On sequences of rational interpolants of the exponential function with unbounded interpolation points
Résumé
We consider sequences of rational interpolants $r_n(z)$ of degree $n$ to the exponential function $e^z$ associated to a triangular scheme of complex points $\{z_{j}^{(2n)}\}_{j=0}^{2n}$, $n>0$, such that, for all $n$, $|z_{j}^{(2n)}|\leq cn^{1-\alpha}$, $j=0,...,2n$, with $0<\alpha\leq 1$ and $c>0$. We prove the local uniform convergence of $r_{n}(z)$ to $e^{z}$ in the complex plane, as $n$ tends to infinity, and show that the limit distributions of the conveniently scaled zeros and poles of $r_{n}$ are identical to the corresponding distributions of the classical Padé approximants. This extends previous results obtained in the case of bounded (or growing like $\log n$) interpolation points. To derive our results, we use the Deift-Zhou steepest descent method for Riemann-Hilbert problems. For interpolation points of order $n$, satisfying $|z_{j}^{(2n)}|\leq cn$, $c>0$, the above results are false if $c$ is large, e.g. $c\geq 2\pi$. In this connection, we display numerical experiments showing how the distributions of zeros and poles of the interpolants may be modified when considering different configurations of interpolation points with modulus of order $n$.