl 28 w(j) p Fg 39 w(one) g(has) p Fh 1324 5229 a(\014) 1324 5288 y(\014) p Fj 1357 5313 a(R) p Fk 1441 5272 a(\033) 1441 5338 y(ij) t(k) p Fm 1556 5313 a(\() p Fl(\015) p Fi 1647 5328 a(1) p Fl 1691 5313 a(;) 17 b(\034) p Fm 11 w(\)) p Fh 1830 5229 a(\014) 1830 5288 y(\014) p Fj 1890 5313 a(\024) 28 b(jI) 7 b(j) p Fl(C) 2202 5246 y(\015) p Fi 2254 5261 a(1) 2298 5210 y(1) p Fk(=r) p Fl 2423 5246 a(j) p Fk 2470 5210 a(\014) p Fl 2524 5246 a(k) p Fk 2579 5210 a(\014) p 2202 5290 431 4 v Fh 2250 5323 a(Q) p Fk 2344 5347 a(r) p Fe 2 w(\000) p Fi(2) p Fk 2344 5427 a(l) p Fi(=1) p Fl 2507 5397 a(i) p Fk 2541 5363 a(\016) 2541 5428 y(l) p Fm 3714 5313 a(\(6) p Fl(:) p Fm(20\)) 1807 5778 y(20.03.02) p 90 rotate dyy eop %%Page: 32 32 32 31 bop Fm 0 60 a(32) p Fg 1660 w(D.) 33 b(Bam) m(busi) p Fn 0 299 a(Lemma) 153 b(6.13.) p Fg 34 w(Assume) p Fl 33 w(i) p Fi 1193 314 a(1) p Fj 1265 299 a(\024) p Fl 29 w(i) p Fi 1405 314 a(2) p Fl 1450 299 a(:::) p Fj 26 w(\024) p Fl 28 w(i) p Fk 1699 314 a(r) p Fe 2 w(\000) p Fi(1) p Fj 1874 299 a(\024) p Fl 28 w(N) p Fg 11 w(,) 32 b(then) i(there) g(exists) p Fl 33 w(C) p Fi 2958 314 a(1) p Fg 3036 299 a(suc) m(h) h(that) p Fj 1816 529 a(R) p Fk 1900 488 a(\033) 1900 554 y(ij) p Fm 1998 529 a(=) p Fj 28 w(;) p Fg 0 760 a(when) p Fl 34 w(j) p Fj 34 w(\025) p Fl 28 w(C) p Fi 511 775 a(1) p Fl 556 760 a(N) p Fn 0 919 a(Pro) s(of.) p Fm 37 w(De\014ning) 751 893 y(\026) p Fn 745 919 a(k) p Fm 36 w(as) p Fh 966 844 a(P) p Fk 1071 869 a(r) p Fe 2 w(\000) p Fi(1) p Fk 1071 949 a(l) p Fi(=1) p Fj 1234 919 a(\006) p Fl(e) p Fk 1357 934 a(i) p Fd 1385 945 a(l) p Fm 1456 919 a(w) m(e) h(ha) m(v) m (e) p Fj 36 w(j) p Fm 1870 893 a(\026) p Fn 1865 919 a(k) p Fj(j) c(\024) p Fl 32 w(r) p Fj 26 w(\000) p Fm 25 w(1,) j(and) i(therefore,) g(for) f(large) f(enough) p Fl 37 w(j) p Fm 41 w(w) m(e) 0 1039 y(ha) m(v) m(e,) p Fh 1343 1074 a(\014) 1343 1133 y(\014) p Fm 1381 1132 a(\026) p Fn 1376 1158 a(k) p Fj 23 w(\001) p Fl 21 w(!) p Fj 26 w(\006) p Fl 23 w(!) p Fk 1758 1173 a(j) p Fh 1800 1074 a(\014) 1800 1133 y(\014) p Fj 1861 1158 a(\025) p Fl 28 w(!) p Fk 2028 1173 a(j) p Fj 2092 1158 a(\000) p Fm 22 w(\() p Fl(r) p Fj 25 w(\000) p Fm 23 w(1\)) p Fl(!) p Fk 2551 1173 a(N) p Fm 0 1344 a(whic) m(h) f(is) f (automatically) d(larger) j(than) h(\() p Fl(r) p Fj 24 w(\000) p Fm 23 w(1\)) p Fl(!) p Fk 1888 1359 a(N) p Fm 1996 1344 a(and) g(also) f(than) p Fl 33 w(\015) p Fi 2674 1359 a(1) p Fl 2719 1344 a(=i) p Fk 2803 1308 a(\034) p Fi 2803 1368 a(1) p Fl 2852 1344 a(:::i) p Fk 2970 1308 a(\034) 2970 1368 y(r) p Fe 2 w(\000) p Fi(1) p Fm 3149 1344 a(if) p Fl 1624 1574 a(!) p Fk 1686 1589 a(j) p Fj 1755 1574 a(\025) p Fm 29 w(2\() p Fl(r) p Fj 24 w(\000) p Fm 23 w(1\)) p Fl(!) p Fk 2270 1589 a(N) p Fm 0 1805 a(whic) m(h) h(is) f(implied) f(b) m(y) i(the) f(h) m(yp) s (othesis,) h(with) f(a) g(suitable) g(c) m(hoice) h(of) p Fl 34 w(C) p Fi 2730 1820 a(1) p Fm 2775 1805 a(.) p 3891 1731 78 4 v 3891 1801 4 70 v 3965 1801 V 3891 1805 78 4 v Fn 0 1964 a(Lemma) 153 b(6.14.) p Fg 34 w(Assume) p Fl 33 w(i) p Fi 1193 1979 a(1) p Fj 1265 1964 a(\024) p Fl 29 w(i) p Fi 1405 1979 a(2) p Fl 1450 1964 a(:::) p Fj 26 