On Blaschke products associated with n-widths

Abstract

Let E be a closed subset of the open unit disk G ¼ fz : jzjo1g; and let m be a positive Borel measure with support supp m ¼ E: Denote by A p the restriction on E of the closed unit ball of the Hardy space H p ðGÞ; 1pppN: In this paper we investigate orthogonality properties of the extremal functions associated with the Kolmogorov, Gelfand, and linear n-widths of A p in L q ðm; EÞ; 1pqoN; qpp: r

Error Estimates and Evaluation of Matrix Functions via the Faber Transform

“…Since B(w) = 1/B(1/w), we, equivalently, may choose the poles to make |B| as small as possible on the interval [1/φ(β), 1/φ(α)]. This kind of minimization problem has received considerable attention in complex approximation theory; see, e.g., [6,36]. From [73,Theorem VIII.3.1], we obtain, for any Blaschke product of the form (6.1), that , (6.9) where cap (E, F) denotes the logarithmic capacity of a two-dimensional condenser with plates E and F; see, e.g., [73,equation (VIII.3.9)].…”

On rational approximation of Markov functions on finite sets

“…Questions of convergence of multipoint Padé approximants (interpolation sequences of rational functions with free poles) were studied by Gonchar and Lopez [17], Totik [35], Stahl and Totik [34]. We also mention papers of Anderson [4], Braess [13], Baratchart, Prokhorov and Saff [6][7][8], Barrett [10], Pekarskii [23] related to rational and meromorphic approximation of Markov functions. In [9] Baratchart, Stahl and Wielonsky gave sharp error rates for the best H 2 approximants (particular Padé approximants) to Markov functions.…”