Correction terms for Padé approximants

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Abstract
Let f(x) be a function of the form f(x) = ∫∞0dφ (u)/(1+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.

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