A geometric approach to the cascade approximation operator for wavelets

Abstract

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\(\mathcal{H}\) be a Hilbert space, and let π be a representation ofL∞(\(\mathbb{T}\)) on\(\mathcal{H}\). LetR be a positive operator inL∞(\(\mathbb{T}\)) such thatR(1) =1, where1 denotes the constant function 1. We study operatorsM on\(\mathcal{H}\) (bounded, but noncontractive) such that $$\pi (f){\rm M} = M\pi (f(z^2 ))andM*\pi (f)M = \pi (R*f),f \in L^\infty (\mathbb{T}),$$ where the * refers to Hilbert space adjoint. We give a complete orthogonal expansion of\(\mathcal{H}\) which reduces π such thatM acts as a shift on one part, and the residual part is\(\mathcal{H}\)(∞) = ∩ n [M n \(\mathcal{H}\)], where [M n \(\mathcal{H}\)] 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 π, we show that, for this wavelet operatorM, the components in the decomposition are unitarily, and canonically, equivalent to spacesL2(E n ) ⊂L2(ℝ), whereEn ⊂ ℝ, n=1,2,3,..., ∞, are measurable subsets which form a tiling of ℝ; i.e., the union is ℝ 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.