Geodesic
Asymptotics of Hadamard determinants and the convergence of rows of Padé approximants for sums of exponentials
Sbornik. Mathematics, Tome 41 (1982) no. 4, pp. 427-441 
A. I. Aptekarev. Asymptotics of Hadamard determinants and the convergence of rows of Padé approximants for sums of exponentials. Sbornik. Mathematics, Tome 41 (1982) no. 4, pp. 427-441. http://geodesic.mathdoc.fr/item/SM_1982_41_4_a1/
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     title = {Asymptotics of {Hadamard} determinants and the convergence of rows of {Pad\'e} approximants for sums of exponentials},
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] H. S. Wall, Analytic theory of continued fraction, Van Nostrand, New York, 1948 | MR | Zbl

[2] O. Perron, Die Lehre von den Kettenbruchen, Br. II, Teubner, Stuttgart, 1957 | MR | Zbl

[3] H. Pade, “Memoire sur les developpements en fractions continues de la fonctial exponential”, Ann. Sci. Ecole Norm. Super, 16:3 (1899), 394–426 | MR

[4] F. Klein, Elementarnaya matematika s tochki zreniya vysshei, t. 1, GTTI, Moskva–Leningrad, 1933

[5] A. Edrei, “Convergence of the complete Pade table of trigonometric functions”, J. Appr. Theory, 15:4 (1975), 278–293 | DOI | MR | Zbl

[6] A. A. Gonchar, “O skhodimosti obobschennykh approksimatsii Pade meromorfnykh funktsii”, Matem. sb., 98(140) (1975), 564–577 | Zbl

[7] Dzh. V. S. Kassels, Vvedenie v teoriyu diofantovykh priblizhenii, IL, Moskva, 1961

[8] A. O. Gelfond, Ischislenie konechnykh raznostei, izd-vo “Nauka”, Moskva, 1967 | MR