Abstract :
Let ψ be a finite positive measure on R, and let Fψ(z) = ∫∞−∞ (dψ,(t)/(z − t)) be its Stieltjes transform. A special multipoint Padé approximation problem for Fψ(z) is studied, where the interpolation points are a finite number of points a1, ..., ap, in R repeated cyclically and the support of ψ is contained in an interval bounded by adjacent interpolation points. For the case p = 3 monotone convergence of each of the subsequences {P3q + m(z)/Q3q + m(z)}, m = 0, 1, 2, of the multipoint Padé approximants {Pn(z)/Qn(z)} is established, and sufficient conditions (involving general moments c(i)j = ∫∞−∞ (dψ(t)/(t−ai)j)) for divergence of the series ∑∞q = 1 | Q3q + m(z)|2 are given.