w(\024) p Fl 28 w(i) p Fk 1699 1979 a(r) p Fe 2 w(\000) p Fi(1) p Fj 1874 1964 a(\024) p Fl 28 w(N) p Fg 11 w(,) 32 b(then) i(one) g(has) p Fh 1207 2108 a(\014) 1207 2168 y(\014) 1207 2228 y(\014) 1207 2287 y(\014) 1207 2347 y(\014) 1207 2407 y(\014) 1270 2218 y([) p Fk 1240 2431 a(j) p Fe 4 w(\025) p Fk(N) p Fj 1427 2312 a(R) p Fk 1511 2271 a(\033) 1511 2337 y(ij) p Fm 1581 2312 a(\() p Fl(\034) 6 b(;) 17 b(\015) p Fi 1767 2327 a(1) p Fm 1810 2312 a(\)) p Fh 1849 2108 a(\014) 1849 2168 y(\014) 1849 2228 y(\014) 1849 2287 y(\014) 1849 2347 y(\014) 1849 2407 y(\014) p Fj 1910 2312 a(\024) p Fl 2027 2245 a(C) p Fj 7 w(jI) 7 b(j) p Fl(\015) p Fi 2274 2260 a(1) 2318 2209 y(1) p Fk(=r) p Fl 2443 2245 a(N) p Fk 2534 2209 a(\014) p Fi 4 w(+1) p 2027 2289 662 4 v Fh 2190 2322 a(Q) p Fk 2284 2346 a(r) p Fe 2 w(\000) p Fi(1) p Fk 2284 2426 a(l) p Fi(=1) p Fl 2447 2396 a(i) p Fk 2481 2362 a(\016) 2481 2426 y(l) p Fl 2734 2312 a(:) p Fm 952 w(\(6) p Fl(:) p Fm(21\)) p Fn 0 2747 a(Pro) s(of.) p Fm 34 w(W) -8 b(e) 34 b(ha) m(v) m(e) p Fh 809 2868 a(\014) 809 2928 y(\014) 809 2987 y(\014) 809 3047 y(\014) 809 3107 y(\014) 809 3167 y(\014) 871 2977 y([) p Fk 842 3191 a(j) t(>N) p Fj 1028 3072 a(R) p Fk 1112 3031 a(\033) 1112 3097 y(ij) p Fm 1183 3072 a(\() p Fl(\034) 6 b(;) 17 b(\015) p Fi 1369 3087 a(1) p Fm 1411 3072 a(\)) p Fh 1450 2868 a(\014) 1450 2928 y(\014) 1450 2987 y(\014) 1450 3047 y(\014) 1450 3107 y(\014) 1450 3167 y(\014) p Fm 1511 3072 a(=) p Fh 1616 2868 a(\014) 1616 2928 y(\014) 1616 2987 y(\014) 1616 3047 y(\014) 1616 3107 y(\014) 1616 3167 y(\014) 1793 2977 y([) p Fk 1649 3191 a(C) p Fb 1706 3201 a(1) p Fk 1745 3191 a(N) 7 b(>j) p Fe 4 w(\025) p Fk(N) p Fj 2065 3072 a(R) p Fk 2149 3031 a(\033) 2149 3097 y(ij) p Fm 2219 3072 a(\() p Fl(\034) f(;) 17 b(\015) p Fi 2405 3087 a(1) p Fm 2448 3072 a(\)) p Fh 2487 2868 a(\014) 2487 2928 y(\014) 2487 2987 y(\014) 2487 3047 y(\014) 2487 3107 y(\014) 2487 3167 y(\014) p Fj 2548 3072 a(\024) p Fk 2704 2946 a(C) p Fb 2761 2956 a(1) p Fk 2800 2946 a(N) p Fh 2716 2977 a(X) p Fk 2653 3191 a(j) p Fi 4 w(=) p Fk(N) p Fi 7 w(+1) p Fh 2939 2987 a(\014) 2939 3047 y(\014) p Fj 2972 3072 a(R) p Fk 3056 3031 a(\033) 3056 3097 y(ij) p Fh 3127 2987 a(\014) 3127 3047 y(\014) p Fm 0 3425 a(and) 34 b(using) f(lemma) e(6.12) i(this) g (giv) m(es) g(\(6.21\).) p 3891 3351 78 4 v 3891 3421 4 70 v 3965 3421 V 3891 3425 78 4 v 199 3545 a(The) h(estimate) e(of) i (the) g(union) g(o) m(v) m(er) p Fl 33 w(j;) 17 b(k) p Fm 36 w(of) 34 b(the) g(measure) f(of) h(the) f(sets) p Fj 34 w(R) p Fk 2991 3508 a(\033) 2991 3573 y(ij) t(k) p Fm 3139 3545 a(is) h(more) e(di\016cult.) 45 b(It) 0 3664 y(follo) m(ws) 33 b(the) h(pro) s(of) f(of) h(lemma) d(8) i(of) g ([18]) f(and) i(lemma) d(3.1) h(of) i([12].) p Fn 0 3824 a(Lemma) 153 b(6.15.) p Fg 34 w(Assume) p Fl 33 w(i) p Fi 1193 3839 a(1) p Fj 1265 3824 a(\024) p Fl 29 w(i) p Fi 1405 3839 a(2) p Fl 1450 3824 a(:::) p Fj 26 w(\024) p Fl 28 w(i) p Fk 1699 3839 a(r) p Fe 2 w(\000) p Fi(2) p Fj 1874 3824 a(\024) p Fl 28 w(N) p Fg 11 w(.) 