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Uniform Approximation of Differentiation Operators by Bounded Linear Operators in the Space Lr

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Abstract

We discuss the problem of the best uniform approximation on the axis, i.e., in the space C(−∞,∞), of the differentiation operator of order k on the class of functions with bounded derivative of order n, 0 < k < n, by bounded linear operators in the space Lr,1 ≤ r < ∞. We give an exact solution of the problem for odd n ≥ 3 and all k, 0 < k < n. For even n, a result close to the best one is obtained. We consider a related inequality between the uniform norm of the kth order derivative of a function, the norm of the function in the space predual for the space of multipliers in Lr, and the L∞-norm of the nth order derivative; this inequality is an analogue of the Kolmogorov inequality.

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Acknowledgement

The author is grateful to the referee, who thoroughly read the manuscript of the paper and made a number of quite useful comments.

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Authors and Affiliations

  1. Institute of Natural Sciences and Mathematics, Ural Federal University, Yekaterinburg, Russia

    V. Arestov

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

    V. Arestov

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  1. V. Arestov
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Correspondence to V. Arestov.

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This work was performed as part of research conducted in the Ural Mathematical Center and also supported by the Russian Foundation for Basic Research (project no. 18-01-00336) and by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University).

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Arestov, V. Uniform Approximation of Differentiation Operators by Bounded Linear Operators in the Space Lr. Anal Math 46, 425–445 (2020). https://doi.org/10.1007/s10476-020-0040-z

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