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Bilinear biorthogonal expansions and the Dunkl kernel on the real line

L. D. Abreu, Ó. Ciaurri, J. L. Varona

TL;DR

This work develops a general framework for bilinear, and in particular biorthogonal, kernel expansions that generalize Paley-Wiener theory to broader kernels and parameters. It first establishes a robust abstract setup for orthogonal and biorthogonal expansions and then applies it to classical Fourier kernels, reproducing sampling formulas and Gegenbauer-type plane-wave expansions. The authors then extend the framework to the Dunkl setting on the real line, deriving a Dunkl analogue of Gegenbauer expansions, a Dunkl-based sampling theorem, and Fourier–Neumann-type expansions, with direct consequences for the Hankel transform. These results yield explicit, uniformly convergent expansions in terms of Dunkl transforms, Bessel functions, and Jacobi polynomials, linking Paley-Wiener theory, special function expansions, and Dunkl analysis with potential applications in spectral theory and signal processing.

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.

Bilinear biorthogonal expansions and the Dunkl kernel on the real line

TL;DR

This work develops a general framework for bilinear, and in particular biorthogonal, kernel expansions that generalize Paley-Wiener theory to broader kernels and parameters. It first establishes a robust abstract setup for orthogonal and biorthogonal expansions and then applies it to classical Fourier kernels, reproducing sampling formulas and Gegenbauer-type plane-wave expansions. The authors then extend the framework to the Dunkl setting on the real line, deriving a Dunkl analogue of Gegenbauer expansions, a Dunkl-based sampling theorem, and Fourier–Neumann-type expansions, with direct consequences for the Hankel transform. These results yield explicit, uniformly convergent expansions in terms of Dunkl transforms, Bessel functions, and Jacobi polynomials, linking Paley-Wiener theory, special function expansions, and Dunkl analysis with potential applications in spectral theory and signal processing.

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.

Paper Structure

This paper contains 17 sections, 5 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Structure
  3. Orthogonal expansions
  4. Biorthogonal expansions
  5. The Fourier kernel
  6. The classical sampling formula
  7. Gegenbauer's plane wave expansion
  8. The Dunkl kernel on the real line
  9. The Dunkl transform
  10. The sampling theorem related to the Dunkl transform
  11. Fourier-Neumann type expansion
  12. Consequences for the Hankel transform
  13. The sampling formula for the Hankel transform.
  14. A version of the Theorem \ref{['thm:Dunkl']} for the Hankel transform.
  15. Technical lemmas
  16. ...and 2 more sections

Key Result

Theorem 1

For each $x\in \Omega$, the following expansionThe condition $t \in I$ in the identity eq:expbilin is not a mistake. Although $K(x,t)$ is defined on $\Omega \times \Omega$, the functions $P_n(t)$ are defined, in general, only on $I$. holds, with respect to $t$, in $L^2(I,d\mu)$: Moreover, $\{S_n\}_{n\in N}$ and $\{T_n\}_{n\in N}$ are a pair of complete biorthogonal sequences in $\mathcal{P}$ such

Theorems & Definitions (14)

  • Remark 1
  • Remark 2
  • Theorem 1
  • proof
  • Lemma 1
  • Remark 3
  • Theorem 2
  • proof
  • Remark 4
  • Remark 5
  • ...and 4 more