About the convergence rate Hermite – Pade approximants of exponential functions
This paper studies uniform convergence rate of Hermite\,--\,Pad\'e approximants (simultaneous Pad\'e 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\...
| Published in: | Известия Саратовского университета. Новая серия: Математика. Механика. Информатика |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Saratov State University
2021-05-01
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| Subjects: | |
| Online Access: | https://mmi.sgu.ru/sites/mmi.sgu.ru/files/text-pdf/2021/05/162-172starovoitov-kechko.pdf |
| Summary: | This paper studies uniform convergence rate of Hermite\,--\,Pad\'e approximants (simultaneous Pad\'e 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\,--\,Pad\'e 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\,--\,Pad\'e 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. |
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| ISSN: | 1816-9791 2541-9005 |