The Spectral Theory of Multiresolution Operators and Applications

Abstract

For any wavelet matrix of rank m and genus g there is an associated wavelet system with a square integrable scaling function and (m-1) wavelet functions; the fundamental scaling equation has mg coefficients. This paper concerns itself with a family of multiresolution operators associated with such a wavelet matrix and the corresponding wavelet system. These operators act on continuous even periodic functions and depend quadratically on the coefficients of the wavelet matrix; they have been studied for rank 2 wavelet systems by Lawton, Cohen, and Eirola in different contexts and under different names (transition operator, wavelet-Galerkin operator, etc.). Lawton and Cohen independently found that a rank 2 wavelet system yields an orthonormal basis if and only if 1 is a nondegenerate eigenvalue of the multiresolution operator. Eirola showed for the rank 2 case that the spectral radius of this operator could be used to explicitly compute the Sobolev smoothness of the scaling function (and hence the wavelet system), and that moreover, this spectral radius could be computed in terms of the spectral radius of a finite-dimensional operator. He found asymptotic formulas for this spectral radius and hence for the Sobolev smoothness as the genus (and number of coefficients in the scaling equation) grows large. This paper reviews the above results and outlines a generalization of Eirola's work to arbitrary rank wavelet systems. In particular we give asymptotic estimates of the Sobolev smoothness of rank 3 scaling functions for large genus by evaluating the kernel of the multiresolution operator at fixed points of an automorphism of the unit circle.