Padé approximants and orthogonal polynomials
Padé approximants are rational functions which may be seen as a generalization of Taylor polynomials. In this lecture, basic facts and main classical theorems on convergence theory of Padé approximants to analytic functions are briefly reviewed. We show the important role played by Logarithmic Potential Theory in the development of this topic. We also consider the relationship between Padé approximants and asymptotic properties of polynomials orthogonal with respect to fixed or varying measures. Finally, some original results about Hermite-Padé approximants to a Nikishin system of functions are presented. Such approximants arise in the study of the transcendence of the number $e$ and are related, as Nikishin systems themselves, to number theory.
Bernardo de la Calle Ysern, Universidad Politécnica de Madrid, Spain
September 13, 2000