Convergence of Rational Interpolants∗

The convergence of (diagonal) sequences of rational interpolants to an analytic function is investigated. Problems connected with their definition are shortly discussed. Results about locally uniform convergence are reviewed. Then the convergence in capacity is studied in more detail. Here, a central place is taken by a theorem about the convergence in capacity of rational interpolants to functions with branch points. The notion of a symmetric domain plays a fundamental role. Apart from very special situations, proofs of the existence of such domains are known so far only for two types of interpolation schemes. 1 Rational Interpolation Interpolating and approximating an analytic function by polynomials or rational functions with prescribed poles is rather well understood and has been studied in great detail by J.L. Walsh (cf. his book [Wal]). In many respects interpolation by rational functions with preassigned poles leads to a theory very similar to that of polynomial interpolation. A rather different situation arises if one considers interpolation by rational functions with free poles. Free poles means here that both, the numerator and the denominator polynomial, are determined by the interpolation conditions, while in case of preassigned poles this is true only for the numerator polynomial. The theoretical background of rational interpolation with free poles is very similar to that of Padé approximants. Actually, Padé approximants are a special type of rational interpolants, they are linearized (also called generalized) rational interpolants with all its interpolation points identical. In the literature rational interpolants with free poles are also known under the name of multi-point Padé approximants. There are good reasons for this somewhat strange terminology. Problems connected with the definition of rational interpolants will be addressed in the next paragraphs. ∗Research supported by the Deutsche Forschungsgemeinschaft (AZ: Sta 299/8–1). †Dedicated to Jean Meinguet on the occasion of his 65 anniversary 1991 Mathematics Subject Classification : 41A20, 41A25, 30E10.