On estimates of the best M–term approximations of functions of the class

A. Kh. Myrzagaliyeva, G. Akishev

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abstract_carpathian_2026_42_1_47-63

https://doi.org/10.37193/CJM.2026.01.04

Published on 30 September 2025

Issue no: Vol 42/2026 no. 1 Tags: Lorentz space, Sobolev class, mixed derivative, trigonometric polynomial, M-term approximation

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Abstract.

The article considers the Lorentz space $L_{p,\tau}(\mathbb{T}^{m})$ of $2\pi$ –periodic functions with $m$ variables and the Sobolev class $W^{\bar{r}}_{p, \tau}$ in $L_{p,\tau}(\mathbb{T}^{m})$ . The main goal of the article is to find the order of the best $M$ –term trigonometric approximation $e_{M}(f)_{p, \tau_2}$ of the class $W^{\bar{r}}_{q, \tau_1}$ for a number of relations between the parameters $p, q, \tau_1, \tau_2\in (1, \infty)$ and $r_j$ , $j=1, \dots, m$ , $\bar{r}=(r_1, \dots, r_m)$ . The article establishes order-sharp estimates for the value $e_{M}(W^{\bar{r}}_{q, \tau_1})_{p, \tau_2}$ in the cases $1<q<2<p<\infty$ , $r_1>\frac1{q}$ and $r_1=\frac1{q}$ , $1<q<2$ and $1<\tau_1<\infty$ or $q=2$ and $2<\tau_1<\infty$ . The exact order of this value are obtained for $p=q$ and $\tau_2=\tau_1=\tau$ , as well as the upper estimate for $1<p<q<\infty$ .