WAVELET BASES FOR A UNITARY OPERATOR

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 T-Z(X):={T(n)x:n is an element of Z, x is an element of X} to be a Riesz basis of its closed linear span [T(Z)X]>. If T-Z(X) and T-Z(Y) are Riesz bases, and [T-Z(X)]subset of[T-Z(Y)], then X is extendable to X' such that T-Z(X') is a Riesz basis of [T-Z(Y)]. The proof provides an algorithm for the construction of Riesz bases for the orthogonal complement of [T-Z(X)] in [T-Z(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.