Wavelet bases for a set of commuting unitary operators

Abstract

Let (U=U1, ..., Ud) be an ordered d-tuple of distinct, pairwise commuting, unitary operators on a complex Hilbert space ℋ, and let X:={x1, ..., xr} ⊂ ℋ such that {Mathematical expression} is a Riesz basis of the closed linear span V0 of {Mathematical expression}. Suppose there is unitary operator D on ℋ such that V0 ⊂DV0 =:V1 and UnD=DUAn for all n ∈ ℤd, where A is a d ×d matrix with integer entries and Δ := det(A) ≠ 0. Then there is a subset Λ in V1, with r(Δ - 1) vectors, such that {Mathematical expression} is a Riesz basis of W0, the orthogonal complement of V0 in V1. The resulting multiscale and decomposition relations can be expressed in a Fourier representation by one single equation, in terms of which the duality principle follows easily. These results are a consequence of an extension, to a set of commuting unitary operators, of Robertson's Theorems on wandering subspace for a single unitary operator [24]. Conditions are given in order that {Mathematical expression} is a Riesz basis of W0. They are used in the construction of a class of linear spline wavelets on a four-direction mesh. © 1993 J.C. Baltzer AG, Science Publishers.