On the K-Functional for the Mixed Generalized Modulus of Smoothness

On the K-Functional for the Mixed Generalized Modulus of Smoothness ISSN 0001-4346, Mathematical Notes, 2018, Vol. 103, No. 2, pp. 319–322. © Pleiades Publishing, Ltd., 2018. Original Russian Text © N. V. Omel’chenko, 2018, published in Matematicheskie Zametki, 2018, Vol. 103, No. 2, pp. 312–315. SHORT COMMUNICATIONS On the K -Functional for the Mixed Generalized Modulus of Smoothness N. V. Omel’chenko “Mathematical Notes,” Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia Received June 11, 2017 DOI: 10.1134/S0001434618010352 Keywords: mixed modulus of smoothness, K -functional. The transition from some modulus of smoothness to the K -functional equivalent to it is one of the widely used methods in the theory of approximation of periodic functions of one or several variables by trigonometric polynomials in the scale of spaces L , 1 ≤ p ≤ +∞. In this paper, the K -functional equivalent to the generalized mixed modulus of smoothness introduced in [1] is constructed. Following [1], we say that a function θ of one variable belongs to the class G if it is 2π-periodic centrally symmetric (θ(−ξ)= θ(ξ) for ξ ∈ R) and continuous; in this case, the series of its Fourier ∧ ∧ coefficients θ = {θ (ν),ν ∈ Z} absolutely converges and θ(0) = 0. We use the http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematical Notes Springer Journals

On the K-Functional for the Mixed Generalized Modulus of Smoothness

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Publisher
Pleiades Publishing
Copyright
Copyright © 2018 by Pleiades Publishing, Ltd.
Subject
Mathematics; Mathematics, general
ISSN
0001-4346
eISSN
1573-8876
D.O.I.
10.1134/S0001434618010352
Publisher site
See Article on Publisher Site

Abstract

ISSN 0001-4346, Mathematical Notes, 2018, Vol. 103, No. 2, pp. 319–322. © Pleiades Publishing, Ltd., 2018. Original Russian Text © N. V. Omel’chenko, 2018, published in Matematicheskie Zametki, 2018, Vol. 103, No. 2, pp. 312–315. SHORT COMMUNICATIONS On the K -Functional for the Mixed Generalized Modulus of Smoothness N. V. Omel’chenko “Mathematical Notes,” Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia Received June 11, 2017 DOI: 10.1134/S0001434618010352 Keywords: mixed modulus of smoothness, K -functional. The transition from some modulus of smoothness to the K -functional equivalent to it is one of the widely used methods in the theory of approximation of periodic functions of one or several variables by trigonometric polynomials in the scale of spaces L , 1 ≤ p ≤ +∞. In this paper, the K -functional equivalent to the generalized mixed modulus of smoothness introduced in [1] is constructed. Following [1], we say that a function θ of one variable belongs to the class G if it is 2π-periodic centrally symmetric (θ(−ξ)= θ(ξ) for ξ ∈ R) and continuous; in this case, the series of its Fourier ∧ ∧ coefficients θ = {θ (ν),ν ∈ Z} absolutely converges and θ(0) = 0. We use the

Journal

Mathematical NotesSpringer Journals

Published: Mar 14, 2018

References

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