V.V. Arestov, M.V. Deikalova. On one generalized translation and the corresponding inequality of different metrics ... P. 40-53

On one generalized translation and the corresponding inequality of different metrics. In this paper, we discuss the properties of the generalized translation operator generated by the system of functions $\left\{ \cos\left(\frac{(2k-1)\pi }{2}t\right)\right\}_{k=1}^\infty$, in the spaces $L^p(0,1)$, $p\ge 1.$ The translation operator is applied to the study of Nikol'skii's inequality between the uniform norm and the $L^p$-norm of polynomials in this system.

Keywords: generalized translation operator, trigonometric polynomial, inequality of different metrics

Received June 5, 2022

Revised July 5, 2022

Accepted July 11, 2022

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. 075-02-2022-874).

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. On one generalized translation and the corresponding inequality of different metrics. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2022, vol. 28, no. 4, pp. 40–53; Proceedings of the Steklov Institute of Mathematics (Suppl. 1), 2022, Vol. 319, Suppl. 1, pp. S30–S42.