Please use this identifier to cite or link to this item: http://hdl.handle.net/10995/101473
Title: Estimates of best approximations of functions with logarithmic smoothness in the lorentz space with anisotropic norm
Authors: Akishev, G.
Issue Date: 2020
Publisher: Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of Russian Academy of Sciences
Citation: Akishev G. Estimates of best approximations of functions with logarithmic smoothness in the lorentz space with anisotropic norm / G. Akishev. — DOI 10.15826/umj.2020.1.002 // Ural Mathematical Journal. — 2020. — Vol. 6. — Iss. 1. — P. 16-29.
Abstract: In this paper, we consider the anisotropic Lorentz space L∗¯p, ¯θ (Im) of periodic functions of m variables. The Besov space B(0,α,τ)¯p, ¯θ of functions with logarithmic smoothness is defined. The aim of the paper is to find an exact order of the best approximation of functions from the class B(0,α,τ)¯p, ¯θ by trigonometric polynomials under different relations between the parameters ¯p, ¯θ, and τ . The paper consists of an introduction and two sections. In the first section, we establish a sufficient condition for a function f ∈ L∗¯p,¯θ(1) (Im) to belong to the space L∗¯p,θ(2) (Im) in the case 1< θ2 < θj(1), j = 1, …, m, in terms of the best approximation and prove its unimprovability on the class Eλ¯p, ¯θ = {f ∈ L∗¯p,¯θ (Im): En(f)¯p,¯θ ≤ λn, n = 0, 1, …}, where En(f)¯p, ¯θ is the best approximation of the function f ∈ L∗¯p,¯θ (Im) by trigonometric polynomials of order n in each variable xj, j = 1, …, m, and λ = {λn} is a sequence of positive numbers λn ↓ 0 as n → +∞. In the second section, we establish order-exact estimates for the best approximation of functions from the class B(0,α,τ)¯p, ¯θ(1) in the space L∗¯p,θ(2) (Im). © 2020, Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of Russian Academy of Sciences. All rights reserved.
Keywords: BEST APPROXIMATION
LORENTZ SPACE
NIKOL’SKII–BESOV CLASS
URI: http://hdl.handle.net/10995/101473
Access: info:eu-repo/semantics/openAccess
SCOPUS ID: 85089116548
PURE ID: 13679248
6e9740cc-fdfc-4e15-97dc-d9fe7010175e
ISSN: 24143952
DOI: 10.15826/umj.2020.1.002
metadata.dc.description.sponsorship: This work was supported by the Competitiveness Enhancement Program of the Ural Federal University (Enactment of the Government of the Russian Federation of March 16, 2013 no. 211, agreement no. 02.A03. 21.0006 of August 27, 2013).
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