A GEOMETRIC APPROACH TO THE CASCADE APPROXIMATION OPERATOR FOR WAVELETS1 Palle E. T. Jorgensen 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 π be a representation of L ∞ (T) on H. Let R be a positive operator in L ∞ (T) such that R (11) = 1 1, where 1 1 denotes the constant function 1. We study operators M on H (bounded, but noncontractive) such thatwhere the * refers to Hilbert space adjoint. We give a complete orthogonal expansion of H which reduces π such that M acts as a shift on one part, and the residual part is H (∞) = 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 * ) = {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 operator M, the components in the decomposition are unitarily, and canonically, equivalent to spaces L 2 (E n ) ⊂ L 2 (R), where E n ⊂ R, n = 0, 1, 2, . . . , ∞, 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.
Contents:1 In wavelet theory, one is given a subband filter, i.e., a function m 0 on the unit circle, satisfying (i)-(iii) from below and one wants to construct a scaling function ϕ (relative to m 0 ), i.e., a nonzero function ϕ on R satisfying the scaling relation ϕ = Mϕ (relation (1.1) in the paper). Here M = M m 0 is the so-called cascade operator defined bywhere a n are Fourier coefficients of the function m 0 . The scaling function ϕ is important because its shifts generate (under some analytic conditions) the so-called multiresolution subspace V = V (ϕ), which is used to construct the wavelets. There are several ways of constructing a scaling function ϕ. The cascade algorithm is one of the possibilities. In this algorithm one picks some simple function h and considers its iterations M n h. Clearly, if the iterations M n h converge to a nonzero function ϕ, the function ϕ satisfies ϕ = Mϕ, so the algorithm gives us a scaling function. We study the problem of convergence of this algorithm in the setting of an abstract Hilbert space. The cascade operator M has some very special structure. The Ruelle operator R, which appears naturally in this type of problem, gives us a way to describe this structure. Namely, the operator M is what we shall call a sub-isometry; see Definition 6.1. In fact the concept depends on a pair (R, π) where R is a Ruelle operator and π is a representation. It turns out that sub-isometries admit an analogue of the Kolmogorov-Wold decomposition for usual isometries. Using this decomposition, we obtain results about convergence of the cascade algorithm in an abstract ...
