Bilinear biorthogonal expansions and the Dunkl kernel on the real line

TL;DR

This paper extends classical Paley-Wiener space concepts using bilinear expansions, providing new series representations for the Dunkl kernel on the real line, unifying sampling and Fourier-Neumann expansions.

Contribution

It introduces a novel bilinear expansion framework for the Dunkl kernel, connecting it with classical sampling and Fourier series expansions.

Findings

Derived a bilinear expansion for the Dunkl kernel.

Unified sampling and Fourier-Neumann expansions within this framework.

Provided new tools for analysis involving Dunkl operators.

Abstract

We study an extension of the classical Paley-Wiener space structure, which is based on bilinear expansions of integral kernels into biorthogonal sequences of functions. The structure includes both sampling expansions and Fourier-Neumann type series as special cases, and it also provides a bilinear expansion for the Dunkl kernel (in the rank 1 case) which is a Dunkl analogue of Gegenbauer's expansion of the plane wave and the corresponding sampling expansions. In fact, we show how to derive sampling and Fourier-Neumann type expansions from the results related to the bilinear expansion for the Dunkl kernel.