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Mat. Sb., 2006, Volume 197, Number 4, Pages 33–52 (Mi msb1184)  

This article is cited in 16 scientific papers (total in 16 papers)

Estimates of derivatives of simplest fractions and other questions

V. I. Danchenko

Vladimir State University

Abstract: The approximation properties of simplest fractions (s.f.'s), that is, of the logarithmic derivatives of complex polynomials, have recently become a subject of intensive research. These properties of s.f.'s prove to have many similarities with those of polynomials. For instance, one has for them analogues of Mergelyan's and Jackson's classical results on uniform polynomial approximation. In connection with approximation by s.f.'s estimates of the Markov–Bernstein kind for derivatives of s.f.'s on various subsets of the complex plane arouse interest. Such estimates are obtained in this paper on circles, straight lines and their intervals, and some applications of these estimates are indicated. Several other questions relating to approximation properties of s.f.'s are also considered.
Bibliography: 28 titles.

DOI: https://doi.org/10.4213/sm1184

Full text: PDF file (581 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2006, 197:4, 505–524

Bibliographic databases:

UDC: 517.53
MSC: 41A20, 41A17
Received: 23.08.2005

Citation: V. I. Danchenko, “Estimates of derivatives of simplest fractions and other questions”, Mat. Sb., 197:4 (2006), 33–52; Sb. Math., 197:4 (2006), 505–524

Citation in format AMSBIB
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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. P. A. Borodin, “Estimates of the Distances to Direct Lines and Rays from the Poles of Simplest Fractions Bounded in the Norm of $L_p$ on These Sets”, Math. Notes, 82:6 (2007), 725–732  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. A. V. Fryantsev, “O chislennoi approksimatsii differentsialnykh polinomov”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 7:2 (2007), 39–43  mathnet  elib
    3. V. I. Danchenko, “Approximation Properties of Sums of the Form $\sum_k\lambda_kh(\lambda_k z)$”, Math. Notes, 83:5 (2008), 587–593  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    4. E. N. Kondakova, “Interpolyatsiya naiprosteishimi drobyami”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 9:2 (2009), 30–37  mathnet  elib
    5. P. A. Borodin, “Approximation by simple partial fractions on the semi-axis”, Sb. Math., 200:8 (2009), 1127–1148  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    6. V. I. Danchenko, E. N. Kondakova, “Chebyshev's alternance in the approximation of constants by simple partial fractions”, Proc. Steklov Inst. Math., 270 (2010), 80–90  mathnet  crossref  mathscinet  zmath  isi  elib
    7. V. I. Danchenko, “Convergence of simple partial fractions in $L_p(\mathbb R)$”, Sb. Math., 201:7 (2010), 985–997  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    8. Danchenko V.I., Chunaev P.V., “Approximation by simple partial fractions and their generalizations”, J. Math. Sci., 176:6 (2011), 844–859  crossref  mathscinet  zmath  elib
    9. V. I. Danchenko, A. E. Dodonov, “Estimates for exponential sums. Applications”, J Math Sci, 2012  crossref  mathscinet
    10. F. D. Kayumov, “Integralnye otsenki dlya proizvodnykh odnolistnykh funktsii”, Uchen. zap. Kazan. un-ta. Ser. Fiz.-matem. nauki, 155, no. 2, Izd-vo Kazanskogo un-ta, Kazan, 2013, 83–90  mathnet
    11. Petr Chunaev, “Least deviation of logarithmic derivatives of algebraic polynomials from zero”, Journal of Approximation Theory, 2014  crossref  mathscinet
    12. V. I. Danchenko, A. E. Dodonov, “Estimates for $L_p$-norms of simple partial fractions”, Russian Math. (Iz. VUZ), 58:6 (2014), 6–15  mathnet  crossref
    13. Chunaev P., Danchenko V., “Quadrature Formulas With Variable Nodes and Jackson-Nikolskii Inequalities For Rational Functions”, J. Approx. Theory, 228 (2018), 1–20  crossref  mathscinet  zmath  isi
    14. M. A. Komarov, “On approximation of the rational functions, whose integral is single-valued on $\mathbb{C}$, by differences of simplest fractions”, Probl. anal. Issues Anal., 7(25), spetsvypusk (2018), 63–71  mathnet  crossref  elib
    15. V. I. Danchenko, M. A. Komarov, P. V. Chunaev, “Ekstremalnye i approksimativnye svoistva naiprosteishikh drobei”, Izv. vuzov. Matem., 2018, no. 12, 9–49  mathnet
    16. 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
  • Математический сборник Sbornik: Mathematics (from 1967)
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