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Mat. Sb., 2009, Volume 200, Number 8, Pages 25–44 (Mi msb7466)  

This article is cited in 14 scientific papers (total in 14 papers)

Approximation by simple partial fractions on the semi-axis

P. A. Borodin

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: This paper investigates the simple partial fractions (that is, the logarithmic derivatives of polynomials) all of whose poles lie within the angular domain $\Lambda_\gamma=ż:\arg z\in(\gamma,2\pi-\gamma)\}$, for any $\gamma\in[0,\pi/2]$. It is shown that they are contained in a proper half-space of the space $L_p({\mathbb R}_+)$ for any $p\in(1,p_0)$ (in particular, they are not dense in this space) and conversely, they are dense in $L_p({\mathbb R}_+)$ for any $p\ge p_0$, where $p_0=(2\pi-2\gamma)/(\pi-2\gamma)$. The distances from the poles of a simple partial fraction $r$ to the semi-axis ${\mathbb R}_+$ are estimated in terms of the degree of the fraction $r$ and its norm in $L_2({\mathbb R}_+)$. The approximation properties of sets of simple partial fractions of degree at most $n$ are investigated, as well as properties of the least deviations $\rho_n(f)$ from these sets for the functions $f\in L_2({\mathbb R}_+)$.
Bibliography: 14 titles.

Keywords: approximation, simple partial fraction, integral metrics.

DOI: https://doi.org/10.4213/sm7466

Full text: PDF file (572 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2009, 200:8, 1127–1148

Bibliographic databases:

UDC: 517.538.5
MSC: 30E10, 41A20
Received: 24.10.2008 and 01.04.2009

Citation: P. A. Borodin, “Approximation by simple partial fractions on the semi-axis”, Mat. Sb., 200:8 (2009), 25–44; Sb. Math., 200:8 (2009), 1127–1148

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. 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
    2. P. V. Chunaev, “On a nontraditional method of approximation”, Proc. Steklov Inst. Math., 270 (2010), 278–284  mathnet  crossref  mathscinet  zmath  isi  elib
    3. I. R. Kayumov, “Convergence of series of simple partial fractions in $L_p(\mathbb R)$”, Sb. Math., 202:10 (2011), 1493–1504  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    4. I. R. Kayumov, “Integral bounds for simple partial fractions”, Russian Math. (Iz. VUZ), 56:4 (2012), 27–37  mathnet  crossref  mathscinet
    5. P. A. Borodin, “Approximation by simple partial fractions with constraints on the poles”, Sb. Math., 203:11 (2012), 1553–1570  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    6. A. V. Kayumova, “Skhodimost ryadov prostykh drobei v $L_p(\mathbb R)$”, Uchen. zap. Kazan. un-ta. Ser. Fiz.-matem. nauki, 154, no. 1, Izd-vo Kazanskogo un-ta, Kazan, 2012, 208–213  mathnet
    7. P. A. Borodin, “Density of a semigroup in a Banach space”, Izv. Math., 78:6 (2014), 1079–1104  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    8. A. R. Alimov, I. G. Tsar'kov, “Connectedness and other geometric properties of suns and Chebyshev sets”, J. Math. Sci., 217:6 (2016), 683–730  mathnet  crossref  mathscinet
    9. A. R. Alimov, I. G. Tsar'kov, “Connectedness and solarity in problems of best and near-best approximation”, Russian Math. Surveys, 71:1 (2016), 1–77  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    10. Tabatabaie S.M., “The problem of density on
      $${L^{2}(G)}$$
      L 2 ( G )”, Acta Math. Hung., 150:2 (2016), 339–345  crossref  mathscinet  zmath  isi  scopus
    11. 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
    12. 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
    13. V. I. Danchenko, M. A. Komarov, P. V. Chunaev, “Ekstremalnye i approksimativnye svoistva naiprosteishikh drobei”, Izv. vuzov. Matem., 2018, no. 12, 9–49  mathnet
    14. Tabatabaie S.M., “The Problem of Density on Commutative Strong Hypergroups”, Math. Rep., 20:3 (2018), 227–232  mathscinet  zmath  isi
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