V.V. Arestov, M.V. Deikalova. A generalized translation operator generated by the sinc function on an interval ... P. 27-48

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 $L^q=L^q((0,1),{\upsilon}),$  $q\ge 1,$ 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 $L^q,$ $1\le q\le\infty.$ 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 $\mathfrak{S}.$

Keywords: generalized translation, sinc function, inequality of different metrics

Received April 14, 2023

Revised May 17, 2023

Accepted May 22, 2023

Funding Agency: 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).

Vitalii Vladimirovich Arestov, Dr. Phys.-Math. Sci., Prof., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Ural Federal University, Yekaterinburg, 620000 Russia, e-mail: vitalii.arestov@urfu.ru

Marina Valer’evna Deikalova, Cand. Sci. (Phys.-Math.), Ural Federal University, Yekaterinburg, 620000 Russia; Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; e-mail: marina.deikalova@urfu.ru

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Cite this article as: V.V. Arestov, M.V. Deikalova. A generalized translation generated by the sinc-function on an interval. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2023, vol. 29, no. 4, pp. 27–48; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2023, Vol. 323, Suppl. 1, pp. S32–S52.