Strong asymptotics of the best rational approximation to the exponential function on a~bounded interval
Sbornik. Mathematics, Tome 215 (2024) no. 12, pp. 1666-1719
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We apply recent findings of complex approximation theory to best rational approximation of degree $n$ to the function $\exp(-(n+\nu)x)$ on a finite interval $[0,c]$. We show that the error norm behaves like the $n$th power of the main approximation rate times the $\nu$th power of a secondary approximation rate. The computation of the first rate is a consequence of works of Gonchar, Rakhmanov and Stahl done in the 1980s; the complete asymptotic description was achieved by Aptekarev in the first years of the 21st century. The solution is given in terms of elliptic integrals of the third kind.
Bibliography: 92 titles.
Keywords:
rational approximation, exponential function, complex potential.
@article{SM_2024_215_12_a3,
author = {A. P. Magnus and J. Meinguet},
title = {Strong asymptotics of the best rational approximation to the exponential function on a~bounded interval},
journal = {Sbornik. Mathematics},
pages = {1666--1719},
publisher = {mathdoc},
volume = {215},
number = {12},
year = {2024},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SM_2024_215_12_a3/}
}
TY - JOUR AU - A. P. Magnus AU - J. Meinguet TI - Strong asymptotics of the best rational approximation to the exponential function on a~bounded interval JO - Sbornik. Mathematics PY - 2024 SP - 1666 EP - 1719 VL - 215 IS - 12 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/SM_2024_215_12_a3/ LA - en ID - SM_2024_215_12_a3 ER -
%0 Journal Article %A A. P. Magnus %A J. Meinguet %T Strong asymptotics of the best rational approximation to the exponential function on a~bounded interval %J Sbornik. Mathematics %D 2024 %P 1666-1719 %V 215 %N 12 %I mathdoc %U http://geodesic.mathdoc.fr/item/SM_2024_215_12_a3/ %G en %F SM_2024_215_12_a3
A. P. Magnus; J. Meinguet. Strong asymptotics of the best rational approximation to the exponential function on a~bounded interval. Sbornik. Mathematics, Tome 215 (2024) no. 12, pp. 1666-1719. http://geodesic.mathdoc.fr/item/SM_2024_215_12_a3/