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

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

M. A. Komarov

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

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

Bibliographic databases:

UDC: 517.538
MSC: 41A20, 41A50
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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• http://mi.mathnet.ru/eng/izv8266
• https://doi.org/10.4213/im8266
• http://mi.mathnet.ru/eng/izv/v79/i3/p3

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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
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
3. P. A. Borodin, “Approximation by sums of shifts of a single function on the circle”, Izv. Math., 81:6 (2017), 1080–1094
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
5. M. A. Komarov, “On approximation by special differences of simplest fractions”, St. Petersburg Math. J., 30:4 (2019), 655–665
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
7. M. A. Komarov, “Estimates of the Best Approximation of Polynomials by Simple Partial Fractions”, Math. Notes, 104:6 (2018), 848–858
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
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