On the convergence of generalized Padé approximants (Q1262488)

Let f be a function whose set of singular points has capacity zero. Among the points may be some branch points so f is not necessarily a single valued function off the set of its singularities. If it were single valued it would be approximable with better than geometric speed by rationals. Let V be a continuum that doesn't divide the plane and let K be a compact set containing the singularities of f such that the condenser capacity C(V,K) is minimal and such that f is single valued on the complement of K. The author shows that interpolating rationals (suited to V) converge in capacity to f in the complement of K and that modifications of these rationals (to remove unwanted poles) converge to f on V with the rate of convergence exp(-2nC(V,K)) conjectured to be best possible for algebraic functions by Goncar.