Ratios of Norms for Polynomials and Connected n-width Problems

Let G ⊂C be a bounded simply connected domain with boundary Γ and let E ⊂G be a regular compact set with connected complement. In this paper we investigate asymptotics of the extremal constants: χn = inf p∈Pkn sup q∈Pn−kn ||pq ||E ||pq ||Γ , n = 1,2, . . . , where ‖ ·‖K is the supremum norm on a compact set K ,Pm is the set of all algebraic polynomials of degree at most m, and kn/n→ θ ∈ [0,1] as n→∞. Subsequently, we obtain asymptotic behavior of the Kolmogorov k-widths, k = kn , of the unit ball A ∞ n of H ∩Pn 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 .