Eestimates 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<θ(1)j,$$j=1,…,m, in terms of the best approximation and prove its unimprovability on the class Eλp¯,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).