On optimal approximation in periodic Besov spaces (Q2633709)

The authors investigate optimal linear approximations (approximation numbers) in the context of isotropic periodic Besov spaces $B_{p,q}^{t}(\mathbb{T}^{d})$ and Besov spaces of dominating mixed smoothness $S_{p,q}^{t}B(\mathbb{T}^{d})$, where $\mathbb{T}^{d}$ denotes the $d$-dimensional torus. Specifically, the authors show that the results on optimal approximation of multivariate periodic Sobolev functions in the $L_{\infty}(\mathbb{T}^{d})$ norm given in [the authors, J. Funct. Anal. 270, No. 11, 4196--4212 (2016; Zbl 1357.46028)] can be replaced by any results on $B_{\infty,1}^{0}(\mathbb{T}^{d})$-approximation and $S_{\infty,1}^{0}B(\mathbb{T}^{d})$-approximation without changing the associated approximation numbers. As the authors point out, this fact is unexpected and has some practical use, since the Littlewood-Paley characterization of the target spaces simplifies the computation of approximation or other $s$-numbers. For $t>\frac{1}{2}$, the authors establish estimates for the approximation numbers of the embedding $I_{d}: S_{1,\infty}^{t}B(\mathbb{T}^{d}) \rightarrow L_{2}(\mathbb{T}^{d})$ which show the dependence of the involved constant on the dimension $d$ and the smoothness $s$. Also, they consider the approximation of tensor products of functions with finite total variation. The authors conclude the paper with some interesting open questions.