44 b(One) 34 b(has) p Fh 1229 3968 a(\014) 1229 4027 y(\014) 1229 4087 y(\014) 1229 4147 y(\014) 1229 4207 y(\014) 1229 4266 y(\014) 1229 4326 y(\014) 1291 4107 y([) p Fd 1274 4309 a(j) p Fc 3 w(\025) p Fd(N) 1284 4366 y(k) q(>j) p Fj 1447 4202 a(R) p Fk 1531 4161 a(\033) 1531 4226 y(ij) t(k) p Fm 1646 4202 a(\() p Fl(\034) 6 b(;) 17 b(\015) p Fi 1832 4217 a(1) p Fm 1875 4202 a(\)) p Fh 1914 3968 a(\014) 1914 4027 y(\014) 1914 4087 y(\014) 1914 4147 y(\014) 1914 4207 y(\014) 1914 4266 y(\014) 1914 4326 y(\014) p Fj 1974 4202 a(\024) p Fl 28 w(C) 2170 4134 y(\015) p Fi 2222 4149 a(1) p Fk 2266 4098 a(\022) r(=r) p Fb 2386 4068 a(2) p Fl 2430 4134 a(N) p Fd 2533 4065 a(\014) s(\022) p 2533 4083 V 2554 4122 a(r) p Fi 2622 4098 a(+1) p 2170 4179 559 4 v Fh 2176 4247 a(Q) p Fk 2270 4272 a(r) p Fe 2 w(\000) p Fi(2) p Fk 2270 4352 a(l) p Fi(=1) p Fl 2433 4322 a(i) p Fd 2479 4225 a(\034) 6 b(\022) p 2479 4236 76 4 v 2480 4287 a(r) p Fb 2515 4272 a(2) p Fe 2566 4252 a(\000) p Fi(2) p Fk(\014) 2467 4352 y(l) p Fg 0 4576 a(with) p Fl 1561 4718 a(\022) p Fm 30 w(:=) 2034 4650 y(1) p 1783 4695 553 4 v 1783 4786 a(2) p Fl(\014) p Fm 27 w(+) 23 b(1) f(+) h(1) p Fl(=r) 2380 4718 y(:) p Fn 0 5063 a(Pro) s(of.) p Fm 38 w(Clearly) 36 b(the) h(di\016culties) h (arise) f(when) h(the) f(sign) g(in) g(fron) m(t) h(of) p Fl 37 w(!) p Fk 2785 5078 a(k) p Fm 2871 5063 a(is) f(opp) s(osite) g (to) g(the) g(sign) g(in) 0 5182 y(fron) m(t) d(of) p Fl 33 w(!) p Fk 419 5197 a(j) p Fm 461 5182 a(.) 44 b(So,) 33 b(w) m(e) h(denote) 459 5451 y(~) p Fj 433 5476 a(R) p Fk 517 5491 a(i;j;k) p Fm 702 5476 a(:=) p Fh 835 5335 a(\032) p Fl 910 5476 a(m) p Fj 28 w(2) 28 b(I) p Fm 69 w(:) p Fh 1330 5391 a(\014) 1330 5451 y(\014) p Fl 1363 5476 a(!) p Fk 1425 5491 a(i) p Fb 1453 5501 a(1) p Fj 1520 5476 a(\006) p Fl 22 w(!) p Fk 1681 5491 a(i) p Fb 1709 5501 a(2) p Fj 1775 5476 a(\006) p Fl 23 w(:::) p Fj 21 w(\006) p Fl 23 w(!) p Fk 2142 5491 a(i) p Fd 2170 5501 a(r) p Fc 2 w(\000) p Fb(2) p Fj 2325 5476 a(\006) p Fm 23 w(\() p Fl(!) p Fk 2526 5491 a(j) p Fj 2590 5476 a(\000) p Fl 22 w(!) p Fk 2751 5491 a(k) p Fm 2800 5476 a(\)) p Fh 2839 5391 a(\014) 2839 5451 y(\014) p Fl 2900 5476 a(<) 3185 5408 y(\015) p Fi 3237 5423 a(1) p 3017 5453 432 4 v Fl 3017 5544 a(i) p Fk 3051 5510 a(\034) p Fi 3051 5571 a(1) p Fl 3101 5544 a(i) p Fk 3135 5510 a(\034) p Fi 3135 5571 a(2) p Fl 3185 5544 a(:::i) p Fk 3303 5510 a(\034) 3303 5571 y(r) p Fe 2 w(\000) p Fi(2) p Fh 3461 5335 a(\033) p Fm 1807 5778 a(20.03.02) p 90 rotate dyy eop %%Page: 33 33 33 32 bop Fg 1222 60 a(Birkho\013) 33 b(normal) f(form) g(for) i(PDE's) p Fm 1222 w(33) 0 299 y(Denote) p Fl 34 w(n) p Fm 27 w(=) p Fl 28 w(k) p Fj 26 w(\000) p Fl 22 w(j) p Fm 6 w(,) f(one) g(has) p Fl 1593 418 a(!) p Fk 1655 433 a(k) p Fj 1726 418 a(\000) p Fl 23 w(!) p Fk 1888 433 a(j) p Fm 1958 418 a(=) p Fl 28 w(n) p Fm 22 w(+) p Fl 22 w(r) p Fk 2289 433 a(j) t(k) p Fm 0 606 a(with) p Fj 1759 726 a(j) p Fl(r) p Fk 1832 741 a(j) t(k) p Fj 1918 726 a(j) 27 b(\024) p Fl 2092 658 a(c) p 2090 703 47 4 v 2090 794 a(j) 2182 726 y(:) p Fm 0 961 a(No) m(w,) 33 b(one) g(can) h(easily) e (v) m(erify) h(that,) g(de\014ning) p Fl 937 1251 a(Q) p Fk 1016 1209 a(n) 1016 1275 y(ij) p Fm 1113 1251 a(:=) p Fh 1246 1110 a(\032) p Fl 1321 1251 a(m) p Fj 28 