Multipoint Padé approximants to complex Cauchy transforms with polar singularities

Abstract

We study diagonal multipoint Padé approximants to functions of the form F ( z ) = ∫ d λ ( t ) z − t + R ( z ) , where R is a rational function and λ is a complex measure with compact regular support included in R , whose argument has bounded variation on the support. Assuming that interpolation sets are such that their normalized counting measures converge sufficiently fast in the weak-star sense to some conjugate-symmetric distribution σ , we show that the counting measures of poles of the approximants converge to σ ̂ , the balayage of σ onto the support of λ , in the weak ∗ sense, that the approximants themselves converge in capacity to F outside the support of λ , and that the poles of R attract at least as many poles of the approximants as their multiplicity and not much more.