On interpolation of holomorphic functions by rational functions

Abstract

Let K⊂C be compact and triangular schemes of nodes in K and poles in C\K be specified. We obtain necessary and sufficient conditions on the distribution of the poles and nodes such that every function f holomorphic on K may be approximated uniformly on K by a sequence of rational functions with the given poles which interpolate ∫ in the given nodes. We then show that certain point systems known to be well suited in other interpolation problems may be used successfully in our case. Suppose G⊂C is a multiply connected domain. We then show that those point systems may be employed similarly to define sets of poles in C\G and sets of nodes in G such that every function ∫ holomorphic in G is the limit of the associated rational interpolants locally uniformly in G.