Estimates for Kolmogorov diameters of classes of functions with conditions on the mixed difference in uniform metric. (Russian) Zbl 0745.41025

Belinskij, Eh. S.

Mat. Zametki 50, No. 5, 147-149 (1991).

The paper is concerned with evaluation (upper and lower) of Kolmogorov diameter \(d_ N(H_ p^ r,L_ \infty)\) for the class \(H^ r_ p\) of all periodic functions \(f\) on the \(n\)-dimensional torus, whose finite differences of order \(\ell\) verify the inequality \(\parallel \Delta^ \ell_ h f\parallel_ p\leq M \cdot \prod \mid h_ j \mid^{r_ j}\), \(h=(h_ j)\) and \(r=(r_ j)\) in \(R^ n\) (only non-zero factors are taken). It is known that for \(2<p\leq \infty\) and \(p^{-1}<r_ 1\leq r_ 2...\leq r_ n\) the set \(H_ p^ r\) can be compactly embedded in \(L_ \infty\), which is the case considered in this paper.

Reviewer: S.Cobzaş (Cluj-Napoca)

MSC:

41A46

Approximation by arbitrary nonlinear expressions; widths and entropy

Keywords:

Kolmogorov diameter