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

MSC:
 41A46 Approximation by arbitrary nonlinear expressions; widths and entropy
Keywords:
Kolmogorov diameter