Chromatic Derivatives and Expansions with Weights

TL;DR

This paper extends chromatic derivatives with weights, enabling expansions around every point for bandlimited functions, and demonstrates their application to Walsh-Fourier and Poisson-wavelet transforms, including in $L^p$ spaces.

Contribution

It introduces a weighted extension of chromatic derivatives, broadening their applicability and providing new examples and analysis in various transform contexts.

Findings

Chromatic expansions are possible around every point with positive kernel functions.

The method is applied to Walsh-Fourier and Poisson-wavelet transforms.

Chromatic expansion in $L^p$-spaces is analyzed.

Abstract

Chromatic derivatives and series expansions of bandlimited functions have recently been introduced in signal processing and they have been shown to be useful in practical applications. We extend the notion of chromatic derivative using varying weights. When the kernel function of the integral operator is positive, this extension ensures chromatic expansions around every points. Besides old examples, the modified method is demonstrated via some new ones as Walsh-Fourier transform, and Poisson-wavelet transform. Moreover the chromatic expansion of a function in some $L^p$-space is investigated.