A generalization of the 2D Sleipian functions

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.