Rational Approximation to the Exponential Function with Complex Conjugate Interpolation Points

Abstract

In this paper, we study asymptotic properties of rational functions that interpolate the exponential function. The interpolation is performed with respect to a triangular scheme of complex conjugate points lying in bounded rectangular domains included in the horizontal strip |Im z |<2 π . Moreover, the height of these domains cannot exceed some upper bound which depends on the type of rational functions. We obtain different convergence results and precise estimates for the error function in compact sets of C that generalize the classical properties of Padé approximants to the exponential function. The proofs rely on, among others, Walsh's theorem on the location of the zeros of linear combinations of derivatives of a polynomial and on Rolle's theorem for real exponential polynomials in the complex domain.