Kolmogorov widths of an intersection of anisotropic finite-dimensional balls in $l_q^k$ for $1\le q\le 2$

Abstract

In this paper, order estimates for the Kolmogorov $n$-widths of an intersection of an arbitrary family of balls $\nu_\alpha B^{\overline{k}}_{\overline{p}_\alpha}$ in $l_q^k$ are obtained for $1\le q\le 2$, $n\le \frac k2$. Here $\overline{p}_\alpha = (p_{\alpha,1}, \, \dots, \, p_{\alpha,d})$, $\overline{k}=(k_1, \, \dots, \, k_d)$, $k=k_1\dots k_d$, $B^{\overline{k}}_{\overline{p}_\alpha}$ is the unit ball with respect to the anisotropic norm given by the vector $\overline{p}_\alpha$.