The best trigonometric approximations and the Kolmogorov diameters of the Besov classes of functions of many variables (Q1335986)

Let \(B^ r_{p,\theta} \subset L_ p (\mathbb{T}^ m)\) be the functional class defined by \[ B^ r_{p,\theta}:= \Biggl\{f \Biggl| \int_{- \pi}^ \pi \Biggr. f dx_ j=0,\;1\leq j\leq m, \text{ and } \sum_{s\in \mathbb{N}^ m} 2^{\langle s,r \rangle\theta} \| \delta_ s (f) \|_ p^ \Theta \leq 1 \Biggr\}. \] Here \(s= (s_ 1,\dots, s_ m)\in \mathbb{N}^ m\) and \(r= (r_ 1,\dots, r_ m)\), \(r_ j>0\) \((1\leq j\leq m)\) and \[ \delta_ s (f;x)= \sum_{2^{s_ j-1}\leq k_ j< 2^{s_ j}} \widehat {f}_ k e^{i\langle k,x \rangle}. \] Basing on the inequalities of E. Belinskij and V. Temlyakov the author obtained sharp in order estimates for the trigonometrical width \(e_ M (B^ r_{p,\theta}, L_ q)\) in case \(1<p\leq 2<q<\infty\) and \({1\over p}- {1\over q}< r_ 1< {1\over p}\), and for the Kolmogorov width \(d_ M (B^ r_{p,\theta}, L_ q)\) in case \(1<p\leq 2<q<\infty\), \(r_ 1> {1\over p}\), and in case \(2\leq p<q <\infty\), \(r_ 1> ({1\over p}- {1\over q}) (1- {2\over q})\).