Ratios of Norms for Polynomials and Connected n-width Problems

Abstract

Let \(G \subset{\mathbb{C}}\) be a bounded simply connected domain with boundary Γ and let \(E \subset G\) be a regular compact set with connected complement. In this paper we investigate asymptotics of the extremal constants: $$\chi_{n}= \inf\limits_{p \in {\mathcal{P}}_{k_{n}}} \sup\limits_{q \in {\mathcal{P}}_{n}-{k_{n}}} \frac{||pq||_E} {||pq||_\Gamma}, \quad n= 1, 2, . . . ,$$ where \(|| \cdot || _{K}\) is the supremum norm on a compact set K, \({\mathcal{P}}_m\) is the set of all algebraic polynomials of degree at most m, and \(k_{n}/n \rightarrow \theta \in [0, 1]\) as \(n \rightarrow \infty\). Subsequently, we obtain asymptotic behavior of the Kolmogorov k-widths, \(k=k_{n}\), of the unit ball An∞ of \(H^{\infty} \cap {\mathcal{P}}_{n}\) restricted to E in C(E), where H∞ is the Hardy space of bounded analytic functions on G and C(E) is the space of continuous functions on E.