Correction terms for Padé approximants

Abstract

Let f(x) be a function of the form f(x)=∫0 ∞dφ(u)/(l + ux), x≥0, where φ(u) is of bounded variation and piecewise differentiable on 0≤u < ∞; and suppose that the [N-1/N] and [N/N] Padé approximants (PA's) to f(x) can be constructed. Then correction terms bN(x) and cN(x) such that [N-1/N]-bN(x) ≤ f(x) ≤ [N/N]+cN(x), x≥0, are considered. Suitable corrections are shown to have the structure x2N {positive constant}/{denominator of corresponding PA}2. The nature of the constants is examined: They vanish when f(x) is a series of Stieltjes, and given appropriate information about f(x) they can be calculated. Applications are considered. Show more