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Izv. RAN. Ser. Mat., 2009, Volume 73, Issue 2, Pages 123–140 (Mi izv2721)  

This article is cited in 23 scientific papers (total in 23 papers)

Approximation by simple partial fractions and the Hilbert transform

V. Yu. Protasov

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

Abstract: We study the problem of approximation of functions in $L_p$ by simple partial fractions on the real axis and semi-axis. A simple partial fraction is a rational function of the form $g(t)=\sum_{k=1}^n\frac1{t-z_k}$, where $z_1,…,z_n$ are complex numbers. We describe the set of functions that can be approximated by simple partial fractions within any accuracy and the set of functions that can be approximated by convex combinations of them (the cone of simple partial fractions). We obtain estimates for the norms of simple partial fractions and conditions for the convergence of function series $\sum_{k=1}^\infty\frac1{t-z_k}$ in the space $L_p$. Our approach is based on the use of the Hilbert transform and the methods of convex analysis.

Keywords: approximation, simple partial fraction, convergence of function series, Hilbert transform, entire function, logarithmic derivative.

DOI: https://doi.org/10.4213/im2721

Full text: PDF file (609 kB)
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English version:
Izvestiya: Mathematics, 2009, 73:2, 333–349

Bibliographic databases:

UDC: 517.538.52+517.444
MSC: 41A20, 46A55, 30E10
Received: 29.08.2007

Citation: V. Yu. Protasov, “Approximation by simple partial fractions and the Hilbert transform”, Izv. RAN. Ser. Mat., 73:2 (2009), 123–140; Izv. Math., 73:2 (2009), 333–349

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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. 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
    2. 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
    3. 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
    4. P. V. Chunaev, “On a nontraditional method of approximation”, Proc. Steklov Inst. Math., 270 (2010), 278–284  mathnet  crossref  mathscinet  zmath  isi  elib
    5. 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
    6. I. R. Kayumov, “Integral bounds for simple partial fractions”, Russian Math. (Iz. VUZ), 56:4 (2012), 27–37  mathnet  crossref  mathscinet
    7. I. R. Kayumov, “A Necessary Condition for the Convergence of Simple Partial Fractions in $L_p(\mathbb R)$”, Math. Notes, 92:1 (2012), 140–143  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    8. 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
    9. 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
    10. M. A. Komarov, “An example of nonuniqueness of a simple partial fraction of the best uniform approximation”, Russian Math. (Iz. VUZ), 57:9 (2013), 22–30  mathnet  crossref
    11. 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
    12. I. R. Kayumov, A. V. Kayumova, “Convergence of the imaginary parts of simplest fractions in $L_p(\mathbb R)$ for $p<1$”, J. Math. Sci. (N. Y.), 202:4 (2014), 553–559  mathnet  crossref
    13. 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
    14. I. R. Kayumov, “On the Convergence of Series in Spaces of Integrable Functions”, Math. Notes, 95:6 (2014), 780–785  mathnet  crossref  crossref  mathscinet  isi  elib
    15. P. Chunaev, “Least deviation of logarithmic derivatives of algebraic polynomials from zero”, J. Approx. Theory, 185 (2014), 98–106  crossref  mathscinet  zmath  isi
    16. F. D. Kayumov, “Integral estimates for derivatives of univalent functions”, Lobachevskii J. Math., 35:4 (2014), 402–408  crossref  mathscinet
    17. M. A. Komarov, “A criterion for the best uniform approximation by simple partial fractions in terms of alternance”, Izv. Math., 79:3 (2015), 431–448  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    18. V. I. Danchenko, L. A. Semin, “Sharp quadrature formulas and inequalities between various metrics for rational functions”, Siberian Math. J., 57:2 (2016), 218–229  mathnet  crossref  crossref  mathscinet  isi  elib
    19. 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
    20. 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
    21. M. A. Komarov, “O priblizhenii spetsialnymi raznostyami naiprosteishikh drobei”, Algebra i analiz, 30:4 (2018), 47–60  mathnet
    22. V. I. Danchenko, M. A. Komarov, P. V. Chunaev, “Ekstremalnye i approksimativnye svoistva naiprosteishikh drobei”, Izv. vuzov. Matem., 2018, no. 12, 9–49  mathnet
    23. Komarov M.A., “Approximation to Constant Functions By Electrostatic Fields Due to Electrons and Positrons”, Lobachevskii J. Math., 40:1, SI (2019), 79–84  crossref  isi  scopus
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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