Prolate spheroidal wavelets and multidimensional chromatic series expansions

Abstract

Chromatic series were originally introduced for bandlimited functions. The n^th chromatic derivative of an analytic function is a linear combination of k^th ordinary derivatives with 0 ≤ k ≤ n, where the coefficients of linear combination are based on a suitable system of orthogonal polynomials. Chromatic derivative and series expansions of bandlimited functions have been used as a replacement for Taylor's series and they have been shown to be more useful in practical signal processing applications than Taylor series. In this paper we have shown that the theory can be extended to prolate spheroidal wavelet series that than combine chromatic series with sampling series in higher dimensions. The multidimensional case has much reacher structure than in the univariate case and will find more applications in image processing and analysis.