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Equivalence of K-functionals and moduli of smoothness generated by the Beltrami-Laplace operator on the spaces \(S^{(p,q)}(\sigma ^{m-1})\)

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Abstract

Some problems in the theory of approximation of functions on the sphere \(\sigma ^{m-1}\) in the metric of \(S^{(p,q)}\) by functions with bounded spectrum, are investigated. We prove analogues of Jackson’s direct theorem for the moduli of smoothness of all orders constructed on the basis of spherical shift. The equivalence between moduli of smoothness and K-functional for the couple \(\left( S^{(p,q)}(\sigma ^{m-1}), W^{r}_{p,q}(\sigma ^{m-1})\right) \) is also shown on the sphere \(\sigma ^{m-1}\).

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Acknowledgements

I would like to thank the anonymous referee for constructive suggestions.

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Authors and Affiliations

  1. Laboratory MC, Polydisciplinary Faculty of Safi, University of Cadi Ayyad, Marrakech, Morocco

    Salah El Ouadih

  2. Laboratory TAGDM, Faculty of Sciences Aïn Chock, University Hassan II, Casablanca, Morocco

    Salah El Ouadih, Radouan Daher, Othman Tyr & Faouaz Saadi

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  1. Salah El Ouadih
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  2. Radouan Daher
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  3. Othman Tyr
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  4. Faouaz Saadi
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Correspondence to Salah El Ouadih.

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El Ouadih, S., Daher, R., Tyr, O. et al. Equivalence of K-functionals and moduli of smoothness generated by the Beltrami-Laplace operator on the spaces \(S^{(p,q)}(\sigma ^{m-1})\). Rend. Circ. Mat. Palermo, II. Ser 71, 445–458 (2022). https://doi.org/10.1007/s12215-020-00587-2

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