On the Unitary Systems Affiliated with Orthonormal Wavelet Theory in n-Dimensions

We consider systems of unitary operators on the complex Hilbert space L(R) of the form U :=UDA , v1 ..., vn :=[DT l1 v1 } } } T ln vn : m, l1 , ..., ln # Z], where DA is the unitary operator corresponding to dilation by an n_n real invertible matrix A and Tv1 , ..., Tvn are the unitary operators corresponding to translations by the vectors in a basis [v1 , ..., vn] for R. Orthonormal wavelets are vectors in L(R) which are complete wandering vectors for U in the sense that [U : U # U] is an orthonormal basis for L2(Rn). 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 (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 , ei , and UDB , ei , 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=CAC. 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 DA is a bilateral shift of infinite multiplicity. 1998 Academic Press