CONVERGENCE OF RATIONAL INTERPOLANTS WITH PREASSIGNED POLES

We study the following problem. Given a domain Omega containing infini ty, is it possible to choose a sequence of polynomials Q(n), n = 1, 2, ..., where Q(n) has degree n, so that the following condition holds: if a function f is analytic in Omega and P-n is the polynomial part of the Laurent expansion of Q(n)f at infinity, then P-n/Q(n) converges t o f, as n tends to infinity, uniformly on bounded closed subsets of Om ega? We get a complete solution of this problem when a is regular for Dirichlet's problem. For irregular domains we obtain some results havi ng independent interest but a main problem remains open: is it possibl e to find such polynomials Q(n) for some irregular domains Omega? (C) 1997 Academic Press.