On a general q-Fourier transformation with nonsymmetric kernels

Richard A. Askey; Mizan Rahman; Sergeǐ K. Suslov

doi:10.1016/0377-0427(95)00259-6

On a general q-Fourier transformation with nonsymmetric kernels

Richard A. Askey, Mizan Rahman, Sergeǐ K. Suslov

Research output: Contribution to journal › Article › peer-review

35 Scopus citations

Abstract

Wiener used the Poisson kernel for the Hermite polynomials to deal with the classical Fourier transform. Askey, Atakishiyev and Suslov used this approach to obtain a q-Fourier transform by using the continuous q-Hermite polynomials. Rahman and Suslov extended this result by taking the Askey-Wilson polynomials, considered to be the most general continuous classical orthogonal polynomials. The theory of q-Fourier transformation is further extended here by considering a nonsymmetric version of the Poisson kernel with Askey-Wilson polynomials. This approach enables us to obtain some new results, for example, the complex and real orthogonalities of these kernels.

Original language	English (US)
Pages (from-to)	25-55
Number of pages	31
Journal	Journal of Computational and Applied Mathematics
Volume	68
Issue number	1-2
DOIs	https://doi.org/10.1016/0377-0427(95)00259-6
State	Published - Apr 22 1996
Externally published	Yes

Keywords

Al-Salam-Chihara polynomials
Askey-Wilson polynomials
Basic hypergeometric series
Fourier transform
Hermite polynomials
Integral transforms
Poisson kernels
q-Fourier transform
q-orthogonal polynomials

ASJC Scopus subject areas

Computational Mathematics
Applied Mathematics

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10.1016/0377-0427(95)00259-6

Cite this

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title = "On a general q-Fourier transformation with nonsymmetric kernels",

abstract = "Wiener used the Poisson kernel for the Hermite polynomials to deal with the classical Fourier transform. Askey, Atakishiyev and Suslov used this approach to obtain a q-Fourier transform by using the continuous q-Hermite polynomials. Rahman and Suslov extended this result by taking the Askey-Wilson polynomials, considered to be the most general continuous classical orthogonal polynomials. The theory of q-Fourier transformation is further extended here by considering a nonsymmetric version of the Poisson kernel with Askey-Wilson polynomials. This approach enables us to obtain some new results, for example, the complex and real orthogonalities of these kernels.",

keywords = "Al-Salam-Chihara polynomials, Askey-Wilson polynomials, Basic hypergeometric series, Fourier transform, Hermite polynomials, Integral transforms, Poisson kernels, q-Fourier transform, q-orthogonal polynomials",

author = "Askey, {Richard A.} and Mizan Rahman and Suslov, {Sergeǐ K.}",

note = "Funding Information: * Corresponding author. E-mail: askey@math.wisc.edu. x This work was supported in part by NSF grant #DMS 93005224 and the Natural Research Council of Canada grant #A6197.",

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AU - Askey, Richard A.

AU - Rahman, Mizan

AU - Suslov, Sergeǐ K.

N1 - Funding Information: * Corresponding author. E-mail: askey@math.wisc.edu. x This work was supported in part by NSF grant #DMS 93005224 and the Natural Research Council of Canada grant #A6197.

PY - 1996/4/22

Y1 - 1996/4/22

N2 - Wiener used the Poisson kernel for the Hermite polynomials to deal with the classical Fourier transform. Askey, Atakishiyev and Suslov used this approach to obtain a q-Fourier transform by using the continuous q-Hermite polynomials. Rahman and Suslov extended this result by taking the Askey-Wilson polynomials, considered to be the most general continuous classical orthogonal polynomials. The theory of q-Fourier transformation is further extended here by considering a nonsymmetric version of the Poisson kernel with Askey-Wilson polynomials. This approach enables us to obtain some new results, for example, the complex and real orthogonalities of these kernels.

AB - Wiener used the Poisson kernel for the Hermite polynomials to deal with the classical Fourier transform. Askey, Atakishiyev and Suslov used this approach to obtain a q-Fourier transform by using the continuous q-Hermite polynomials. Rahman and Suslov extended this result by taking the Askey-Wilson polynomials, considered to be the most general continuous classical orthogonal polynomials. The theory of q-Fourier transformation is further extended here by considering a nonsymmetric version of the Poisson kernel with Askey-Wilson polynomials. This approach enables us to obtain some new results, for example, the complex and real orthogonalities of these kernels.

KW - Al-Salam-Chihara polynomials

KW - Askey-Wilson polynomials

KW - Basic hypergeometric series

KW - Fourier transform

KW - Hermite polynomials

KW - Integral transforms

KW - Poisson kernels

KW - q-Fourier transform

KW - q-orthogonal polynomials

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ER -

On a general q-Fourier transformation with nonsymmetric kernels

Abstract

Keywords

ASJC Scopus subject areas

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