Journal
AXIOMS
Volume 11, Issue 10, Pages -Publisher
MDPI
DOI: 10.3390/axioms11100558
Keywords
Gegenbauer polynomials; multivariate orthogonal polynomials; Hahn polynomials; Fourier transform; Parseval's identity; hypergeometric function
Categories
Funding
- TUBITAK Research Grant [120F140]
- Agencia Estatal de Investigacion (AEI) of Spain [PID2020-113275GB-I00]
- European Community fund FEDER
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This paper explores the Fourier transform of orthogonal polynomial systems and focuses on the Hermite functions and multivariate orthogonal polynomials on the unit ball. By using the Fourier transform and Parseval's identity, a new family of orthogonal functions is introduced.
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.
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