2 3 A pr 2 00 8 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. Mathematics Subject Classification (2000). 30E10, 41A16.