On the Unitary Systems Affiliated with Orthonormal Wavelet Theory inn-Dimensions

Abstract

We consider systems of unitary operators on the complex Hilbert spaceL2(Rn) of the form U≔UDA,Tv1,…,Tvn≔{DmTl1v1…Tlnvn:m,l1,…,ln∈Z}, whereDAis the unitary operator corresponding to dilation by ann×nreal invertible matrixAandTv1,…,Tvnare the unitary operators corresponding to translations by the vectors in a basis {v1,…,vn} for Rn. Orthonormal waveletsψare vectors inL2(Rn) which are complete wandering vectors for U in the sense that {Uψ:U∈U} is an orthonormal basis forL2(Rn). It has recently been established that wheneverAhas 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 (DA,Tv1,…,Tvn) 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 UDA,Tei, UDB,Tei, each corresponding to translation along the coordinate axes, are unitarily equivalent if and only if there is a matrixCwith integer entries and determinant ±1 such thatB=C−1AC. 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 realn×nmatrixAto have the property that the dilation unitary operatorDAis a bilateral shift of infinite multiplicity.