Rational Interpolation of the Exponential Function

TL;DR: In this article, it was shown that locally uniformly in the complex plane C, where the normalization Qm,n (0) = 1 has been imposed, one can obtain sharp estimates for the error |ez − Rm n (z)| when z ∈ K. These results generalize properties of the classical Pade approximation.

Abstract: Let m, n be nonnegative integers and B (m+n) be a set of m + n + 1 real interpolation points (not necessarily distinct). Let Rm,n = P m,n/Qm.n be the unique rational function with deg Pm,n ≤ m, deg Qm,n ≤ n, that interpolates ex in the points of B (m+n). If m = mv , n = nv with mv + nv → ∞, and mv / nv → λ as v → ∞, and the sets B (m+n) are uniformly bounded, we show that locally uniformly in the complex plane C, where the normalization Qm,n (0) = 1 has been imposed. Moreover, for any compact set K ⊂ C we obtain sharp estimates for the error |ez — Rm,n (z)| when z ∈ K. These results generalize properties of the classical Pade approximants. Our convergence theorems also apply to best (real) Lp rational approximants to ex on a finite real interval.