Best Error Bounds for Padé Approximants to Convergent Series of Stieltjes

Abstract

We consider series of Stieltjes with a nonzero radius of convergence R. We establish by way of Padé approximants the allowed range of values for such functions at any point in the cut (−∞ < z ≤ − R) complex plane when a finite number of Taylor-series coefficients are known. The previous results for z real and positive are sharpened. We investigate the fitting problem and again give error bounds throughout the cut complex plane and we give necessary and sufficient conditions that the set of values fitted be values of a series of Stieltjes with radius of convergence at least R.