Elsevier

Advances in Mathematics

Volume 159, Issue 2, 10 May 2001, Pages 143-228
Advances in Mathematics

Regular Article
Minimality of the Data in Wavelet Filters

WITH AN APPENDIX BY BRIAN TREADWAY
https://doi.org/10.1006/aima.2000.1958Get rights and content
Under an Elsevier user license
open archive

Abstract

Orthogonal wavelets, or wavelet frames, for L2(R) are associated with quadrature mirror filters (QMF), a set of complex numbers which relate the dyadic scaling of functions on R to the Z-translates. In this paper, we show that generically, the data in the QMF-systems of wavelets are minimal, in the sense that the data cannot be nontrivially reduced. The minimality property is given a geometric formulation in the Hilbert space ℓ2(Z), and it is then shown that minimality corresponds to irreducibility of a wavelet representation of the algebra O2; and so our result is that this family of representations of O2 on the Hilbert space ℓ2(Z) is irreducible for a generic set of values of the parameters which label the wavelet representations.

Keywords

wavelet
Cuntz algebra
representation
orthogonal expansion
quadrature mirror filter
isometry in Hilbert space

Cited by (0)

Communicated by R. D. Mauldin

f1

E-mail: [email protected]

1

Partially supported by the National Science Foundation under Grants DMS-9700130 and INT-9722779.