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A Generalized Translation Operator Generated by the Sinc Function on an Interval

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Abstract

We discuss the properties of the generalized translation operator generated by the system of functions \(\mathfrak{S}=\{{(\sin k\pi x)}/{(k\pi x)}\}_{k=1}^{\infty}\) in the spaces \(L^{q}=L^{q}((0,1),{\upsilon})\), \(q\geq 1\), on the interval \((0,1)\) with the weight \(\upsilon(x)=x^{2}\). We find an integral representation of this operator and study its norm in the spaces \(L^{q}\), \(1\leq q\leq\infty\). The translation operator is applied to the study of Nikol’skii’s inequality between the uniform norm and the \(L^{q}\)-norm of polynomials in the system \(\mathfrak{S}\).

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Funding

This work was performed as a part of the research conducted in the Ural Mathematical Center and supported by the Ministry of Education and Science of the Russian Federation (agreement no. 075-02-2023-913).

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Authors and Affiliations

  1. Ural Federal University, 620000, Yekaterinburg, Russia

    V. V. Arestov & M. V. Deikalova

  2. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 620108, Yekaterinburg, Russia

    V. V. Arestov & M. V. Deikalova

Authors
  1. V. V. Arestov
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  2. M. V. Deikalova
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Correspondence to V. V. Arestov or M. V. Deikalova.

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Translated from Trudy Instituta Matematiki i Mekhaniki UrO RAN, Vol. 29, No. 4, pp. 27 - 48, 2023 https://doi.org/10.21538/0134-4889-2023-29-4-27-48.

Translated by M. Deikalova

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Arestov, V.V., Deikalova, M.V. A Generalized Translation Operator Generated by the Sinc Function on an Interval. Proc. Steklov Inst. Math. 323 (Suppl 1), S32–S52 (2023). https://doi.org/10.1134/S0081543823060032

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