Energy Moments in Time and Frequency for Two-Scale Difference Equation Solutions and Wavelets

Abstract

This paper indicates how to find energy moments in direct and Fourier space of a solution to the functional equation $u(x) = \sum_{k = 0}^{N - 1} {2c_k u(2x - k)} $ and shows that the Sobolev regularity of u is determined by the spectral radius of a matrix defined from the coefficients $(c_k )$. The results are applied to compactly supported orthonormal wavelets.