The convergence of Pade approximants to functions with branch points

[摘要] 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 capacity of Fade approximants [m/n] with m + n --> infinity, m/n --> 1, is investigated. Two types of assumptions are considered: [n the first case the function f to be approximated has to have all its singularities 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 the function f has to be analytic in a domain possessing a certain symmetry 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 determined by the minimality of the capacity of the complement of D, in the second 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 possible for convergence in capacity. In addition to the convergence results the asymptotic distribution of zeros and poles of the approximants is studied. (C) 1997 Academic Press.

[发布日期] 1997-11-01