w(2) 28 b(I) p Fm 69 w(:) p Fj 60 w(j) p Fn(k) p Fj 23 w(\001) p Fl 21 w(!) p Fm 26 w(+) p Fl 23 w(n) p Fj(j) f(\024) p Fl 2489 1183 a(\015) p Fi 2541 1198 a(1) p 2321 1228 432 4 v Fl 2321 1319 a(i) p Fk 2355 1285 a(\034) p Fi 2355 1345 a(1) p Fl 2405 1319 a(i) p Fk 2439 1285 a(\034) p Fi 2439 1345 a(2) p Fl 2489 1319 a(:::i) p Fk 2607 1285 a(\034) 2607 1345 y(r) p Fe 2 w(\000) p Fi(2) p Fm 2787 1251 a(+) p Fl 2900 1183 a(c) p 2898 1228 47 4 v 2898 1319 a(j) p Fh 2957 1110 a(\033) p Fm 0 1572 a(one) c(has) 364 1546 y(~) p Fj 339 1572 a(R) p Fk 423 1587 a(i;j;j) p Fi 4 w(+) p Fk(n) p Fj 711 1572 a(\032) p Fl 29 w(Q) p Fk 896 1535 a(n) 896 1597 y(ij) p Fm 988 1572 a(\(for) f(details) h (see) g([3]\),) g(remarking) e(also) h(that,) i(for) p Fl 23 w(j) p Fj 33 w(\025) p Fl 28 w(j) p Fi 3069 1587 a(0) p Fm 3137 1572 a(one) e(has) p Fl 23 w(Q) p Fk 3554 1535 a(n) 3554 1597 y(ij) p Fj 3652 1572 a(\032) p Fl 28 w(Q) p Fk 3836 1535 a(n) 3836 1597 y(ij) p Fb 3897 1607 a(0) p Fm 3941 1572 a(,) 0 1691 y(w) m(e) 34 b(obtain) p Fh 1479 1701 a(\014) 1479 1761 y(\014) 1479 1821 y(\014) 1479 1880 y(\014) 1479 1940 y(\014) 1479 2000 y(\014) 1512 1811 y([) p Fk 1549 2023 a(j) p Fm 1665 1880 a(~) p Fj 1640 1905 a(R) p Fk 1724 1920 a(i;j;j) p Fi 4 w(+) p Fk(n) p Fh 1985 1701 a(\014) 1985 1761 y(\014) 1985 1821 y(\014) 1985 1880 y(\014) 1985 1940 y(\014) 1985 2000 y(\014) p Fj 2046 1905 a(\024) p Fh 2151 1701 a(\014) 2151 1761 y(\014) 2151 1821 y(\014) 2151 1880 y(\014) 2151 1940 y(\014) 2151 2000 y(\014) p Fl 2184 1905 a(Q) p Fk 2263 1864 a(n) 2263 1930 y(ij) p Fb 2324 1940 a(0) p Fj 2390 1905 a([) p Fh 2478 1705 a(0) 2478 1884 y(@) 2596 1811 y([) p Fk 2566 2023 a(j) t() p Fm 28 w(1) p Fg 33 w(there) i(exists) p Fl 33 w(C) p Fg 40 w(suc) m(h) h(that) p Fh 1200 4132 a(\014) 1200 4192 y(\014) 1200 4252 y(\014) 1200 4311 y(\014) 1200 4371 y(\014) p Fe 1277 4182 a(1) p Fh 1262 4212 a([) p Fk 1233 4424 a(i) p Fb 1261 4434 a(1) p Fi 1300 4424 a(=1) p Fe 1462 4182 a(1) p Fh 1446 4212 a([) p Fk 1418 4424 a(i) p Fb 1446 4434 a(2) p Fi 1485 4424 a(=1) p Fl 1602 4306 a(:::) p Fe 1747 4182 a(1) p Fh 1731 4212 a([) p Fk 1703 4424 a(i) p Fd 1731 4434 a(r) p Fi 1770 4424 a(=1) p Fh 1888 4212 a([) p Fk 1919 4421 a(\033) p Fj 2015 4306 a(R) p Fk 2099 4265 a(\033) 2099 4331 y(i) p Fh 2153 4132 a(\014) 2153 4192 y(\014) 2153 4252 y(\014) 2153 4311 y(\014) 2153 4371 y(\014) p Fj 2214 4306 a(\024) p Fl 28 w(C) 7 b(\015) p Fi 2449 4321 a(1) 2494 4265 y(1) p Fk(=r) p Fj 2618 4306 a(jI) g(j) p Fg 0 4645 a(\(see) 34 b(\(6.13\)) e(for) h(the) h(de\014nition) f(of) p Fj 34 w(R) p Fk 1452 4608 a(\033) 1452 4671 y(i) p Fg 1506 4645 a(\).) p Fn 0 4864 a(Pro) s(of.) p Fm 34 w(By) g(\(6.18\)) e (one) j(has) p Fh 556 5020 a(\014) 556 5080 y(\014) 556 5140 y(\014) 556 5199 y(\014) 556 5259 y(\014) p Fe 633 5070 a(1) p Fh 618 5100 a([) p Fk 589 5312 a(i) p Fb 617 5322 a(1) p Fi 656 5312 a(=0) p Fe 818 5070 a(1) p Fh 802 5100 a([) p Fk 774 5312 a(i) p Fb 802 5322 a(2) p Fi 841 5312 a(=1) p Fl 958 5194 a(:::) p Fe 1103 5070 a(1) p Fh 1087 5100 a([) p Fk 1059 5312 a(i) p Fd 1087 5322 a(r) p Fi 1126 5312 a(=1) p Fh 1244 5100 a([) p Fk 1275 5309 a(\033) p Fj 1371 5194 