Multipoint Padé approximants to complex Cauchy transforms with polar singularities

We study diagonal multipoint Padé approximants to functions of the form F(z)=∫dλ(t)z−t+R(z), where RR is a rational function and λλ is a complex measure with compact regular support included in RR, 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 σσ, we show that the counting measures of poles of the approximants converge to σ̂, the balayage of σσ onto the support of λλ, in the weak∗∗ sense, that the approximants themselves converge in capacity to FF outside the support of λλ, and that the poles of RR attract at least as many poles of the approximants as their multiplicity and not much more.