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 \(L_{\bar{p}, \bar\theta}^{*}(\mathbb{I}^{m})\) of periodic functions of \(m\) variables. The Besov space \(B_{\bar{p}, \bar\theta}^{(0, \alpha, \tau)}\) 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 \(B_{\bar{p}, \bar\theta}^{(0, \alpha, \tau)}\) by trigonometric polynomials under different relations between the parameters \(\bar{p}, \bar\theta,\) and \(\tau\).

The paper consists of an introduction and two sections. In the first section, we establish a sufficient condition for a function \(f\in L_{\bar{p}, \bar\theta^{(1)}}^{*}(\mathbb{I}^{m})\) to belong to the space \(L_{\bar{p}, \theta^{(2)}}^{*}(\mathbb{I}^{m})\) in the case \(1{<\theta^{2}<\theta_{j}^{(1)}},$ ${j=1,\ldots,m},\) in terms of the best approximation and prove its unimprovability on the class \(E_{\bar{p},\bar{\theta}}^{\lambda}=\{f\in L_{\bar{p},\bar{\theta}}^{*}(\mathbb{I}^{m})\colon
{E_{n}(f)_{\bar{p},\bar{\theta}}\leq\lambda_{n},}\) \({n=0,1,\ldots\},}\) where \(E_{n}(f)_{\bar{p},\bar{\theta}}\) is the best approximation of the function \(f \in L_{\bar{p},\bar{\theta}}^{*}(\mathbb{I}^{m})\) by trigonometric polynomials of order \(n\) in each variable \(x_{j},\) \(j=1,\ldots,m,\) and \(\lambda=\{\lambda_{n}\}\) is a sequence of positive numbers \(\lambda_{n}\downarrow0\) as \(n\to+\infty\). In the second section, we establish order-exact estimates for the best approximation of functions from the class \(B_{\bar{p}, \bar\theta^{(1)}}^{(0, \alpha, \tau)}\) in the space \(L_{\bar{p}, \theta^{(2)}}^{*}(\mathbb{I}^{m})\).