Estimates of best approximations of functions with logarithmic smoothness in the lorentz space with anisotropic norm

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.