Characterizations of wavelet bases and frames in Hilbert spaces

Abstract

Let U = (Ui,...,Ud) be an ordered d-tuple of distinct commuting unitary operators on a complex Hilbert space H, and Y = {y 1,...,ys}a finite subset of H. Let UZd (Y) = {Unyj : n ∈ Zd,j = 1,...,s}, and let Φ(θ) be the s by s matrix function (sequences presented) defined on the d-dimensional torus. We obtain characterizations, in terms of the matrix function Φ(θ), for the set Uzd (Y) to be (1) a Bessel sequence; or (2) a (tight) frame; or (3) a Riesz basis for its closed linear span in H. Connections with other related work will also be discussed.