On the existence of trigonometric Hermite – Jacobi approximations and non-linear Hermite – Chebyshev approximations

  • Alexander P. Starovoitov Francisk Skorina Gomel State University, 104 Savieckaja Street, Gomiel 246028, Belarus
  • Elena P. Kechko Francisk Skorina Gomel State University, 104 Savieckaja Street, Gomiel 246028, Belarus
  • Tatyana M. Osnach Francisk Skorina Gomel State University, 104 Savieckaja Street, Gomiel 246028, Belarus
DOI: https://doi.org/10.33581/2520-6508-2023-2-6-17

Abstract

In this paper, analogues of algebraic Hermite – Padé approximations are defined, being trigonometric Hermite – Padé approximations and Hermite – Jacobi approximations. Examples of functions are represented for which trigonometric Hermite – Jacobi approximations exist but are not the same as trigonometric Hermite – Padé approximations. Similar examples are made for linear and non-linear Hermite – Chebyshev approximations, which are multiple analogues of linear and non-linear Padé – Chebyshev approximations. Each type of examples follows from the well-known representations for the numerator and denominator of fractions, introduced by C. Hermite when proving the transcendence of number e.

Author Biographies

Alexander P. Starovoitov, Francisk Skorina Gomel State University, 104 Savieckaja Street, Gomiel 246028, Belarus

doctor of science (physics and mathematics), full professor; professor at the department of mathematical analysis and differential equations, faculty of mathematics and technologies of programming

Elena P. Kechko, Francisk Skorina Gomel State University, 104 Savieckaja Street, Gomiel 246028, Belarus

PhD (physics and mathematics); associate professor at the department of computational mathematics and programming, faculty of mathematics and technologies of programming

Tatyana M. Osnach, Francisk Skorina Gomel State University, 104 Savieckaja Street, Gomiel 246028, Belarus

postgraduate student at the department of mathematical analysis and differential equations, faculty of mathematics and technologies of programming

References

  1. Nikishin EM, Sorokin VN. Ratsional’nye approksimatsii i ortogonal’nost’ [Rational approximations and orthogonality]. Moscow: Nauka; 1988. 256 p. Russian.
  2. Baker GA Jr, Graves-Morris P. Padé approximants. London: Addison-Wesley Publishing Company; 1981. 2 volumes (Rota G-C, editor. Encyclopedia of mathematics and its applications; volume 13–14). Russian edition: Baker G Jr, Graves-Morris P. Approksimatsii Pade. 1. Osnovy teorii. 2. Obobshcheniya i prilozheniya. Rakhmanov EA, Suetin SP, translators; Gonchar AA, editor. Moscow: Mir; 1986. 502 p.
  3. Frobenius G. Ueber Relationen zwischen den Näherungsbrüchen von Potenzreihen. Journal für die reine und angewandte Mathematik. 1881;90:1–17. DOI: 10.1515/crll.1881.90.1.
  4. Jacobi CGJ. Über die Darstellung einer Reihe gegebner Werthe durch eine gebrochne rationale Function. Journal für die reine und angewandte Mathematik. 1846;30:127–156. DOI: 10.1515/crll.1846.30.127.
  5. Hermite M. Sur la fonction exponentielle. Comptes Rendus de l’Académie des Sciences. 1873;77:18–24, 74–79, 226–233, 285–293.
  6. Starovoitov AP, Ryabchenko NV. [On determinantal representations of Hermite – Padé polynomials]. Trudy Moskovskogo matematicheskogo obshchestva. 2022;83(1):17–36. Russian.
  7. Labych YuA, Starovoitov AP. [Trigonometric Padé approximations for functions with regularly decreasing Fourier coefficients]. Matematicheskii sbornik. 2009;200(7):107–130. Russian. DOI: 10.4213/sm4523.
  8. Suetin SP. [The Padé approximations and efficient analytic continuation of a power series]. Uspekhi matematicheskikh nauk. 2002;57(1):45–142. Russian. DOI: 10.4213/rm475.
  9. Suetin SP. [On the existence of non-linear Padé – Chebyshev approximations for analytic functions]. Matematicheskie zametki. 2009;86(2):290–303. Russian. DOI: 10.4213/mzm5262.
  10. Medvedeva VYu, Rouba YA. Rational interpolation of a function |x|α with Chebyshev – Markov nodes of the first kind. Journal of the Belarusian State University. Mathematics and Informatics. 2023;1:6–19. Russian. DOI: 10.33581/2520-6508-2023-1-6-19.
Published
2023-07-25
Keywords: trigonometric series, Fourier sums, trigonometric Padé approximations, Hermite – Padé polynomials, Padé – Chebyshev approximations
Supporting Agencies This work was supported by the state programme of scientific research «Convergence-2025».
How to Cite
Starovoitov, A. P., Kechko, E. P., & Osnach, T. M. (2023). On the existence of trigonometric Hermite – Jacobi approximations and non-linear Hermite – Chebyshev approximations. Journal of the Belarusian State University. Mathematics and Informatics, 2, 6-17. https://doi.org/10.33581/2520-6508-2023-2-6-17
Issue
No 2 (2023): Journal of the Belarusian State University. Mathematics and Informatics
Section
Real, Complex and Functional Analysis

Copyright (c) 2023 Journal of the Belarusian State University. Mathematics and Informatics

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

The authors who are published in this journal agree to the following:

  1. The authors retain copyright on the work and provide the journal with the right of first publication of the work on condition of license Creative Commons Attribution-NonCommercial. 4.0 International (CC BY-NC 4.0).
  2. The authors retain the right to enter into certain contractual agreements relating to the non-exclusive distribution of the published version of the work (e.g. post it on the institutional repository, publication in the book), with the reference to its original publication in this journal.
  3. The authors have the right to post their work on the Internet (e.g. on the institutional store or personal website) prior to and during the review process, conducted by the journal, as this may lead to a productive discussion and a large number of references to this work. (See The Effect of Open Access.)