Rational Interpolation of the Exponential Function

Let m, n be nonnegative integers and a<m+n) be a set of m + n + 1 real in~lation points (not necessarily distinct). Let Rm.n = P m.n / Qm.n be the unique rational function with degPm.n ::; m, deg Qm,n :$n, that in~lates e'" in the points of a<nr+n). Ifm = mv, n = nv with mv + nv --+ 00, and mv /nv --+ A as v --+ 00, and the sets a<nr+n) are uniformly bounded, we show that Pm.n(z) --+ ~/(l+)'), Qm.n(z) --+ e-z/(l+).) locally uniformly in the complex plane C, where the normalization Qm,n(O) = 1 has been imposed. Moreover, for any compact set K C C we obtain sharp estimates for the error leZ -Rm.n(z)1 whenz E K. These results generalize properties of the classical Fade apProximants. Our convergence theorems also apply to best (real) Lp ratiQnal approximants to e'" on a finite reaJl interval.