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A positive product formula of integral kernels of $k$-Hankel transforms

Wentao Teng

TL;DR

This work develops a positive radial product formula for the integral kernels of the $k$-Hankel transform $F_{k,1}$ within the $(k,a)$-generalized Fourier framework. It shows that the generalized spherical mean $M_f$ can be represented by a compactly supported probability measure $\sigma_{x,t}^{k,1}$ under the hypothesis $2\langle k\rangle+N-2>0$, and it derives a radial product formula linking $B_{k,1}$ with this measure through the Bessel function $j_{2\langle k\rangle+N-2}$. The paper further establishes a weak Huygens principle for the deformed wave equation $u_{tt}-2|x|\Delta_k u=0$, and provides a distributional framework for the associated generalized Fourier transforms and translations. By connecting Dunkl theory with $(k,1)$-generalized Fourier analysis, the results advance understanding of positivity, spherical means, and wave propagation in these deformed harmonic analysis settings.

Abstract

The $k$-Hankel transform $F_{k,1}$ (or the $(k,1)$-generalized Fourier transform) is the Dunkl analogue of the unitary inversion operator in the minimal representation of a conformal group initiated by T. Kobayashi and G. Mano. It is one of the two most significant cases in $(k,a)$-generalized Fourier transforms. We will establish a positive radial product formula for the integral kernels of $F_{k,1}$. Such a product formula is equivalent to a representation of the generalized spherical mean operator in terms of the probability measure $σ_{x,t}^{k,1}(ξ)$. We will then study the representing measure $σ_{x,t}^{k,1}(ξ)$ and analyze the support of the measure, and derive a weak Huygens's principle for the deformed wave equation in $(k,1)$-generalized Fourier analysis.

A positive product formula of integral kernels of $k$-Hankel transforms

TL;DR

This work develops a positive radial product formula for the integral kernels of the -Hankel transform within the -generalized Fourier framework. It shows that the generalized spherical mean can be represented by a compactly supported probability measure under the hypothesis , and it derives a radial product formula linking with this measure through the Bessel function . The paper further establishes a weak Huygens principle for the deformed wave equation , and provides a distributional framework for the associated generalized Fourier transforms and translations. By connecting Dunkl theory with -generalized Fourier analysis, the results advance understanding of positivity, spherical means, and wave propagation in these deformed harmonic analysis settings.

Abstract

The -Hankel transform (or the -generalized Fourier transform) is the Dunkl analogue of the unitary inversion operator in the minimal representation of a conformal group initiated by T. Kobayashi and G. Mano. It is one of the two most significant cases in -generalized Fourier transforms. We will establish a positive radial product formula for the integral kernels of . Such a product formula is equivalent to a representation of the generalized spherical mean operator in terms of the probability measure . We will then study the representing measure and analyze the support of the measure, and derive a weak Huygens's principle for the deformed wave equation in -generalized Fourier analysis.

Paper Structure

This paper contains 9 sections, 12 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dunkl theory
  4. The $(k,a)$-generalized Fourier transform
  5. ${\left\|\cdot\right\|}_\infty$-Density of the space $W_{k,1}\left(\mathbb{R}^N\right)$ in $C_0(\mathbb{R}^N\backslash{\left\{0\right\}})$
  6. The positivity of spherical mean operator
  7. The representing measure $\sigma_{x,t}^{k,1}$ of the generalized spherical mean
  8. Solution to the deformed wave equation
  9. Appendix: The metric $d(x,y)$

Key Result

Theorem 1.1

1). Assume $2\left\langle k\right\rangle+N-2>0$. For $a=1$, the spherical mean operator $f\mapsto M_f$ is positivity-preserving on $C_0(\mathbb{R}^N)$, i.e., if $f\in C_0(\mathbb{R}^N)$ and $f\geq 0$ on $\mathbb R^N$, then $M_f\geq 0$ on $\mathbb R^N\times \mathbb R^+$. 2). Under the conditions in 1 where $M^1(\mathbb{R}^N)$ stands for the space of Borel probability measures. The measure $\sigma_{

Theorems & Definitions (19)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • proof
  • Proposition 4.1
  • Lemma 4.2
  • ...and 9 more