Estimates for approximation characteristics of Nikol'skii–Besov classes of functions with mixed smoothness in the space Bq,1${B_{q,1}}$

K. V. Pozharska and A. S. Romanyuk

Mathematische Nachrichten, 2025, vol. 298, issue 9, 3114-3134

Abstract: Exact‐order estimates are obtained for some approximation characteristics of the classes of periodic multivariate functions with mixed smoothness (the Nikol'skii–Besov classes Bp,θr$B^{\bm{r}}_{p, \theta }$) in the space Bq,1$B_{q,1}$, 1≤p,q≤∞$1 \le p, q \le \infty$, 1≤θ≤∞$1\le \theta \le \infty$, whose norm is stronger than the Lq$L_q$‐norm. It is shown that in the multivariate case (in contrast to the univariate) in most of the considered situations the obtained estimates differ in order from the corresponding estimates in the space Lq$L_q$. Besides, a significant progress is made in estimates for the considered approximation characteristics of the classes Bp,θr$B^{\bm{r}}_{p, \theta }$ in the space Bq,1$B_{q, 1}$ comparing to the known estimates in the space Lq$L_q$.

Date: 2025
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