A generalization of the 2D Sleipian functions

TL;DR

This paper explores new families of special functions that extend 2D Sleipian functions, analyzing their spectral properties, differential operators, and Fourier transforms to advance understanding in mathematical analysis.

Contribution

It introduces generalized 2D Sleipian functions, investigates their spectral properties, and provides explicit Fourier transform expressions for related polynomials.

Findings

Spectral properties of the generalized functions are derived.

Differential operators commuting with integral operators are identified.

Explicit Fourier transforms of Disk and Gegenbauer polynomials are provided.

Abstract

The main content of this work is devoted to study various explicit family of special functions generalizing the famous 2D Sleipain functions, founded in 1960's by D. Slepian and his co-authors. As a consequence, many desirable spectral properties of the corresponding weighted finite Fourier transform are deduced from the rich literature. In particular, similar aspect related to Slepian's seminal papers is the investigation of differential operators that commute with appropriate integral operators are given. Finally, we provided the reader with some analytic expressions for the Fourier transforms of the Disk polynomials and the two variables Gegenbauer polynomials.