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 Izv. RAN. Ser. Mat., 2017, Volume 81, Issue 3, Pages 109–133 (Mi izv8548)

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

M. A. Komarov

Abstract: In the problem of approximating real functions $f$ by simple partial fractions of order $ė n$ on closed intervals $K=[c-\varrho,c+\varrho]\subset\mathbb{R}$, we obtain a criterion for the best uniform approximation which is similar to Chebyshev's alternance theorem and considerably generalizes previous results: under the same condition $z_j^*\notin B(c,\varrho)= ż\colon|z-c|\le\varrho\}$ on the poles $z_j^*$ of the fraction $\rho^*(n,f,K;x)$ of best approximation, we omit the restriction $k=n$ on the order $k$ of this fraction. In the case of approximation of odd functions on $[-\varrho,\varrho]$, we obtain a similar criterion under much weaker restrictions on the position of the poles $z_j^*$: the disc $B(0,\varrho)$ is replaced by the domain bounded by a lemniscate contained in this disc. We give some applications of this result. The main theorems are extended to the case of weighted approximation. We give a lower bound for the distance from $\mathbb{R}^+$ to the set of poles of all simple partial fractions of order $ė n$ which are normalized with weight $2\sqrt x$ on $\mathbb{R}^+$ (a weighted analogue of Gorin's problem on the semi-axis).

Keywords: simple partial fraction, approximation, alternance, uniqueness, disc, odd function, lemniscate.

DOI: https://doi.org/10.4213/im8548

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English version:
Izvestiya: Mathematics, 2017, 81:3, 568–591

Bibliographic databases:

UDC: 517.538
MSC: 41A20, 41A50
Revised: 05.05.2016

Citation: M. A. Komarov, “A criterion for the best uniform approximation by simple partial fractions in terms of alternance. II”, Izv. RAN. Ser. Mat., 81:3 (2017), 109–133; Izv. Math., 81:3 (2017), 568–591

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/izv8548
• https://doi.org/10.4213/im8548
• http://mi.mathnet.ru/eng/izv/v81/i3/p109

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This publication is cited in the following articles:
1. M. A. Komarov, “On approximation by special differences of simplest fractions”, St. Petersburg Math. J., 30:4 (2019), 655–665
2. P. A. Borodin, “Approximation by Sums of the Form $\sum_k\lambda_kh(\lambda_kz)$ in the Disk”, Math. Notes, 104:1 (2018), 3–9
3. V. I. Danchenko, M. A. Komarov, P. V. Chunaev, “Ekstremalnye i approksimativnye svoistva naiprosteishikh drobei”, Izv. vuzov. Matem., 2018, no. 12, 9–49
4. M. A. Komarov, “Estimates of the Best Approximation of Polynomials by Simple Partial Fractions”, Math. Notes, 104:6 (2018), 848–858
5. Komarov M.A., “Approximation to Constant Functions By Electrostatic Fields Due to Electrons and Positrons”, Lobachevskii J. Math., 40:1, SI (2019), 79–84
6. P. A. Borodin, “Zhadnye priblizheniya proizvolnym mnozhestvom”, Izv. RAN. Ser. matem., 84:2 (2020), 43–59
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