A family of Fourier transform's eigenfunctions

TL;DR

This paper introduces a new family of Fourier eigenfunctions that are radial and indexed by dimension, with potential applications in optical and thermodynamic systems, expanding the known set of eigenfunctions.

Contribution

It presents explicit constructions of Fourier eigenfunctions for various dimensions, including non-standard distributional eigenfunctions for higher dimensions.

Findings

Explicit eigenfunctions for d=1,2,3

Non-standard eigenfunctions for d>3

Potential applications in optical and thermodynamic systems

Abstract

This paper presents a family of Fourier eigenfunctions indexed by the space dimension d. These eigenfunctions are radial and built upon some generalized exponential integral function. For d=1,2,3, they are integrable or square integrable and give new explicit examples of Fourier eigenfunctions in the usual or Fourier-Plancherel sense. For d>3, the functions are examples of non standard eigenfunctions, i.e. eigenfunctions in the sense of distribution. The discovery of these eigenfunctions stems from research in thermal lens spectroscopy, at the intersection of thermodynamics and optics. Their use could simplify the analysis of thermo-optical systems, paving the way for applications in optical computing, material studies and thermodynamic.