Estimates of Kolmogorov diameters of the classes of periodic functions of several variables with low order of smoothness (Q1093854)

Let \(\tilde W^ r_ p\) be the class of \(2\pi\)-periodical functions \(x(t_ 1,...,t_ n)\) whose means are equal to 0. Each function has r-th Weil fractional derivatives which are bounded by 1 in \(\tilde L_ p\). The author derives the next estimates of Kolmogorov diameters \(d_ N(\tilde W^ r_ p,\tilde L_ q)\) \[ (N^{-q/2}\log^{\ell - 1}N)^{r_ 1-1/p+1/q}\ll d_ N(\tilde W^ r_ p,\tilde L_ q)\quad \ll \] \[ \ll \quad (N^{-q/2}\log^{\ell -1}N)^{r_ 1- 1/p+1/q}(\log^{\ell -1}N)^{1/2-1/q}, \] where \(r=R^ n\), \(r_ 1=...=r_{\ell}<r_{\ell +1}\leq...\leq r_ n\) and \(2\leq p<q<\infty\), \(1/p-1/q<r_ 1<\beta\) or \(1<p\leq 2<q<\infty\), \(1/p-1/q<r_ 1<1/p\).