Wavelet bases for a unitary operator

Abstract

Let T be a unitary operator on a complex Hilbert space ℋ, and X, Y be finite subsets of ℋ. We give a necessary and sufficient condition for TZ(X): {Tnx: n ∈ Z, x ∈ X} to be a Riesz basis of its closed linear span 〈TZ(X)〉. If TZ(X) and TZ(Y) are Riesz bases, and 〈TZ(X)〉⊂〈TZ(Y)〉, then X is extendable to X′ such that TZ(X′) is a Riesz basis of TZ(Y) The proof provides an algorithm for the construction of Riesz bases for the orthogonal complement of 〈TZ(X)〉 in 〈TZ(Y)〉. In the case X consists of a single B-spline, the algorithm gives a natural and quick construction of the spline wavelets of Chui and Wang [2, 3]. Further, the duality principle of Chui and Wang in [3] and [4] is put in the general setting of biorthogonal Riesz bases in Hilbert space.