Multipoint Padé approximants to complex Cauchy transforms with polar singularities

We study diagonal multipoint Padé approximants to functions of the form F (z) = Z dλ(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 b 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. AMS Classification (MSC2000): primary 41A20, 41A30, 42C05; secondary 30D50, 30D55, 30E10, 31A15.