Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya
General information
Latest issue
Forthcoming papers
Impact factor
Guidelines for authors
License agreement
Submit a manuscript

Search papers
Search references

Latest issue
Current issues
Archive issues
What is RSS

Izv. RAN. Ser. Mat.:

Personal entry:
Save password
Forgotten password?

Izv. RAN. Ser. Mat., 2015, Volume 79, Issue 3, Pages 3–22 (Mi izv8266)  

This article is cited in 8 scientific papers (total in 8 papers)

A criterion for the best uniform approximation by simple partial fractions in terms of alternance

M. A. Komarov

Vladimir State University

Abstract: We consider the problem of best uniform approximation of real continuous functions $f$ by simple partial fractions of degree at most $n$ on a closed interval $S$ of the real axis. We get analogues of the classical polynomial theorems of Chebyshev and de la Vallée-Poussin. We prove that a real-valued simple partial fraction $R_n$ of degree $n$ whose poles lie outside the disc with diameter $S$, is a simple partial fraction of the best approximation to $f$ if and only if the difference $f-R_n$ admits a Chebyshev alternance of $n+1$ points on $S$. Then $R_n$ is the unique fraction of best approximation. We show that the restriction on the poles is unimprovable. Particular cases of the theorems obtained have been stated by various authors only as conjectures.

Keywords: simple partial fraction, approximation, alternance, uniqueness, the Haar condition.

DOI: https://doi.org/10.4213/im8266

Full text: PDF file (642 kB)
References: PDF file   HTML file

English version:
Izvestiya: Mathematics, 2015, 79:3, 431–448

Bibliographic databases:

UDC: 517.538
MSC: 41A20, 41A50
Received: 11.06.2014
Revised: 30.01.2015

Citation: M. A. Komarov, “A criterion for the best uniform approximation by simple partial fractions in terms of alternance”, Izv. RAN. Ser. Mat., 79:3 (2015), 3–22; Izv. Math., 79:3 (2015), 431–448

Citation in format AMSBIB
\by M.~A.~Komarov
\paper A criterion for the best uniform approximation by simple partial fractions in terms of alternance
\jour Izv. RAN. Ser. Mat.
\yr 2015
\vol 79
\issue 3
\pages 3--22
\jour Izv. Math.
\yr 2015
\vol 79
\issue 3
\pages 431--448

Linking options:
  • http://mi.mathnet.ru/eng/izv8266
  • https://doi.org/10.4213/im8266
  • http://mi.mathnet.ru/eng/izv/v79/i3/p3

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru

    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles
    Cycle of papers

    This publication is cited in the following articles:
    1. M. A. Komarov, “Best Approximation Rate of Constants by Simple Partial Fractions and Chebyshev Alternance”, Math. Notes, 97:5 (2015), 725–737  mathnet  crossref  crossref  mathscinet  isi  elib
    2. M. A. Komarov, “A criterion for the best uniform approximation by simple partial fractions in terms of alternance. II”, Izv. Math., 81:3 (2017), 568–591  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    3. P. A. Borodin, “Approximation by sums of shifts of a single function on the circle”, Izv. Math., 81:6 (2017), 1080–1094  mathnet  crossref  crossref  adsnasa  isi  elib
    4. M. A. Komarov, “Approximation by linear fractional transformations of simple partial fractions and their differences”, Russian Math. (Iz. VUZ), 62:3 (2018), 23–33  mathnet  crossref  isi
    5. M. A. Komarov, “On approximation by special differences of simplest fractions”, St. Petersburg Math. J., 30:4 (2019), 655–665  mathnet  crossref  mathscinet  isi  elib
    6. V. I. Danchenko, M. A. Komarov, P. V. Chunaev, “Extremal and approximative properties of simple partial fractions”, Russian Math. (Iz. VUZ), 62:12 (2018), 6–41  mathnet  crossref  isi
    7. M. A. Komarov, “Estimates of the Best Approximation of Polynomials by Simple Partial Fractions”, Math. Notes, 104:6 (2018), 848–858  mathnet  crossref  crossref  mathscinet  isi  elib
    8. M. A. Komarov, “On the rate of approximation in the unit disc of $H^1$-functions by logarithmic derivatives of polynomials with zeros on the boundary”, Izv. Math., 84:3 (2020), 437–448  mathnet  crossref  crossref  mathscinet  isi  elib
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
    Number of views:
    This page:613
    Full text:259
    First page:35

    Contact us:
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2021