Padé approximant bounds for the difference of two series of Stieltjes

Abstract

Over the complex plane cut along the negative real axis let f (z) be an analytic function of the form f (z) =∫∞0dφ (u)/(1+uz), where φ (u) is of bounded variation on 0⩽u<∞. Suppose that the first (N+1) moments defined by fn=∫∞0undφ (u), n=0,1,...,N, exist and are known. Suppose further that dφ (u) undergoes at most p sign changes, where 0⩽p⩽N, as u goes from zero to infinity, to locations of an associated set of points u=ul, l=1,2,...,p, being known. Then, by means of a simple modification to the customary Padé approximant method, best possible upper and lower bounds can be imposed on f (x) for all 0<x<∞. Similarly, for given z in the cut complex plane, a best possible inclusion region can be imposed upon f (z). Finally it is shown that a best possible inclusion region can be imposed upon f (z) when the constraint that the set of points u=ul, l=1,2,...,p, is known is either relaxed or disposed of altogether.