Generalizations of Chromatic Derivatives and Series Expansions

Chromatic series expansions of bandlimited functions provide an alternative representation to the Whittaker-Shannon-Kotel'nikov sampling series. Chromatic series share similar properties with Taylor series insofar as the coefficients of the expansions, which are called chromatic derivatives, are based on the ordinary derivatives of the function. Chromatic derivatives are linear combinations of ordinary derivatives in which the coefficients of the combinations are related to a system of orthonormal polynomials. But unlike Taylor series, chromatic series have more useful applications in signal processing because they represent bandlimited functions more efficiently than Taylor series. The goal of this article is to generalize the notion of chromatic derivatives and then extend chromatic series expansions to a larger class of signals than the class of bandlimited signals. In particular, we extend chromatic series to signals given by integral transforms other than the Fourier transform, such as the Laplace and Hankel transforms.