Kolmogorov n-Widths of Function Classes Induced by a Non-Degenerate Differential Operator: A Convex Duality Approach

TL;DR

This paper determines the asymptotic behavior of Kolmogorov n-widths for certain function classes defined by non-degenerate differential operators, using convex duality methods to solve a long-standing open problem.

Contribution

It introduces a convex duality approach to compute the asymptotic order of Kolmogorov n-widths for classes induced by non-degenerate differential operators, resolving a longstanding open problem.

Findings

Derived the asymptotic order of Kolmogorov n-widths for the class U^{[P]}_2

Applied convex analytical tools to the problem

Solved the problem for non-degenerate differential operators

Abstract

Let $P(D)$ be the differential operator induced by a polynomial $P$, and let ${U^{[P]}_2}$ be the class of multivariate periodic functions $f$ such that $\|P(D)(f)\|_2\leq 1$. The problem of computing the asymptotic order of the Kolmogorov $n$-width $d_n({U^{[P]}_2},L_2)$ in the general case when ${U^{[P]}_2}$ is compactly embedded into $L_2$ has been open for a long time. In the present paper, we use convex analytical tools to solve it in the case when $P(D)$ is non-degenerate.