Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)


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Starovoitov A. P., Kechko E. P. About the convergence rate Hermite – Pade approximants of exponential functions. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2021, vol. 21, iss. 2, pp. 162-172. DOI: 10.18500/1816-9791-2021-21-2-162-172, EDN: KQHABT

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31.05.2021
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KQHABT

About the convergence rate Hermite – Pade approximants of exponential functions

Autors: 
Starovoitov Alexander Pavlovich, Gomel State University named after Francisk Skorina, Belarus
Kechko Elena P., Gomel State University named after Francisk Skorina, Belarus
Abstract: 
This paper studies uniform convergence rate of Hermite – Pade approximants (simultaneous Pade approximants) $\{\pi^j_{n,\overrightarrow{m}}(z)\}_{j=1}^k$ for a system of exponential functions $\{e^{\lambda_jz}\}_{j=1}^k$, where $\{\lambda_j\}_{j=1}^k$ are different nonzero complex numbers. In the general case a research of the asymptotic properties of Hermite – Pade approximants is a rather complicated problem. This is due to the fact that in their study mainly asymptotic methods are used, in particular, the saddle-point method. An important phase in the application of this method is to find a special saddle contour (the Cauchy integral theorem allows to choose an integration contour rather arbitrarily), according to which integration should be carried out. Moreover, as a rule, one has to repy only on intuition. In this paper, we propose a new method to studying the asymptotic properties of Hermite – Pade approximants, that is based on the Taylor theorem and heuristic considerations underlying the Laplace and saddle-point methods, as well as on the multidimensional analogue of the Van Rossum identity that we obtained. The proved theorems complement and generalize the known results by other authors.
Acknowledgments: 
This work was supported by the Ministry of Education of the Republic of Belarus within the state program of scientific research for 2016–2020 and the Belarusian Republican Foundation for Fundamental Research (project No. F18M-025).
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Received: 
03.01.2020
Accepted: 
14.05.2020
Published: 
31.05.2021