THE CONVERGENCE OF PADE APPROXIMANTS TO FUNCTIONS WITH BRANCH-POINTS

Pade approximants are a natural generalization of Taylor polynomials; however instead of polynomials now rational functions are used for the development of a given function. In this article the convergence in c apacity of Fade approximants [m/n] with m + n --> infinity, m/n --> 1, is investigated. Two types of assumptions are considered: [n the firs t case the function f to be approximated has to have all its singulari ties in a compact set E subset of or equal to C of capacity zero (the function may be multi-valued in (C) over bar\E). In the second case th e function f has to be analytic in a domain possessing a certain symme try property (this notion is defined and discussed below). It is shown that close-to-diagonal sequences of Fade approximants [m/n] converge to f in capacity in a domain D that can be determined in various ways. In the case of the first type of assumptions the domain D is determin ed by the minimality of the capacity of the complement of D, in the se cond case the domain D is determined by a symmetry property. The rate of convergence is determined, and it is shown that this rate is best p ossible for convergence in capacity. In addition to the convergence re sults the asymptotic distribution of zeros and poles of the approximan ts is studied. (C) 1997 Academic Press.