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

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 iden...

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Bibliographic Details
Published inAxioms Vol. 11; no. 10; p. 558
Main Authors Güldoğan Lekesiz, Esra, Aktaş, Rabia, Area, Iván
Format Journal Article
LanguageEnglish
Published Basel MDPI AG 01.10.2022
Subjects
Eigenvectors
Fourier transform
Fourier transformations
Fourier transforms
Gegenbauer polynomials
Hahn polynomials
hypergeometric function
Investigations
Mathematical analysis
Mathematical research
Multivariate analysis
multivariate orthogonal polynomials
Orthogonal functions
Orthogonality
Parseval’s identity
Partial differential equations
Polynomials
Germany
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Summary: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.
ISSN:2075-1680
2075-1680
DOI:10.3390/axioms11100558