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Discrete Jacobi–Dunkl Transform and Approximation Theorems

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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 \( \mathbb {L}_{2}^{(\alpha ,\beta )} \). 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.

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All data generated or analyzed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

The authors would be grateful to the referees for useful comments and suggestions

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  1. Department of Mathematics, Faculty of Sciences Aïn Chock, Laboratory of Fundamental and Applied Mathematics, University Hassan II, Casablanca, Morocco

    Othman Tyr & Radouan Daher

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  1. Othman Tyr
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  2. Radouan Daher
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Correspondence to Othman Tyr.

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Tyr, O., Daher, R. Discrete Jacobi–Dunkl Transform and Approximation Theorems. Mediterr. J. Math. 19, 224 (2022). https://doi.org/10.1007/s00009-022-02132-0

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