Pure and Applied Mathematics Journal

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Approximation of Functions by Singular Integrals

Received: 2 November 2014    Accepted: 13 November 2014    Published: 17 November 2014
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Abstract

In this work questions on approximation of locally summable functions by singular integrals are investigated. Was estimated the rate of approximation in terms of various metric characteristics describing the structural properties of the given function.

DOI 10.11648/j.pamj.20140306.11
Published in Pure and Applied Mathematics Journal (Volume 3, Issue 6, December 2014)
Page(s) 113-120
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Approximation, Singular Integrals, Bounded Mean Oscillation, Vanishing Mean Oscillation

References
[1] Butzer P.L., Nessel R.J. Fourier analysis and approximation. Vol.1: One-Dimensional Theory. New York and London, 1971.
[2] Calderon A.P., Zygmund A. On the existence of certain singular integrals. Acta. Math., 1952, v.88, pp.85-139.
[3] Gadzhiev N.M., Rzaev R.M. On the order of locally summable functions approximation by singular integrals. Funct. Approx. Comment. Math., 1992, v.20, pp.35-40.
[4] Garnett J.B. Bounded analytic functions. Academic Press Inc., New York, 1981.
[5] Golubov B.I. On asymptotics of multiple singular integrals for differentiable functions. Matem. Zametki, 1981, v.30, No5, pp.749-762 (Russian).
[6] Janson S. On functions with conditions on the mean oscillation. Ark. math., 1976, v.14, No2, pp. 189-196.
[7] John F., Nirenberg L. On functions of bounded mean oscillation. Comm. Pure Appl. Math., 1961, v.14, pp.415-426.
[8] Kerman R.A. Pointwise convergent approximate identities of dilated radially decreasing kernels. Proc. Amer. Math. Soc., 1987, v.101, No1, pp.41-44.
[9] Rzaev R.M. On approximation of essentially continuous functions by singular integrals. Izv. Vuzov. Matematika, 1989, No3, pp.57-62 (Russian).
[10] Rzaev R.M. On approximation of locally summable functions by singular integrals in terms of mean oscillation and some applications. Preprint Inst. Phys. Natl. Acad. Sci. Azerb., 1992, №1, p.1-43 (Russian).
[11] Rzaev R.M. A multidimensional singular integral operator in the spaces defined by conditions on the -th order mean oscillation. Dokady Mathematics, 1997, v.56, No2, pp.747-749.
[12] Rzaev R.M., Aliyeva L.R. On local properties of functions and singular integrals in terms of the mean oscillation. Cent. Eur. J. Math., 2008, v.6, No4, p.595-609.
[13] Sarason D. Functions of vanishing mean oscillation. Trans. Amer. Math. Soc., 1975, v.207, pp. 391-405.
[14] Spanne S. Some function spaces defined using the mean oscillation over cubes. Ann. Scuola Norm. Sup. Pisa, 1965, v.19, No4, pp.593-608.
[15] Stein E.M., Singular integrals and differentiability properties of functions. Princeton University Press. Princeton, New J., 1970.
[16] Stein E.M., Weiss G. Introduction to Fourier Analysis on Euclidean spaces. Princeton University Press. Princeton, New J., 1971.
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    Rahim M. Rzaev, Gulnara Kh. Mammadova, Mansur Sh. Maharramov. (2014). Approximation of Functions by Singular Integrals. Pure and Applied Mathematics Journal, 3(6), 113-120. https://doi.org/10.11648/j.pamj.20140306.11

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    ACS Style

    Rahim M. Rzaev; Gulnara Kh. Mammadova; Mansur Sh. Maharramov. Approximation of Functions by Singular Integrals. Pure Appl. Math. J. 2014, 3(6), 113-120. doi: 10.11648/j.pamj.20140306.11

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    AMA Style

    Rahim M. Rzaev, Gulnara Kh. Mammadova, Mansur Sh. Maharramov. Approximation of Functions by Singular Integrals. Pure Appl Math J. 2014;3(6):113-120. doi: 10.11648/j.pamj.20140306.11

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  • @article{10.11648/j.pamj.20140306.11,
      author = {Rahim M. Rzaev and Gulnara Kh. Mammadova and Mansur Sh. Maharramov},
      title = {Approximation of Functions by Singular Integrals},
      journal = {Pure and Applied Mathematics Journal},
      volume = {3},
      number = {6},
      pages = {113-120},
      doi = {10.11648/j.pamj.20140306.11},
      url = {https://doi.org/10.11648/j.pamj.20140306.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20140306.11},
      abstract = {In this work questions on approximation of locally summable functions by singular integrals are investigated. Was estimated the rate of approximation in terms of various metric characteristics describing the structural properties of the given function.},
     year = {2014}
    }
    

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Author Information
  • Institute of Mathematics and Mechanics of National Academy of Sciences of Azerbaijan, Baku, AZ1141, Azerbaijan; Department of Mathematics and Informatics of Azerbaijan State Pedagogical University, Baku, AZ1000, Azerbaijan

  • Department of Mathematics and Informatics of Ganja State University, Ganja, Azerbaijan

  • Institute of Mathematics and Mechanics of National Academy of Sciences of Azerbaijan, Baku, AZ1141, Azerbaijan

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