Weighted extremal domains and best rational approximation

Abstract

Let f be holomorphically continuable over the complex plane except for finitely many branch points contained in the unit disk. We prove that best rational approximants to f of degree n, in the L 2 -sense on the unit circle, have poles that asymptotically distribute according to the equilibrium measure on the compact set outside of which f is single-valued and which has minimal Green capacity in the disk among all such sets. This provides us with n-th root asymptotics of the approximation error. By conformal mapping, we deduce further estimates in approximation by rational or meromorphic functions to f in the L 2 -sense on more general Jordan curves encompassing the branch points. The key to these approximation-theoretic results is a characterization of extremal domains of holomorphy for f in the sense of a weighted logarithmic potential, which is the technical core of the paper.2000 Mathematics Subject Classification. 42C05, 41A20, 41A21.

Minimax principle and lower bounds in H2-rational approximation

“…That the infimum is indeed attained in the right hand side of (10) follows from (8) and the fact that RAB(n) has a solution. Define A f , the Hankel operator with symbol f , by…”

“…It may seem artificial to favor the hyperbolic geodesic segment linking the branchpoints among all possible cuts. However, this one turns out to attract almost all poles of best rational approximants (see [8] for this and generalizations to finitely many branchpoints) and also of best meromorphic approximants (see [6,Theorem 10.1] and Corollary 3 below), which makes it in some sense the natural singular set of the function. We need additional facts from AAK theory that shed light on singular vectors of Hankel operators with continuous symbol.…”

“…The present paper is, in part, a sequel to [8]. In the latter reference best L 2 and L ∞ rational approximants are compared in the n-th root sense, whereas here we compare them in norm.…”

“…Let us mention Peller's converse theorems on the speed of rational approximation [33], Glover's construction of near-best uniform rational approximants [15], Parfenov's solution of a conjecture by Gonchar on the degree of rational approximation to holomorphic functions on compact subsets of the domain of analyticity [31], the Gonchar-Rakhmanov estimates in uniform rational approximation to sectionally holomorphic functions off an S-contour, and its generalization to best L 2 and L p approximants in [8,42].…”

Existence of compact sets with minimum capacity in problems of rational approximation of multivalued analytic functions

“…We note that in the case treated in [1] an additional assumption about the mutual arrangement of E and A f was imposed. In the general case, when no additional assumptions are made, the answer in the problem can be negative even for m = 1: in place of (1) we may have the inequality…”