Kolmogorov widths of balls in mixed norms: the case of rigidity

TL;DR

This paper characterizes when certain mixed-norm balls are rigid, meaning they are poorly approximated by low-dimensional linear subspaces, advancing the understanding of their Kolmogorov widths.

Contribution

It determines the parameter conditions for rigidity of mixed-norm balls, settling a key case in the estimation of their widths.

Findings

Identified parameter sets where balls are rigid in mixed norms.

Provided a new construction for approximation in exceptional cases.

Established the qualitative behavior of widths for large s, b.

Abstract

We describe the set of parameters $(p_1,p_2,q_1,q_2)$ such that the balls $B_{q_1,q_2}^{s,b}$ are rigid in $\ell_{q_1,q_2}^{s,b}$ metric i.e. they are poorly approximated by linear subspaces of dimension $\le (1-\varepsilon)sb$, for large $s, b$. Thus we have settled an important qualitative case in the problem of estimating widths of balls in mixed norms. The proof combines lower bounds from our previous papers and a new construction for the approximation by linear subspaces in the so-called exceptional case.