a(R) p Fk 1455 5153 a(\033) 1455 5219 y(i) p Fh 1509 5020 a(\014) 1509 5080 y(\014) 1509 5140 y(\014) 1509 5199 y(\014) 1509 5259 y(\014) p Fj 1570 5194 a(\024) p Fh 1766 5100 a(X) p Fk 1675 5312 a(i) p Fb 1703 5322 a(1) p Fk 1742 5312 a(;:::;i) p Fd 1890 5322 a(r) p Fk 1929 5312 a(;\033) p Fj 2018 5194 a(j) o(R) p Fk 2129 5153 a(\033) 2129 5219 y(i) p Fj 2184 5194 a(j) 27 b(\024) p Fl 28 w(C) 7 b(\015) p Fi 2474 5209 a(1) 2519 5153 y(1) p Fk(=r) p Fj 2643 5194 a(jI) g(j) p Fm(2) p Fk 2810 5153 a(r) p Fh 2871 5024 a( ) p Fe 2982 5070 a(1) p Fh 2950 5100 a(X) p Fk 2957 5312 a(i) p Fi(=0) p Fm 3136 5127 a(1) p 3122 5172 79 4 v Fl 3122 5263 a(i) p Fk 3156 5234 a(\016) p Fh 3212 5024 a(!) p Fk 3291 5045 a(r) p Fl 3385 5194 a(;) p Fm 0 5539 a(from) 33 b(whic) m(h,) g(remarking) f(that) h(the) g(series) h (is) g(con) m(v) m(ergen) m(t) g(one) g(has) f(the) h(thesis.) p 3891 5466 78 4 v 3891 5535 4 70 v 3965 5535 V 3891 5539 78 4 v 1807 5778 a(20.03.02) p 90 rotate dyy eop %%Page: 34 34 34 33 bop Fm 0 60 a(34) p Fg 1660 w(D.) 33 b(Bam) m(busi) p Fn 0 299 a(Remark) 192 b(6.17.) p Fg 43 w(Here) 42 b(w) m(e) g(did) g (not) g(assume) g(that) f(the) h(indexes) h(are) f(smaller) e(than) p Fl 43 w(N) p Fg 11 w(;) 45 b(on) d(the) 0 418 y(con) m(trary) 30 b(w) m(e) h(considered) h(them) d(as) i(v) -6 b(arying) 29 b(from) h(zero) g(to) g(in\014nit) m(y) g(th) m(us) i(this) e(theorem) g (ensures) i(that) 0 538 y(the) i(condition) f(\(3.1\)) f(is) h (ful\014lled) h(for) f(almost) f(all) g(masses.) p Fn 0 738 a(Remark) 153 b(6.18.) p Fg 34 w(The) 34 b(condition) p Fl 33 w(\016) d(>) p Fm 29 w(1) p Fg 33 w(explicitly) g(reads) p Fl 34 w(\034) 39 b(>) p Fm 28 w(2) p Fl(r) p Fi 2703 702 a(2) p Fm 2769 738 a(+) p Fl 23 w(r) p Fg 3 w(.) p Fm 199 899 a(Analogously) 32 b(one) i(has) p Fn 0 1059 a(Theorem) 152 b(6.19.) p Fg 34 w(Assume) p Fl 1694 1142 a(\034) 11 b(\022) p 1694 1186 105 4 v 1700 1277 a(r) p Fi 1748 1249 a(2) p Fj 1832 1209 a(\000) p Fm 23 w(2) p Fl(\014) 33 b(>) p Fm 28 w(1) p Fl 33 w(;) p Fm 1427 w(\(6) p Fl(:) p Fm(23\)) p Fg 0 1429 a(then) p Fh 898 1443 a(\014) 898 1502 y(\014) 898 1562 y(\014) 898 1622 y(\014) 898 1682 y(\014) 898 1741 y(\014) p Fk 980 1522 a(N) p Fh 960 1552 a([) p Fk 931 1764 a(i) p Fb 959 1774 a(1) p Fi 998 1764 a(=0) p Fk 1164 1522 a(N) p Fh 1144 1552 a([) p Fk 1116 1764 a(i) p Fb 1144 1774 a(2) p Fi 1183 1764 a(=0) p Fl 1300 1647 a(:::) p Fk 1493 1522 a(N) p Fh 1473 1552 a([) p Fk 1401 1764 a(i) p Fd 1429 1774 a(r) p Fc 2 w(\000) p Fb(1) p Fi 1556 1764 a(=0) p Fh 1674 1552 a([) p Fk 1705 1761 a(\033) p Fh 1881 1552 a([) p Fk 1801 1766 a(j) p Fe 4 w(\025) p Fk(N) p Fi 7 w(+1) p Fj 2088 1647 a(R) p Fk 2172 1606 a(\033) 2172 1671 y(ij) p Fh 2243 1443 a(\014) 2243 1502 y(\014) 2243 1562 y(\014) 2243 1622 y(\014) 2243 1682 y(\014) 2243 1741 y(\014) p Fj 2304 1647 a(\024) p Fl 28 w(C) p Fj 7 w(jI) 7 b(j) p Fl(\015) p Fi 2656 1662 a(1) 2700 1606 y(1) p Fk(=r) p Fl 2825 1647 a(N) p Fk 2916 1606 a(\014) p Fi 4 w(+1) p Fg 0 1953 a(and) 34 b(moreo) m(v) m(er) p Fh 652 2086 a(\014) 652 2146 y(\014) 652 2206 y(\014) 652 2265 y(\014) 652 2325 y(\014) 652 2385 y(\014) p Fk 734 2166 a(N) p Fh 714 2196 a([) p Fk 