Estimates for the best approximations of functions from the Nikol'skii-Besov class in the Lorentz space by trigonometric polynomials

G. A. Akishev

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 26 (2020) no. 2, pp. 5-27

Résumé

We consider spaces of periodic functions of many variables, specifically, the Lorentz space $L_{p, \tau}(\mathbb{T}^{m})$ and the Nikol'skii–Besov space $S_{p, \tau, \theta}^{\bar{r}}B$, and study the best approximation of a function $f \in L_{p, \tau}(\mathbb{T}^{m})$ by trigonometric polynomials with the numbers of harmonics from a step hyperbolic cross. Sufficient conditions are established for a function $f \in L_{p, \tau_{1}}(\mathbb{T}^{m})$ to belong to a space $L_{q, \tau_{2}}(\mathbb{T}^{m})$ in the cases $1 $, $1 \tau_{1}, \tau_{2} \infty$ and $p = q$, $ 1 \tau_{2} \tau_{1} \infty$. Estimates for the best approximations of functions from the Nikol'skii–Besov class $S_{p, \tau_{1}, \theta}^{\bar{r}}B$ in the norm of the space $L_{q, \tau_{2}}(\mathbb{T}^{m})$ are derived for different relations between the parameters $p$, $q$, $\tau_{1}$, $\tau_{2}$, and $\theta$. For some relations between these parameters, it is shown that the estimates are exact.

Détail

Keywords: Lorentz space, trigonometric polynomial, best approximation, hyperbolic cross.
Mots-clés : Nikol'skii–Besov class