English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We consider sequences of rational interpolants $r_n(z)$ of degree $n$ to theexponential 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 thecomplex plane, as $n$ tends to infinity, and show that the limit distributionsof the conveniently scaled zeros and poles of $r_{n}$ are identical to thecorresponding distributions of the classical Pad\'e approximants. This extendsprevious results obtained in the case of bounded (or growing like $\log n$)interpolation points. To derive our results, we use the Deift-Zhou steepestdescent 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 numericalexperiments showing how the distributions of zeros and poles of theinterpolants may be modified when considering different configurations ofinterpolation points with modulus of order $n$.

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