Kolmogorov widths of the class $W_1^1$

TL;DR

This paper determines the decay rate of Kolmogorov widths for univariate Sobolev classes of integer smoothness in L_q spaces, revealing a precise asymptotic behavior.

Contribution

It establishes the exact asymptotic order of Kolmogorov widths for the class W_1^1 in L_q spaces, completing the classical case analysis.

Findings

Kolmogorov widths decay as n^{-1/2} log n for 2<q<∞

Provides a complete characterization of decay rates for univariate Sobolev classes

Advances understanding of approximation properties of Sobolev spaces

Abstract

We prove that $d_n(W^1_1,L_q)\asymp n^{-1/2}\log n$, $2<q<\infty$. This completes the study of orders of decay of Kolmogorov widths for the classical case of the univariate Sobolev classes of integer smoothness.