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 the cascade algorithm in wavelet theory. Let $ H $ be a Hilbert space, and let $ \pi $ be a representation of $ L^\infty(T) $ on $ H $. Let $ R $ be a positive operator in $ L^\infty(T) $ such that $ R(1)=1 $, where $ 1 $ denotes the constant function $ 1 $. We study operators $ M $ on $ H $ (bounded, but non-contractive) such that $ \pi(f)M=M\pi(f(z^2)) $ and $ M^* \pi(f)M=\pi(R^* f) $, $ f \in L^\infty (T) $, where the $ * $ refers to Hilbert space adjoint. We give a complete orthogonal expansion of $ H $ which reduces $ \pi $ such that $ M $ acts as a shift on one part, and the residual part is $ H^{(\infty)}=\bigcap_n[M^n H] $, where $ [M^n H] $ is the closure of the range of $ M^n $. The shift part is present, we show, if and only if $ \ker(M^*) \neq \{0\} $. We apply the operator-theoretic results to the refinement operator (or cascade algorithm) from wavelet theory. Using the representation $ \pi $, we show that, for this wavelet operator $ M $, the components in the decomposition are unitarily, and canonically, equivalent to spaces $ L^2(E_n) \subset L^2(R) $, where $ E_n \subset R $, $ n=0,1,2,...,\infty $, are measurable subsets which form a tiling of $ R $; i.e., the union is $ R $ up to zero measure, and pairwise intersections of different $ E_n $'s have measure zero. We prove two results on the convergence of the cascade algorithm, and identify singular vectors for the starting point of the algorithm.