On sequences of rational interpolants of the exponential function with unbounded interpolation points

TL;DR

This paper proves the convergence and zero-pole distribution of rational interpolants of the exponential function with unbounded interpolation points, extending previous bounded-point results using Riemann-Hilbert analysis.

Contribution

It establishes convergence and zero-pole distribution results for rational interpolants with points growing like a power of n, extending classical Padé approximant analysis.

Findings

Rational interpolants converge locally uniformly to e^z.

Zero and pole distributions match those of Padé approximants.

Results hold for points growing like n^{1- ext{alpha}} with 0<alpha≤1.

Abstract

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 Pad\'e 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 $n$.