686 2408 a(i) p Fb 714 2418 a(1) p Fi 753 2408 a(=0) p Fk 919 2166 a(N) p Fh 899 2196 a([) p Fk 870 2408 a(i) p Fb 898 2418 a(2) p Fi 937 2408 a(=0) p Fl 1055 2290 a(:::) p Fk 1248 2166 a(N) p Fh 1228 2196 a([) p Fk 1156 2408 a(i) p Fd 1184 2418 a(r) p Fc 2 w(\000) p Fb(1) p Fi 1311 2408 a(=0) p Fh 1428 2196 a([) p Fk 1459 2405 a(\033) p Fh 1635 2196 a([) p Fk 1555 2409 a(j) p Fe 4 w(\025) p Fk(N) p Fi 7 w(+1) p Fh 1926 2196 a([) p Fk 1843 2410 a(k) p Fe 2 w(\025) p Fk(N) p Fi 7 w(+1) p Fj 2137 2290 a(R) p Fk 2221 2249 a(\033) 2221 2315 y(ij) t(k) p Fh 2336 2086 a(\014) 2336 2146 y(\014) 2336 2206 y(\014) 2336 2265 y(\014) 2336 2325 y(\014) 2336 2385 y(\014) p Fj 2397 2290 a(\024) p Fl 28 w(C) p Fj 7 w(jI) 7 b(j) p Fl(\015) p Fi 2749 2305 a(1) p Fk 2793 2249 a(\022) r(=r) p Fb 2913 2219 a(2) p Fl 2957 2290 a(N) p Fd 3060 2216 a(\014) s(\022) p 3060 2234 78 4 v 3081 2273 a(r) p Fi 3149 2249 a(+1) p Fl 3288 2290 a(:) p Fn 0 2837 a(Remark) 153 b(6.20.) p Fg 34 w(Eq.) 43 b(\(6.23\)) 32 b(is) h(implied) f(b) m(y) p Fl 34 w(\034) 38 b(>) 28 b(C) 7 b(r) p Fi 2205 2801 a(4) p Fg 2250 2837 a(.) p Fn 0 3012 a(Pro) s(of) 39 b(of) g(theorem) e(6.5.) p Fm 45 w(Just) d(tak) m(e) p Fl 32 w(\015) p Fi 1606 3027 a(1) p Fm 1678 3012 a(=) p Fl 29 w(\015) 6 b(=) -6 b(N) p Fi 1977 2975 a(4) p Fk(r) p Fb 2056 2945 a(3) p Fm 2132 3012 a(and) 33 b(apply) g(theorem) g(6.19.) p 3891 2938 V 3891 3008 4 70 v 3965 3008 V 3891 3012 78 4 v Fn 0 3172 a(Remark) 129 b(6.21.) p Fg 29 w(Remark) 27 b(that) g(in) i(the) f (de\014nition) h(of) f(strongly) f(non) m(v) -6 b(anishing) 29 b(frequencies) h(app) s(ear) 0 3292 y(the) d(quan) m(tities) e(in) i (the) f(de\014nition) h(of) f(the) h(sets) p Fj 27 w(R) p Fk 1870 3255 a(\033) 1870 3320 y(ij) t(k) p Fg 2010 3292 a(with) p Fl 26 w(r) p Fm 11 w(+) 8 b(2) p Fg 27 w(instead) 26 b(of) p Fl 27 w(r) p Fg 3 w(.) 41 b(Actualy) 25 b(the) i(quan) m (tities) 0 3411 y(en) m(tering) g(in) g(the) f(p) s(erturbativ) m(e) h (construction) g(are) g(those) g(w) m(e) f(estimated) g(in) h(this) f (section.) 42 b(If) 27 b(one) g(w) m(an) m(ts) 0 3531 y(to) j(\014t) h(in) g(the) g(de\014nition) g(of) g(strongly) f(non) m (v) -6 b(anishing) 32 b(frequencies) h(one) e(should) g(just) h (substitute) f(in) g(the) 0 3650 y(estimates) h(of) i(this) f(section) p Fl 34 w(r) p Fm 25 w(+) 22 b(2) p Fg 33 w(to) p Fl 33 w(r) p Fg 3 w(.) p Fn 0 3851 a(Remark) 114 b(6.22.) p Fg 26 w(The) 25 b(theory) g(of) g(this) f(subsection) j(directly) d (applies) h(also) g(to) f(the) h(case) h(of) f(frequencies) 0 3970 y(of) 33 b(the) h(form) p Fl 1660 4093 a(!) p Fk 1722 4108 a(j) p Fm 1792 4093 a(=) p Fh 1897 4013 a(p) p 1997 4013 312 4 v Fl 1997 4093 a(\026) p Fk 2057 4108 a(j) p Fm 2121 4093 a(+) p Fl 23 w(m) p Fg 0 4296 a(with) p Fl 33 w(\026) p Fk 287 4311 a(j) p Fj 357 4296 a(\030) p Fl 28 w(j) p Fk 509 4260 a(d) p Fg 588 4296 a(and) p Fl 34 w(d) 27 b(>) p Fm 29 w(1) p Fg(.) p Fn 1715 5017 a(References) p Fm 44 5138 a([1]) 49 b(D.) 36 b(Bam) m(busi,) h(N.N.) e (Nekhoroshev:) p Ff 50 w(A) k(pr) -5 b(op) g(erty) 39 b(of) f(exp) -5 b(onential) 38 b(stability) g(in) g(nonline) -5 b(ar) 39 b(wave) 199 5258 y(e) -5 b(quation) 35 b(ne) -5 b(ar) 36 b(the) f(fundamental) i(line) -5 b(ar) 35 b(mo) -5 b(de) p Fm(.) 