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 Mat. Zametki, 2016, Volume 100, Issue 3, Pages 421–432 (Mi mz10860)

Mixed Generalized Modulus of Smoothness and Approximation by the “Angle” of Trigonometric Polynomials

K. V. Runovskii, N. V. Omel'chenko

Sevastopol Branch of the M.V. Lomonosov Moscow State University

Abstract: The notion of general mixed modulus of smoothness of periodic functions of several variables in the spaces $L_p$ is introduced. The proposed construction is, on the one hand, a natural generalization of the general modulus of smoothness in the one-dimensional case, which was introduced in a paper of the first author and in which the coefficients of the values of a given function at the nodes of a uniform lattice are the Fourier coefficients of a $2\pi$-periodic function called the generator of the modulus; while, on the other hand, this construction is a generalization of classical mixed moduli of smoothness and of mixed moduli of arbitrary positive order. For the modulus introduced in the paper, in the case $1 \le p \le +\infty$, the direct and inverse theorems on the approximation by the “angle” of trigonometric polynomials are proved. The previous estimates of such type are obtained as direct consequences of general results, new mixed moduli are constructed, and a universal structural description of classes of functions whose best approximation by “angle” have a certain order of convergence to zero is given.

Keywords: generalized modulus of smoothness, mixed modulus of smoothness, approximation by “angle,” direct and inverse approximation theory.

 Funding Agency Grant Number Russian Foundation for Basic Research 15-01-01236 à This work was supported by the Russian Foundation for Basic Research within the framework of research grant no. 15-01-01236-a.

DOI: https://doi.org/10.4213/mzm10860

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English version:
Mathematical Notes, 2016, 100:3, 448–457

Bibliographic databases:

UDC: 517.51

Citation: K. V. Runovskii, N. V. Omel'chenko, “Mixed Generalized Modulus of Smoothness and Approximation by the “Angle” of Trigonometric Polynomials”, Mat. Zametki, 100:3 (2016), 421–432; Math. Notes, 100:3 (2016), 448–457

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/mz10860
• https://doi.org/10.4213/mzm10860
• http://mi.mathnet.ru/eng/mz/v100/i3/p421

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This publication is cited in the following articles:
1. N. V. Omel'chenko, “On the $K$-Functional for the Mixed Generalized Modulus of Smoothness”, Math. Notes, 103:2 (2018), 319–322
2. N. V. Omel'chenko, K. V. Runovskii, “Realizations of Mixed Generalized $K$-Functionals”, Math. Notes, 107:2 (2020), 353–356
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