Rational approximation to the exponential function with complex conjugate interpolation points

In this paper. we study asymptotic properties of rational functions that in terpolate the exponential function. The interpolation is performed with res pect to a triangular scheme of complex conjugate points lying in bounded re ctangular domains included in the horizontal strip \ Imz \ < 2 pi. 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 gene ralize the classical properties of Pade approximants to the exponential fun ction. 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 Rol le's theorem for real exponential polynomials in the complex domain. (C) 20 01 Academic Press.