Convergence of Rational Interpolants with Preassigned Poles Original Research Article

We study the following problem. Given a domainΩcontaining infinity, is it possible to choose a sequence of polynomialsQn,n=1, 2, ..., whereQnhas degreen, so that the following condition holds: if a functionfis analytic inΩandPnis the polynomial part of the Laurent expansion ofQnfat infinity, thenPn/Qnconverges tof, asntends to infinity, uniformly on bounded closed subsets ofΩ? We get a complete solution of this problem whenΩis regular for Dirichletʹs problem. For irregular domains we obtain some results having independent interest but a main problem remains open: is it possible to find such polynomialsQnfor some irregular domainsΩ?