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

This article is cited in 6 scientific papers (total in 6 papers)

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

M. A. Komarov

Vladimir State University

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 мол_а
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).


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

Bibliographic databases:

UDC: 517.538
Received: 24.02.2014
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
\by M.~A.~Komarov
\paper Best Approximation Rate of Constants by Simple Partial Fractions and Chebyshev Alternance
\jour Mat. Zametki
\yr 2015
\vol 97
\issue 5
\pages 718--732
\jour Math. Notes
\yr 2015
\vol 97
\issue 5
\pages 725--737

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    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  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    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  mathnet  crossref  isi
    3. 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
    4. 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
    5. 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
    6. M. A. Komarov, “Approximation to constant functions by electrostatic fields due to electrons and positrons”, Lobachevskii J. Math., 40:1, SI (2019), 79–84  crossref  isi  scopus
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