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 Izv. Saratov Univ. Math. Mech. Inform., 2021, Volume 21, Issue 2, Pages 162–172 (Mi isu883)

Scientific Part
Mathematics

A. P. Starovoitov, E. P. Kechko

Francisk Skorina Gomel State University, 104 Sovetskaya St., Gomel 246019, Belarus

Abstract: This paper studies uniform convergence rate of Hermite – Padé approximants (simultaneous Padé 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 – Padé 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 – Padé 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.

Key words: Hermite integrals, Hermite – Padé approximants, system of exponential functions, asymptotic equality, saddle-point method.

 Funding Agency Grant Number Ministry of Education of the Republic of Belarus Belarusian Republican Foundation for Fundamental Research Ô18Ì-025 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)

DOI: https://doi.org/10.18500/1816-9791-2021-21-2-162-172

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UDC: 517.538.52+517.538.53
Revised: 14.05.2020

Citation: A. P. Starovoitov, E. P. Kechko, “About the convergence rate Hermite – Padé approximants of exponential functions”, Izv. Saratov Univ. Math. Mech. Inform., 21:2 (2021), 162–172

Citation in format AMSBIB
\Bibitem{StaKec21} \by A.~P.~Starovoitov, E.~P.~Kechko \paper About the convergence rate Hermite -- Pad\'e approximants of~exponential functions \jour Izv. Saratov Univ. Math. Mech. Inform. \yr 2021 \vol 21 \issue 2 \pages 162--172 \mathnet{http://mi.mathnet.ru/isu883} \crossref{https://doi.org/10.18500/1816-9791-2021-21-2-162-172} \elib{https://elibrary.ru/item.asp?id=45797870}