On the convergence of two-point partial Padé approximants for meromorphic functions of Stieltjes type

Let μ be a (possibly complex) measure on R+=[0,∞) such that ∫xnd|μ|(x)<+∞,n∈Z. Let r denote a rational function whose poles lie in C\R+ and r(∞)=0. We consider two-point rational interpolants to the function f(z)=∫dμ(x)z−x+r(z), where some poles are prescribed in advance and the others are left free. We show that if the prescribed poles are chosen conveniently, then sequences of two-point rational approximants converge geometrically to f on compact subsets of C\R+ away from the poles of r. Estimates of the rate of convergence along with some numerical experiments are also given.