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 Mat. Zametki, 2015, Volume 97, Issue 5, Pages 718–732 (Mi mz10470)

Best Approximation Rate of Constants by Simple Partial Fractions and Chebyshev Alternance

M. A. Komarov

Abstract: We consider the problem of interpolation and best uniform approximation of constants $c\ne 0$ by simple partial fractions $\rho_n$ of order $n$ on an interval $[a,b]$. (All functions and numbers considered are real.) For the case in which $n>4|c|(b-a)$, we prove that the interpolation problem is uniquely solvable, obtain upper and lower bounds, sharp in order $n$, for the interpolation error on the set of all interpolation points, and show that the poles of the interpolating fraction lie outside the disk with diameter $[a,b]$. We obtain an analog of Chebyshev's classical theorem on the minimum deviation of a monic polynomial of degree $n$ from a constant. Namely, we show that, for $n>4|c|(b-a)$, the best approximation fraction $\rho_n^*$ for the constant $c$ on $[a,b]$ is unique and can be characterized by the Chebyshev alternance of $n+1$ points for the difference $\rho_n^*-c$. For the minimum deviations, we obtain an estimate sharp in order $n$.

Keywords: best approximation of constants, simple partial fraction, Chebyshev alternance.

 Funding Agency Grant Number Russian Foundation for Basic Research 12-01-31471 ìîë_à Ministry of Education and Science of the Russian Federation 14.B37.21.0369 This work was supported by the Ministry of Education and Science of the Russian Federation (grant no. 14.B37.21.0369) and by the Russian Foundation for Basic Research (grant no. 12-01-31471 mol_a).

DOI: https://doi.org/10.4213/mzm10470

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English version:
Mathematical Notes, 2015, 97:5, 725–737

Bibliographic databases:

UDC: 517.538
Revised: 21.10.2014

Citation: M. A. Komarov, “Best Approximation Rate of Constants by Simple Partial Fractions and Chebyshev Alternance”, Mat. Zametki, 97:5 (2015), 718–732; Math. Notes, 97:5 (2015), 725–737

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/mz10470
• https://doi.org/10.4213/mzm10470
• http://mi.mathnet.ru/eng/mz/v97/i5/p718

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. 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
2. M. A. Komarov, “Approximation by linear fractional transformations of simple partial fractions and their differences”, Russian Math. (Iz. VUZ), 62:3 (2018), 23–33
3. M. A. Komarov, “On approximation by special differences of simplest fractions”, St. Petersburg Math. J., 30:4 (2019), 655–665
4. V. I. Danchenko, M. A. Komarov, P. V. Chunaev, “Ekstremalnye i approksimativnye svoistva naiprosteishikh drobei”, Izv. vuzov. Matem., 2018, no. 12, 9–49
5. M. A. Komarov, “Estimates of the Best Approximation of Polynomials by Simple Partial Fractions”, Math. Notes, 104:6 (2018), 848–858
6. Komarov M.A., “Approximation to Constant Functions By Electrostatic Fields Due to Electrons and Positrons”, Lobachevskii J. Math., 40:1, SI (2019), 79–84
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