A geometric approach to the cascade approximation operator for wavelets

This paper is devoted to an approximation problem for operators in Hilbert space, that appears when one tries to study geometrically thecascade algorithm in wavelet theory. Let be a Hilbert space, and let be a representation ofL ( ) on . LetR be a positive operator inL ( ) such thatR(1) =1, where1 denotes the constant function 1. We study operatorsM on (bounded, but noncontractive) such that where the * refers to Hilbert space adjoint. We give a complete orthogonal expansion of which reduces such thatM acts as a shift on one part, and the residual part is ⁽⁾ = _n M _n ], where M ⁿ ] is the closure of the range ofM ⁿ . The shift part is present, we show, if and only if ker (M ^*){0}. We apply the operator-theoretic results to the refinement operator (or cascade algorithm) from wavelet theory. Using the representation , we show that, for this wavelet operatorM, the components in the decomposition are unitarily, and canonically, equivalent to spacesL ²(E _n ) L ²(), whereE _n , n=1,2,3,..., , are measurable subsets which form a tiling of ; i.e., the union is up to zero measure, and pairwise intersections of differentE _n 's have measure zero. We prove two results on the convergence of the cascale algorithm, and identify singular vectors for the starting point of the algorithm.Terminology used in the paper     the one-torus -   Haar measure on the torus - Z   the Zak transform - X=ZXZ ^–1   transformation of operators -   a given Hilbert space -   a representation ofL ( ) on - R   the Ruelle operator onL ( ) - M   an operator on - R ^*,M ^*   adjoint operators Work supported in part by the U.S. National Science Foundation.