Abstract:
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.
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