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

TL;DR

This paper provides order estimates for Kolmogorov n-widths of intersections of anisotropic balls in finite-dimensional l_q spaces for 1 ≤ q ≤ 2, advancing understanding of their geometric approximation properties.

Contribution

It introduces new order estimates for Kolmogorov widths of intersections of anisotropic balls in l_q^k spaces, generalizing previous results to arbitrary families and anisotropic norms.

Findings

Derived order estimates for Kolmogorov n-widths

Extended results to arbitrary families of anisotropic balls

Applicable for 1 ≤ q ≤ 2 and n ≤ k/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 $\overline{p}_\alpha$.