Approximation of classes of periodic functions in one and many variables from the Nikol’skii – Besov and Sobolev spaces

• Datum: 2022
• Hauptverfasser: Romanyuk, A. S., Yanchenko, S. Ya., Романюк, А. С., Янченко, С. Я.
• Format: Artikel
• Sprache: Ukrainisch
• Veröffentlicht: Institute of Mathematics, NAS of Ukraine 2022
• Online Zugang: https://umj.imath.kiev.ua/index.php/umj/article/view/7141
• Назва журналу: Ukrains’kyi Matematychnyi Zhurnal

UDC 517.51 In this paper, we obtain exact-order estimates for the best orthogonal trigonometric approximations of the Nikol'skii-Besov classes $B^{\boldsymbol{r}}_{1,\theta}(\mathbb{T}^d),$ $1\leq\theta\leq\infty,$ of periodic functions of one and many variables with dominating mixed smoothness in the space $B_{\infty,1}(\mathbb{T}^d)$.In the multidimensional case, $d\geq 2,$ we establish exact-order estimates for approximations of the mentioned classes of functions by their step-hyperbolic Fourier sums and find the orthoprojection width orders in the same space. The behavior of corresponding approximation characteristics of the Sobolev classes $W^{\boldsymbol{r}}_{1,\boldsymbol{\alpha}}\left(\mathbb{T}^d\right)$ for $d\in\{1,2\}$ is also studied.