ON ESTIMATES OF M-TERM APPROXIMATIONS ON CLASSES OF FUNCTIONS WITH BOUNDED MIXED DERIVATIVE IN THE LORENTZ SPACE

Abstract

The paper considers spaces of periodic functions of several variables, namely, the Lorentz space $L_{q, \tau }(\mathrm {T}^{m})$, the class of functions with bounded mixed fractional derivative $W_{q, \tau }^{\overline{r}}$, $1< q, \tau < \infty$, and studies the order of the best M-term approximation of a function $f \in L_{p, \tau }(\mathrm {T}^{m})$ by trigonometric polynomials. The article consists of the introduction, the main part, and the conclusion. In the introduction, we introduce basic concepts, definitions, and necessary statements for the proof of the main results. You can also find information about previous results on the topic. In the main part, we establish exact-order estimates for the best M-term approximations of functions of the class $W_{q, \tau _{1}}^{\overline{r}}$ in the norm of the space $L_{p, \tau _{2 }}(\mathrm {T}^{m})$ for various relations between the parameters $p, q, \tau _{1}, \tau _{2}$.