Estimates of best approximations of functions with logarithmic smoothness in the Lorentz space with anisotropic norm. (English) Zbl 1464.46025

Akishev, Gabdolla

Ural Math. J. 6, No. 1, 16-29 (2020).

Summary: 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 \(L_{\bar{p}, \theta^{(2)}}^*(\mathbb{I}^m)\).

MSC:

46E30	Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
42A10	Trigonometric approximation

Keywords:

Lorentz space; Nikol’skii-Besov class; best approximation

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