Quadrature on the half-line and two-point Padé approximants to Stieltjes functions. Part I. Algebraic aspects

Abstract

Let f( z) be a Stieltjes function with asymptotic expansions L 0 and L ∞ at z = 0 and z = ∞, respectively. Let [ k n ] denote the two-point Padé approximant of type ( m, n) which matches k of the coefficients of the series L 0 and which takes its remaining interpolation conditions from L ∞. We discuss in this paper the algebraic aspects of this problem. We shall emphasize the relation between quadrature formulas and two-point Padé approximants and derive expressions for the error. In a subsequent paper we shall consider the convergence aspects of these approximants. For example, the positivity of the error, which is obtained here, will result in the monotonic convergence of certain sequences of two-point Padé approximants, a property which is well known in the case of classical Padé approximants.