Palle E. T. Jorgensen
Minimality of the data in wavelet filters
(2091K, LaTeX2e amsart class; 69 pages, 2 tables, 4 figures, 12 pages of plots (total 162 EPS graphics))

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 QMF-systems 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.