We consider systems of unitary operators on the complex Hilbert space L2 (Rn) of the form U :=UDA , Tv1 , ..., Tvn :=[DmT l1 v1 }}} Tln 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 Rn . Orthonormal wavelets are vectors in L2 (Rn ) 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 , Tei , and UDB, Tei , 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&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 real n_n matrix A to have the property that the dilation unitary operator DA is a bilateral shift of infinite multiplicity.
Ionascu, Eugen J., "On the Unitary Systems Affiliated with Orthonormal Wavelet Theory in n-Dimensions" (1998). Faculty Bibliography. 850.