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A $q$-linear analogue of the plane wave expansion

Luís Daniel Abreu, Óscar Ciaurri, Juan Luis Varona

TL;DR

This work constructs a $q$-linear analogue of Gegenbauer's plane wave expansion by framing a $q$-Dunkl-type kernel through Bilinear Biorthogonal Expansions. The authors define a $q$-Fourier–Dunkl transform $\\mathcal{F}_{\\alpha,q}$ with kernel $E_{\\alpha}(ix;q^{2})$, build a Paley–Wiener-type space $PW_{\\alpha,q}$, and develop biorthogonal families from generalized little $q$-Gegenbauer polynomials and $q$-Neumann functions. The main result provides an explicit expansion of the kernel $E_{\\alpha}(ixt;q^{2})$ in terms of $\\mathcal{J}_{\\alpha+\\beta,n}(x;q^{2})$ and $C_{n}^{(\\beta+1/2,\\alpha+1/2)}(t;q^{2})$, along with an orthogonal $q$-Fourier–Neumann series for functions in $PW_{\\alpha,q}$, with uniform convergence on compact sets. As a byproduct, the paper yields $q$-analogues of Neumann series and a robust framework for $q$-harmonic analysis on Paley–Wiener spaces, extending plane wave expansions to discrete $q$-grid settings and connecting to prior $q$-quadratic cases and Dunkl-type transforms.

Abstract

We obtain a $q$-linear analogue of Gegenbauer's expansion of the plane wave. It is expanded in terms of the little $q$-Gegenbauer polynomials and the \textit{third} Jackson $q$-Bessel function. The result is obtained by using a method based on bilinear biorthogonal expansions.

A $q$-linear analogue of the plane wave expansion

TL;DR

This work constructs a -linear analogue of Gegenbauer's plane wave expansion by framing a -Dunkl-type kernel through Bilinear Biorthogonal Expansions. The authors define a -Fourier–Dunkl transform with kernel , build a Paley–Wiener-type space , and develop biorthogonal families from generalized little -Gegenbauer polynomials and -Neumann functions. The main result provides an explicit expansion of the kernel in terms of and , along with an orthogonal -Fourier–Neumann series for functions in , with uniform convergence on compact sets. As a byproduct, the paper yields -analogues of Neumann series and a robust framework for -harmonic analysis on Paley–Wiener spaces, extending plane wave expansions to discrete -grid settings and connecting to prior -quadratic cases and Dunkl-type transforms.

Abstract

We obtain a -linear analogue of Gegenbauer's expansion of the plane wave. It is expanded in terms of the little -Gegenbauer polynomials and the \textit{third} Jackson -Bessel function. The result is obtained by using a method based on bilinear biorthogonal expansions.

Paper Structure

This paper contains 14 sections, 5 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. The method of Bilinear Biorthogonal Expansions
  3. Preliminaries on $q$-special functions
  4. Application of the method
  5. The integral transform
  6. The space $\mathcal{P}$
  7. The biorthogonal functions
  8. Generalized little $q$-Gegenbauer polynomials
  9. $q$-Fourier-Neumann type series
  10. Main result
  11. Proof of formula \ref{['eq:qlinearplanewave']}
  12. Technical lemmas
  13. Some integrals involving $q$-Bessel functions
  14. Proof of Lemma \ref{['lem:qF-PQ']}

Key Result

Theorem 1

For each $x\in \Omega$, the following expansion The 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 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 $\mathcal{P}$, in such a way th

Theorems & Definitions (11)

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