Journal of computational and applied mathematics vol:65 issue:1-3 pages:57-72
Let f(z) be a Stieltjes function with asymptotic expansions L0 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.