The convergence of Padé approximants to functions with branch points
The dates of receipt of the original and revised versions of this paper tell a story: the original was received in 1983, and the revised in 1997. In the fourteen years in between, the author's ideas have revolutionized not only the convergence theory of Padé approximation, but also the study of complex orthogonal polynomials, rational approximation, and numerous extremal problems. The reasons for the delay are administrative; it is a delight to finally have this polished paper in print. He presents here one approach to analyzing the convergence of sequences of Padé approximants to functions with branchpoints. There are two classes of functions considered. The first is analytic but not necessarily single-valued in the plane except for singularities in a set of capacity \(0\). The second class of functions is analytic in a domain possessing a symmetry property. In both cases, the author establishes convergence in capacity, with the exact rate, of sequences of diagonal Padé approximants in a certain domain. Outside this domain, there is divergence in capacity. The actual domain of convergence is completely characterized, and in the case of functions whose branchpoints have a simple structure, may be described with the aid of quadratic differentials. The proofs rely heavily on asymptotic properties of orthogonal polynomials in the complex plane, with respect to an ``inner product'' that does not have the usual positivity properties, and employ sophisticated potential theory. It is good to have this classic in print: for many years it and its ideas were quoted and the question always was: but where did it or will it appear? Now, we may all quote actual page numbers.
scientific article; zbMATH DE number 1100511
scientific article