Actual problems of the approximation theory in metrics of discrete spaces on sets of summable periodic and almost periodic functions

2024/07/05 by A. S. Serdyuk, Serdyuk, Anatolii, Andrii Shidlich +1 · 1 citation
Mathematics · Computer Science · #Mathematical Approximation and Integration #Advanced Computational Techniques in Science and Engineering #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2407.04329

Abstract

This review paper highlights the main aspects of the development of research related to the solution of extreme problems in the theory of approximation in the spaces \mathcal S^p and B\mathcal S^p of periodic and almost periodic summable functions, respectively, where the l_p-norms of the sequences of Fourier coefficients are finite. In particular, the review contains the results known so far concerning the best, best n-term approximations and widths of classes of functions of one and many variables defined by means of ψ-derivatives and generalized moduli of smoothness in the spaces \mathcal S^p and B\mathcal S^p. Particular attention is paid to the development of studies related to the derivation of direct and inverse approximation theorems in these spaces.

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• Extremal problem of area approximation

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• Approximation by linear methods of classes of (ψ,β)-differentiable functions
• Direct and inverse theorems on the approximation of almost periodic functions in Besicovitch-Stepanets spaces
• Direct and inverse approximation theorems in the Besicovitch-Museilak-Orlicz spaces of almost periodic functions
• Strong approximation of almost periodic functions
• Approximation by Fourier sums in classes of Weyl--Nagy differentiable functions with high exponent of smoothness