Padé approximants, density of rational functions in [formula omitted] and smoothness of the integration operator

Abstract

First we establish some generic universalities for Padé approximants in the closure X∞(Ω) in the space A∞(Ω) of all rational functions with poles off Ω¯. The closure Ω¯ of the domain Ω⊂C is taken with respect to the finite plane C. Next we give sufficient conditions on Ω so that X∞(Ω)=A∞(Ω). Some of these conditions imply that, even if the boundary ∂Ω of a Jordan domain Ω has infinite length, the integration operator on Ω preserves H∞(Ω) and A(Ω) as well. We also give an example of a Jordan domain Ω and a function f∈A(Ω), such that its antiderivative is not bounded on Ω. Finally we restate these results for Volterra operators on the open unit disc D and we complete them by some generic results.