Asymptotics of Hadamard determinants and the convergence of rows of Pad\'e approximants for sums of exponentials

A. I. Aptekarev

Sbornik. Mathematics, Tome 41 (1982) no. 4, pp. 427-441

Résumé

The asymptotics of Hadamard determinants $\Delta_{n,m}$ (of dimension $m\times m$) for arbitrary fixed $m$ and $n\to\infty$ for the function $f(z)=\sum^k_{t=1}e^{\lambda_tz}$ are studied, where $\{\lambda_t\}^k_{t=1}$ are distinct complex numbers with unit modulus. A theorem on the convergence of the $(s\cdot p)$th row of the Padé table for the function $f(z)=\sum^k_{t=1}e^{\lambda_tz}$ ($\{\lambda_t\}^k_{t=1}$ are arbitrary distinct complex numbers) in the topology of $H(\mathbf C)$ holds for arbitrary natural numbers $p$ and $s$ equal to the number of $\lambda_t$ with modulus that is maximal among the $\{\lambda_t\}^k_{t=1}$. Bibliography: 8 titles.