Kolmogorov widths of Besov classes $B^1_{1,\theta}$ and products of octahedra

TL;DR

This paper determines the decay rates of Kolmogorov widths for certain Besov classes related to $W^1_1$, using bounds for widths of octahedron products, extending Kashin's theorem.

Contribution

It provides new asymptotic decay rates for Kolmogorov widths of Besov classes and generalizes Kashin's theorem to product of octahedra in a specific norm.

Findings

Decay rates for Kolmogorov widths of Besov classes are established.

A lower bound for widths of product of octahedra is derived.

The results extend Kashin's theorem on widths of octahedra.

Abstract

In this paper we find the orders of decay for Kolmogorov widths of some Besov classes related to $W^1_1$ (the behaviour of the widths for $W^1_1$ remains unknown): $$ d_n(B^1_{1,\theta}[0,1],L_q[0,1])\asymp n^{-1/2}\log^{\max(\frac12,1-\frac{1}{\theta})}n,\quad 2<q<\infty. $$ The proof relies on the lower bound for widths of product of octahedra in a special norm (maximum of two weighted $\ell_q$ norms). This bound generalizes the theorem of B.S.~Kashin on widths of octahedra in $\ell_q^N$.