Multidimensional chromatic derivatives and series expansions

Abstract

Chromatic derivatives and series expansions of bandlimited functions have recently been introduced as an alternative representation to the Taylor series, and they have been shown to be more useful in practical signal processing applications than in the Taylor series. Although chromatic series were originally introduced for bandlimited functions, they have now been extended to a larger class of functions. The n-th chromatic derivative of an analytic function is a linear combination of the k-th ordinary derivatives with 0 ≤ k ≤ n, where the coefficients of the linear combination are based on a suitable system of orthogonal polynomials. The goal of this article is to extend chromatic derivatives and series to higher dimensions. This is of interest not only because the associated multivariate orthogonal polynomials have much reacher structure than in the univariate case, but also because we believe that the multidimensional case will find natural applications to fields such as image processing and analysis.