Fourier transform and related integral transforms in superspace

TL;DR

This paper extends classical Fourier, fractional Fourier, and Radon transforms to superspace, introducing new kernels and properties, and constructs an eigenfunction basis using generalized Hermite polynomials.

Contribution

It introduces a natural symplectic fermionic kernel for Fourier transforms in superspace and develops an eigenfunction basis using generalized Hermite polynomials.

Findings

Fermionic Fourier kernel has a symplectic structure

Basic properties of the transforms are established

Eigenfunction basis constructed using superspace Hermite polynomials

Abstract

In this paper extensions of the classical Fourier, fractional Fourier and Radon transforms to superspace are studied. Previously, a Fourier transform in superspace was already studied, but with a different kernel. In this work, the fermionic part of the Fourier kernel has a natural symplectic structure, derived using a Clifford analysis approach. Several basic properties of these three transforms are studied. Using suitable generalizations of the Hermite polynomials to superspace (see [H. De Bie, F. Sommen, Hermite and Gegenbauer polynomials in superspace using Clifford analysis, J. Phys. A 40 (2007) 10441-10456]) an eigenfunction basis for the Fourier transform is constructed.