On the convergence of multivariate Padé approximants

While the concept of Padé approximant is essentially several centuries old, its multivariate version dates only from the early seventies. In the last century many univariate convergence results were proven, describing the approximation power for several function classes. It is not our aim to give a general review of the univariate case, but to discuss only these theorems that have a multivariate counterpart. The first section summarizes the theorems under discussion, in a univariate framework. The second and third section discuss the multivariate versions of these theorems, for different approaches to the multivariate Padé approximation problem. 1 Convergence of univariate Padé approximants. Given a function f(z), through its series expansion at a certain point in the complex plane, the Padé approximant [n/m] of degree n in the numerator and m in the denominator for f is defined by