On estimates of the order of approximation of functions of several variables in the anisotropic Lorentz-Karamata space

Abstract

In this paper we consider anisotropic Lorentz-Karamata space $2\pi$ of periodic functions of $m$ variables and Nikol'skii--Besov's class . In this paper, we establish order-sharp estimates of the best approximation by trigonometric polynomials with harmonic numbers from the step hyperbolic cross of functions from the Nikol'skii - Besov class in the norm of the anisotropic Lorentz-Karamata space.

15 pages. Comments 1: In Lemma 1, " $1\le {\theta }_j\le \infty$ " and $A\{j:\frac{{\gamma }_j}{{\gamma }_j^{{}^{\prime }}}=\delta ,j=1,\dots ,m\}$ are replaced with "$1\le {\theta }_j<\infty$" and $A=\{j:\frac{{\gamma }_j}{{\gamma }_j^{{}^{\prime }}}=\delta ,j=1,\dots ,m\}$. Comments 2: The statement of Theorem 1 is refined. arXiv admin note: text overlap with arXiv:2106.07188