 00436 Palle E. T. Jorgensen
 Minimality of the data in wavelet filters
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Nov 7, 00

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Abstract. Orthogonal wavelets, or wavelet frames, for $L^{2}\left( \mathbb{R}\right) $
are associated with quadrature mirror filters (QMF), a
set of complex numbers which relate the dyadic scaling of functions on
$\mathbb{R}$ to the $\mathbb{Z}$translates.
In this paper, we show that generically, the data in the QMFsystems of
wavelets is minimal, in the sense that it cannot be nontrivially reduced. The
minimality property is given a geometric formulation in the Hilbert space
$\ell^{2}\left( \mathbb{Z}\right) $, and it is then shown that minimality
corresponds to irreducibility of a wavelet representation of the algebra
$\mathcal{O}_{2}$; and so our result is that this family of representations of
$\mathcal{O}_{2}$ on the Hilbert space $\ell^{2}\left( \mathbb{Z}\right) $
is irreducible for a generic set of values of the parameters which label the
wavelet representations.
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