Abstract
We give a solution of the problem on the best uniform approximation on the number axis of the first-order differentiation operator on the class of functions with bounded second derivative by linear operators bounded in the space L2. This is one of the few cases of the exact solution of the problem on the approximation of the differentiation operator in some space with the use of approximating operators that are bounded in another space. We obtain a related exact inequality between the uniform norm of the derivative of a function, the variation of the Fourier transform of the function, and the L∞-norm of its second derivative. This inequality can be regarded as a nonclassical variant of the Hadamard-Kolmogorov inequality.
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Funding
This work was 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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Russian Text © The Author(s), 2018, published in Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2018, Vol. 24, No. 4, pp. 34–56.
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Arestov, V.V. Best Uniform Approximation of the Differentiation Operator by Operators Bounded in the Space L2. Proc. Steklov Inst. Math. 308 (Suppl 1), 9–30 (2020). https://doi.org/10.1134/S0081543820020029
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DOI: https://doi.org/10.1134/S0081543820020029