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Moduli of Smoothness Related to the Laplace-Operator

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Abstract

We introduce and study a series of new moduli of smoothness in the multivariate case in \(L_p\)-spaces of periodic functions. The main focus lies on the case \(0<p<1\). We prove a direct Jackson-type estimate and provide necessary and sufficient conditions with respect to the dimension \(d\) and to integrability \(p\) for the equivalence of these moduli and polynomial \(K\)-functionals related to the Laplace-operator. As a consequence we obtain an inverse Bernstein-type estimate. Moreover, we are able to characterize the approximation error in case of approximation by families of linear polynomial operators which are generated by Bochner–Riesz kernels in terms of the introduced moduli.

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Acknowledgments

This research was supported by AvH-Foundation.

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Authors and Affiliations

  1. Lomonosov Moscow State University Sevastopol Branch, Sevastopol, 99001, Russian Federation

    Konstantin Runovski

  2. Friedrich Schiller University, 07737, Jena, Germany

    Hans-Jürgen Schmeisser

Authors
  1. Konstantin Runovski
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  2. Hans-Jürgen Schmeisser
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Correspondence to Hans-Jürgen Schmeisser.

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Communicated by Paul Butzer.

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Runovski, K., Schmeisser, HJ. Moduli of Smoothness Related to the Laplace-Operator. J Fourier Anal Appl 21, 449–471 (2015). https://doi.org/10.1007/s00041-014-9373-y

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  • DOI: https://doi.org/10.1007/s00041-014-9373-y

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