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

This article is cited in 7 scientific papers (total in 7 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)
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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
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    Citing articles on Google Scholar: Russian citations, English citations
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    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, “O priblizhenii spetsialnymi raznostyami naiprosteishikh drobei”, Algebra i analiz, 30:4 (2018), 47–60  mathnet
    6. V. I. Danchenko, M. A. Komarov, P. V. Chunaev, “Ekstremalnye i approksimativnye svoistva naiprosteishikh drobei”, Izv. vuzov. Matem., 2018, no. 12, 9–49  mathnet
    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  isi  elib
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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