 03353 Vojkan Jaksic and Yoram Last
 A New Proof of Poltoratskii's Theorem
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Jul 30, 03

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Abstract. We provide a new simple proof to the celebrated theorem of Poltoratskii
concerning ratios of Borel transforms of measures. That is, we show
that for any complex Borel measure $\mu$ on $\R$ and any
$f\in L^1(\R,d\mu)$,
$\lim_{\epsilon\to 0}(F_{f\mu}(E+i\epsilon)/F_{\mu}(E+i\epsilon))
= f(E)$ a.e. w.r.t. $\musing$, where $\musing$ is the part of $\mu$
which is singular with respect to Lebesgue measure and $F$ denotes a
Borel transform, namely, $F_{f\mu}(z) = \int (xz)^{1}f(x)\,d\mu(x)$
and $F_{\mu}(z) = \int (xz)^{1}d\mu(x)$.
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