The Convergence of Padé Approximants to Functions with Branch Points

Abstract

Padé approximants are a natural generalization ofTaylor polynomials; however instead of polynomials now rationalfunctions are used for the development of a given function.In this article the convergence in capacity of Padé approximants[m/n] withm+n→∞,m/n→1,is investigated. Two types of assumptions are considered: Inthe first case the functionfto be approximated has to haveall its singularities in a compact setE⊆C of capacityzero (the function may be multi-valued in C\E).In the second case the functionfhas to be analytic in a domainpossessing a certain symmetry property (this notion is definedand discussed below). It is shown that close-to-diagonal sequencesof Padé approximants [m/n] converge tofincapacity in a domainDthat can be determined in various ways.In the case of the first type of assumptions the domainDisdetermined by the minimality of the capacity of the complementofD, in the second case the domainDis determined by a symmetryproperty. The rate of convergence is determined, and it is shownthat this rate is best possible for convergence in capacity.In addition to the convergence results the asymptotic distributionof zeros and poles of the approximants is studied. © 1997 Academic Press.