On the Unitary Systems Affiliated with Orthonormal Wavelet Theory inn-Dimensions
✍ Scribed by Eugen J Ionascu; David R Larson; Carl M Pearcy
- Publisher
- Elsevier Science
- Year
- 1998
- Tongue
- English
- Weight
- 359 KB
- Volume
- 157
- Category
- Article
- ISSN
- 0022-1236
No coin nor oath required. For personal study only.
✦ Synopsis
We consider systems of unitary operators on the complex Hilbert space L 2 (R n ) of the form U :=U D A , T v 1 , ...,
where D A is the unitary operator corresponding to dilation by an n_n real invertible matrix A and T v 1 , ..., T v n are the unitary operators corresponding to translations by the vectors in a basis [v 1 , ..., v n ] for R n . Orthonormal wavelets are vectors in L 2 (R n ) which are complete wandering vectors for U in the sense that [U : U # U] is an orthonormal basis for L 2 (R n ). It has recently been established that whenever A has the property that all of its eigenvalues have absolute values strictly greater than one (the expansive case) then U has orthonormal wavelets. The purpose of this paper is to determine when two (n+1)-tuples of the form (D A , T v 1 , ..., T v n ) give rise to the ``same wavelet theory.'' In other words, when is there a unitary transformation of the underlying Hilbert space that transforms one of these unitary systems onto the other? We show, in particular, that two systems U D A , T e i , and U D B , T e i , each corresponding to translation along the coordinate axes, are unitarily equivalent if and only if there is a matrix C with integer entries and determinant \1 such that B=C &1 AC. This means that different expansive dilation factors nearly always yield unitarily inequivalent wavelet theories. Along the way we establish necessary and sufficient conditions for an invertible real n_n matrix A to have the property that the dilation unitary operator D A is a bilateral shift of infinite multiplicity.
1998 Academic Press
1. Introduction
The mathematical concept of an orthonormal wavelet in L 2 (R) has become extremely useful in practical applications to signal processing involving filtering, detection, data compression, etc. In fact, the use of wavelet technology in signal processing is now a big business, and is growing rapidly.