On the correlation structure of multiplicity M scaling functions and wavelets

Abstract

Studies the autocorrelation and cross-correlation structure of the scaling and wavelet functions associated with compactly supported orthonormal wavelet bases. These correlation structures play an important role in both wavelet-based interpolation and in answering the question of existence of scale-limited signals. It is shown how to efficiently compute the correlation functions approximately at the M-adic rationals. Furthermore, when the tight frame is, in particular, an orthonormal wavelet basis, it is shown that the approximations involved in the computation of the samples of the correlations cancel in a manner to make the computations exact. For the case of orthonormal wavelet bases, an attempt is made to give a complete description of the zeros at the M-adic rationals. An interesting fact that arises from the analysis is that all the correlation functions possible have infinitely many zeros in their support. >