The real-pole sandwich for rational approximations and oscillation equations

Rational approximations to the exponential function with real poles only are studied with respect to stability at infinity and maximal order. Along each half line in the parameter space it is shown that these two properties occur in an alternating way (or do not occur at all). As an application of the general results the special approximations withq-fold poles at ± ^–1 only, real, are studied. A short proof of the superconvergence for collocation methods is also given.Dedicated to W. Gröbner on the occasion of his 80th birthday.