44 b(Ph) m(ysica) 33 b(D) p Fn 33 w(122) p Fm(,) h(73-104) f(\(1998\).) 44 5420 y([2]) 49 b(D.) 27 b(Bam) m(busi:) p Ff 41 w(Nekhor) -5 b(oshev) 28 b(the) -5 b(or) g(em) 30 b(for) g(smal) 5 b(l) 31 b(amplitude) g(solutions) f(in) g(nonline) -5 b(ar) 30 b(Schr\177) -51 b(odin-) 199 5539 y(ger) 36 b(e) -5 b(quations.) p Fm 43 w(Math.) 44 b(Z.) p Fn 33 w(130) p Fm(,) 33 b(345{387,) f(\(1999\).) 1807 5778 y(20.03.02) p 90 rotate dyy eop %%Page: 35 35 35 34 bop Fg 1222 60 a(Birkho\013) 33 b(normal) f(form) g(for) i(PDE's) p Fm 1222 w(35) 44 299 y([3]) 49 b(D.) 30 b(Bam) m(busi:) p Ff 42 w(On) j(long) g(time) f(stability) h(in) f(Hamiltonian) h(p) -5 b(erturb) g(ations) 32 b(of) g(nonr) -5 b(esonant) 32 b(line) -5 b(ar) 199 418 y(PDE's,) p Fm 32 w(Nonlinearit) m(y) p Fn 32 w(12) p Fm(,) 33 b(823-850,) f(\(1999\).) 44 579 y([4]) 49 b(G.) 23 b(Benettin,) h(L.) f(Galgani,) g(A.) f(Giorgilli:) p Ff 37 w(A) k(Pr) -5 b(o) g(of) 25 b(of) g(Nekhor) -5 b(oshev's) 24 b(The) -5 b(or) g(em) 26 b(for) f(the) h(Stability) 199 699 y(Times) 36 b(in) f(Ne) -5 b(arly) 36 b(Inte) -5 b(gr) g(able) 36 b(Hamiltonian) f(Systems) p Fm 34 w(Cel.) 44 b(Mec) m(h.) p Fn 45 w(37) p Fm(,) 34 b(1-25,) e(\(1985\).) 44 860 y([5]) 49 b(G.) 42 b(Benettin,) i(J.) e(F) -8 b(r\177) -50 b(ohlic) m(h,) 44 b(A.) e(Giorgilli:) p Ff 60 w(A) i(Nekhor) -5 b(oshev-typ) g(e) 42 b(The) -5 b(or) g(em) 44 b(for) g(Hamiltonian) 199 980 y(Systems) 34 b(with) e(In\014nitely) i(Many) g(De) -5 b(gr) g(e) g(es) 32 b(of) g(F) -8 b(r) j(e) g(e) g(dom.) p Fm 43 w(Comm) m(un.) 42 b(Math.) h(Ph) m(ys.) p Fn 44 w(119) p Fm(,) 31 b(95-108) 199 1099 y(\(1988\).) 44 1260 y([6]) 49 b(J.) 35 b(Bourgain:) p Ff 46 w(Construction) i(of) g (appr) -5 b(oximative) 36 b(and) h(almost) h(p) -5 b(erio) g(dic) 35 b(solutions) i(of) g(p) -5 b(erturb) g(e) g(d) 199 1380 y(line) g(ar) 36 b(Schr\177) -51 b(odinger) 34 b(and) i(wave) f(e) -5 b(quation) p Fm 33 w(GAF) -11 b(A) p Fn 33 w(6) p Fm 33 w(201{230) 32 b(\(1995\).) 44 1541 y([7]) 49 b(J.) 30 b(Bourgain:) p Ff 42 w(Quasi-p) -5 b(erio) g(dic) 31 b(solutions) i(of) g(Hamiltonian) g(p) -5 b(erturb) g(ations) 32 b(of) g(2D) h(line) -5 b(ar) 33 b(Schr\177) -51 b(o-) 199 1660 y(dinger) 35 b(e) -5 b(quations.) p Fm 43 w(Ann.) 45 b(of) 33 b(Math.) p Fn 44 w(148) p Fm(,) h(363{439,) d(\(1998\).) 44 1822 y([8]) 49 b(J.) 31 b(Bourgain,) p Ff 30 w(On) i(di\013usion) g(in) g(high-dimensional) e(Hamiltonian) i(systems) g(and) g(PDE.) p Fm 30 w(J.) d(Anal.) 199 1941 y(Math.,) p Fn 33 w(80) p Fm(,) j(1{35) g(\(2000\).) 44 2102 y([9]) 49 b(J.) 28 b(Bourgain:) 41 b(Nonlinear) 27 b(Sc) m(hr\177) -50 b(odinger) 30 b(equations.) 42 b(In) 28 b(\\Hyp) s(erb) s(olic) f(equations) h(and) g (frequency) 199 2222 y(in) m(teractions";) 46 b(L.) c(Ca\013arelli,) g (E.) f(W) -8 b(einan) 42 b(editors.) 70 b(IAS/P) m(ark) 41 b(Cit) m(y) g(mathematics) e(series) k(5.) 199 2341 y(American) 33 b(Mathematical) e(So) s(ciet) m(y) i(\(Pro) m(vidence,) h(Rho) s(de) g (Island) f(1999\).) -6 2502 y([10]) 49 b(L.) 26 b(Chierc) m(hia,) i(J.) e(Y) -8 b(ou:) p Ff 40 w(KAM) 30 b(tori) e(for) h(1D) g(nonline) -5 b(ar) 29 b(wave) g(e) -5 b(quations) 29 b(with) g(p) -5 b(erio) g(dic) 27 b(b) -5 b(oundary) 199 2622 y(c) g(onditions) p Fm(.) 