Fourier Transform of the Orthogonal Polynomials on the Unit Ball and Continuous Hahn Polynomials

Abstract

Some systems of univariate orthogonal polynomials can be mapped into other families by the Fourier transform. The most-studied example is related to the Hermite functions, which are eigenfunctions of the Fourier transform. For the multivariate case, by using the Fourier transform and Parseval’s identity, very recently, some examples of orthogonal systems of this type have been introduced and orthogonality relations have been discussed. In the present paper, this method is applied for multivariate orthogonal polynomials on the unit ball. The Fourier transform of these orthogonal polynomials on the unit ball is obtained. By Parseval’s identity, a new family of multivariate orthogonal functions is introduced. The results are expressed in terms of the continuous Hahn polynomials.

Keywords

• Gegenbauer polynomials
• multivariate orthogonal polynomials
• Hahn polynomials
• Fourier transform
• Parseval’s identity
• hypergeometric function