00-436 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)) 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 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.

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