Ratios of Norms for Polynomials and Connected n-width Problems

TL;DR

This paper studies the asymptotic behavior of Kolmogorov k-widths for polynomial sets constrained by supremum norms on specific domains, advancing understanding of approximation limits in complex analysis.

Contribution

It introduces new asymptotic estimates for Kolmogorov k-widths of polynomial sets on connected compact subsets of simply connected domains.

Findings

Derived asymptotic formulas for Kolmogorov k-widths

Established connections between polynomial norms and geometric properties of domains

Provided insights into n-width problems in complex approximation theory

Abstract

Let G be a bounded simply connected domain and E be a regular compact subset of G with connected complement. We investigate the asymptotic behavior of the Kolmogorov k-width, k=k(n), of the set of polynomials of degree at most n having the supremum norm at most 1 on G restricted to E in the space of continuous functions on E.