43 b(Comm.) e(Math.) k(Ph) m(ys.) p Fn 44 w(211) p Fm(,) 34 b(497{525) d(\(2000\).) -6 2783 y([11]) 49 b(W.) 42 b(Craig,) g(C.E.) f(W) -8 b(a) m(yne:) p Ff 61 w(Newton) e('s) 42 b(Metho) -5 b(d) 43 b(and) g(Perio) -5 b(dic) 41 b(Solutions) j(of) f (Nonline) -5 b(ar) 42 b(Wave) 199 2902 y(Equations) p Fm 33 w(Comm.) f(Pure) 34 b(and) g(Appl.) 44 b(Math.) p Fn 44 w(46) p Fm 34 w(1409-1501) 32 b(\(1993\).) -6 3064 y([12]) 49 b(S.B.) 29 b(Kuksin:) 42 b(Nearly) 29 b(In) m(tegrable) g (In\014nite-Dimensional) g(Hamiltonian) e(Systems,) i(Lect.) 44 b(Notes) 199 3183 y(Math.) g(1556) 33 b(\(Springer) h(1994\)) -6 3344 y([13]) 49 b(S.B.) 37 b(Kuksin,) j(J.) e(P\177) -50 b(osc) m(hel:) p Ff 54 w(Invariant) 40 b(Cantor) g(manifolds) g(of) f (quasi-p) -5 b(erio) g(dic) 38 b(oscil) 5 b(lations) 39 b(for) 199 3464 y(a) d(nonline) -5 b(ar) 35 b(Schr\177) -51 b(odinger) 35 b(e) -5 b(quation) p Fm 32 w(Ann.) 45 b(of) 33 b(Math.) p Fn 44 w(142) p Fm 34 w(149{179) f(\(1995\).) -6 3625 y([14]) 49 b(A.) 39 b(Morbidelli,) h(A.) e(Giorgilli:) p Ff 54 w(Sup) -5 b(er) g(exp) g(onential) 41 b(stability) f(of) h(KAM) g (tori.) p Fm 61 w(J.) e(Statist.) 61 b(Ph) m(ys.) p Fn 199 3744 a(78) p Fm(,) 34 b(1607{1617) d(\(1995\).) -6 3905 y([15]) 49 b(N.N.) 27 b(Nekhoroshev:) p Ff 41 w(Behaviour) j(of) g (Hamiltonian) h(systems) f(close) g(to) h(inte) -5 b(gr) g(able.) p Fm 43 w(F) d(unct.) 42 b(Anal.) 199 4025 y(and) 34 b(Appl.) p Fn 44 w(5) p Fm 34 w(338{339) d(\(1971\).) -6 4186 y([16]) 49 b(N.N.) 31 b(Nekhoroshev:) p Ff 44 w(Exp) -5 b(onential) 34 b(estimate) g(of) g(the) h(stability) f(time) h(of) f(ne) -5 b(ar) 35 b(inte) -5 b(gr) g(able) 33 b(Hamil-) 199 4306 y(tonian) i(systems.) p Fm 46 w(Russ.) 45 b(Math.) f(Surv) m(eys.) p Fn 45 w(32) p Fm 34 w(\(6\)) 32 b(1-65) h(\(1977\).) -6 4467 y([17]) 49 b(L.) 35 b(Niederman:) p Ff 46 w(Nonline) -5 b(ar) 36 b(Stability) h(ar) -5 b(ound) 37 b(an) f(El) 5 b(liptic) 37 b(Equilibrium) f(Point) g(in) h(a) f(Hamilto-) 199 4586 y(nian) f(System) p Fm 34 w(Nonlinearit) m(y) p Fn 32 w(11) p Fm(,) f(1465{1479,) d(\(1998\).) -6 4747 y([18]) 49 b(J.) e(P\177) -50 b(osc) m(hel:) p Ff 71 w(A) 48 b(KAM{The) -5 b(or) g(em) 48 b(for) f(some) h(Partial) f (Di\013er) -5 b(ential) 47 b(Equations) p Fm 46 w(Ann.) 84 b(Scuola) 199 4867 y(Norm.) 43 b(Sup.) i(Pisa) 33 b(Cl.) 43 b(Sci.) p Fn 44 w(23) p Fm(,) 34 b(119{148) e(\(1996\).) -6 5028 y([19]) 49 b(J.) 36 b(P\177) -50 b(osc) m(hel:) p Ff 50 w(Quasi{p) -5 b(erio) g(dic) 36 b(solutions) i(for) f(a) h (nonline) -5 b(ar) 38 b(wave) g(e) -5 b(quation.) p Fm 51 w(Commen) m(t.) 50 b(Math.) 199 5147 y(Helv.) p Fn 88 w(71) p Fm(,) 33 b(269{296,) f(\(1996\).) -6 5308 y([20]) 49 b(J.) 40 b(Xu,) h(J.) e(Y) -8 b(ou,) 41 b(Q.) f(Qiu:) p Ff 58 w(Invariant) h(tori) h(for) f(ne) -5 b(arly) 42 b(inte) -5 b(gr) g(able) 41 b(Hamiltonian) g(systems) h(with) 199 5428 y(de) -5 b(gener) g(acy) p Fm 33 w(Math.) 44 b(Z.) p Fn 33 w(226) p Fm(,) 33 b(375{387) f(\(1997\).) 1807 5778 y(20.03.02) p 90 rotate dyy eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF