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Discrete Jacobi–Dunkl Transform and Approximation Theorems

Discrete Jacobi–Dunkl Transform and Approximation Theorems This paper uses some basic notions and results from the discrete harmonic analysis associated with the Jacobi–Dunkl operator to study some problems in the theory of approximation of functions in the space L2(α,β)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ \mathbb {L}_{2}^{(\alpha ,\beta )} $$\end{document}. Analogs of the direct Jackson theorems of approximations for the modulus of smoothness (of arbitrary order) constructed using the translation operators which was defined by Vinogradov are proved. In conclusion of this work, we show that the modulus of smoothness and the K-functionals constructed from the Sobolev-type space corresponding to the Jacobi–Dunkl Laplacian operator are equivalent. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mediterranean Journal of Mathematics Springer Journals

Discrete Jacobi–Dunkl Transform and Approximation Theorems

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References (38)

Publisher
Springer Journals
Copyright
Copyright © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022. Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
ISSN
1660-5446
eISSN
1660-5454
DOI
10.1007/s00009-022-02132-0
Publisher site
See Article on Publisher Site

Abstract

This paper uses some basic notions and results from the discrete harmonic analysis associated with the Jacobi–Dunkl operator to study some problems in the theory of approximation of functions in the space L2(α,β)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ \mathbb {L}_{2}^{(\alpha ,\beta )} $$\end{document}. Analogs of the direct Jackson theorems of approximations for the modulus of smoothness (of arbitrary order) constructed using the translation operators which was defined by Vinogradov are proved. In conclusion of this work, we show that the modulus of smoothness and the K-functionals constructed from the Sobolev-type space corresponding to the Jacobi–Dunkl Laplacian operator are equivalent.

Journal

Mediterranean Journal of MathematicsSpringer Journals

Published: Oct 1, 2022

Keywords: Jacobi–Dunkl operator; Jacobi polynomials; discrete Jacobi–Dunkl transform; Jacobi–Dunkl translation operator; Jacobi–Dunkl Laplacian operator; K-functionals; modulus of smoothness; 41A36; 44A20; 42C05; 33C45

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