The kernel of the radially deformed Fourier transform

TL;DR

This paper derives explicit formulas for the kernel of the radially deformed Fourier transform in even dimensions for specific parameter values, revealing boundedness in two dimensions and expanding understanding of this integral transform.

Contribution

The paper provides new explicit formulas for the kernel of the radially deformed Fourier transform for even dimensions and specific parameters, extending previous results.

Findings

Kernel formulas derived for even dimensions and specific parameters

Kernel is bounded in two dimensions

Advances understanding of the radially deformed Fourier transform

Abstract

The radially deformed Fourier transform, introduced in [S. Ben Said, T. Kobayashi and B. Orsted, Laguerre semigroup and Dunkl operators, Compositio Math.], is an integral transform that depends on a numerical parameter $a \in R^{+}$. So far, only for $a=1$ and $a=2$ the kernel of this integral transform is determined explicitly. In the present paper, explicit formulas for the kernel of this transform are obtained when the dimension is even and $a = 2/n$ with $n \in N$. As a consequence, it is shown that the integral kernel is bounded in dimension 2.