Skip to main content
SpringerLink
Log in

On Weighted Convergence of Double Singular Integral Operators Involving Summation

  • Chapter
  • First Online:
Applied Mathematical Analysis: Theory, Methods, and Applications

Part of the book series: Studies in Systems, Decision and Control ((SSDC,volume 177))

  • 777 Accesses

Abstract

This chapter consists of five sections. First section is devoted to introduction part in which the description of the problem is presented and theoretical background is given. In the second section, the preliminary concepts which are utilized in the sequel are introduced. Then, the conditions under which double singular integral operators involving summation are well-defined in the space of Lebesgue measurable functions defined on different sets are presented. In the third section, Fatou type convergences of handled operators are discussed. In the fourth section, the rate of convergences with respect to obtained approximations in the preceding section are established. In the last section, we present some concluding remarks.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 189.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 249.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Alexits, G.: Convergence Problems of Orthogonal Series. Translated from the German by I. Földer. International Series of Monographs in Pure and Applied Mathematics, vol. 20. Pergamon Press, New York (1961)

    Chapter  Google Scholar 

  2. Aral, A.: On convergence of singular integrals with non-isotropic kernels. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 50(1–2), 83–93 (2001)

    Google Scholar 

  3. Almali, S.E.: On approximation properties of non-convolution type of integral operators for non-integrable function. Adv. Math. Sci. J. 5(2), 219–230 (2016)

    MATH  Google Scholar 

  4. Almali, S.E.: On approximation properties for non-linear integral operators. New Trends Math. Sci. 5(4), 123–129 (2017)

    Article  MathSciNet  Google Scholar 

  5. Almali, S.E., Gadjiev, A.D.: On approximation properties of certain multidimensional nonlinear integrals. J. Nonlinear Sci. Appl. 9(5), 3090–3097 (2016)

    Article  MathSciNet  Google Scholar 

  6. Almali, S.E., Uysal, G., Mishra, V.N., Güller, Ö.Ö.: On singular integral operators involving power nonlinearity. Korean J. Math. 25(4), 483–494 (2017)

    MathSciNet  Google Scholar 

  7. Bardaro, C.: On approximation properties for some classes of linear operators of convolution type. Atti Sem. Mat. Fis. Univ. Modena 33(2), 329–356 (1984)

    MathSciNet  MATH  Google Scholar 

  8. Bardaro, C., Vinti, G.: On approximation properties of certain nonconvolution integral operators. J. Approx. Theory 62(3), 358–371 (1990)

    Article  MathSciNet  Google Scholar 

  9. Bardaro, C., Musielak, J., Vinti, G.: Nonlinear Integral Operators and Applications. De Gruyter Series in Nonlinear Analysis and Applications, vol. 9. Walter de Gruyter & Co., Berlin (2003)

    Google Scholar 

  10. Bracewell, R.: The Fourier Transform and its Applications, 3rd edn. McGraw-Hill Science, New York (1999)

    MATH  Google Scholar 

  11. Butzer, P.L., Nessel, R.J.: Fourier Analysis and Approximation, vol. I. Academic Press, New York (1971)

    Book  Google Scholar 

  12. Carasso, A.S.: Singular integrals, image smoothness, and the recovery of texture in image deblurring. SIAM J. Appl. Math. 64(5), 1749–1774 (2004)

    Article  MathSciNet  Google Scholar 

  13. Dirac, P.A.M.: The Principles of Quantum Mechanics, 4th edn. Oxford University Press, London (1958)

    MATH  Google Scholar 

  14. Fatou, P.: Series trigonometriques et series de Taylor. Acta Math. 30(1), 335–400 (1906)

    Article  MathSciNet  Google Scholar 

  15. Gadjiev, A.D.: On nearness to zero of a family of nonlinear integral operators of Hammerstein. Izv. Akad. Nauk Azerbaĭdžan, SSR Ser. Fiz.-Tehn. Mat. Nauk 2, 32–34 (1966)

    Google Scholar 

  16. Gadjiev, A.D.: The order of convergence of singular integrals which depend on two parameters. In: Special problems of functional analysis and their applications to the theory of differential equations and the theory of functions. Izdat. Akad. Nauk Azerbaidžan. SSR., Baku 40–44 (1968)

    Google Scholar 

  17. Ghorpade, S.R., Limaye, B.V.: A Course in Multivariable Calculus and Analysis. Springer, New York (2010)

    Book  Google Scholar 

  18. Gripenberg, G., Londen, S.O., Staffans, O.: Volterra Integral and Functional Equations. Encyclopedia of Mathematics and its Applications, vol. 34. Cambridge University Press, Cambridge (1990)

    Google Scholar 

  19. Hobson, E.W.: The Theory of Functions of a Real Variable and the Theory of Fourier’s Series, vol. 1. Cambridge University Press, England (1921)

    MATH  Google Scholar 

  20. Karsli, H.: On Fatou type convergence of convolution type double singular integral operators. Anal. Theory Appl. 31(3), 307–320 (2015)

    MathSciNet  MATH  Google Scholar 

  21. Labsker, L.G., Gadjiev, A.D.: On some classes of double singular integrals. Izv. Akad. Nauk Azerbaĭdžan. SSR Ser. Fiz.-Mat. Tehn. Nauk 4, 37–54 (1962)

    Google Scholar 

  22. Lebesgue, H.: Sur les intégrales singulières. Annales de la faculté des sciences de Toulouse Sér. 3(1), 25–117 (1909)

    Article  Google Scholar 

  23. Lenze, B.: On multidimensional Lebesgue–Stieltjes convolution operators. Multivariate Approximation Theory, IV (Oberwolfach, 1989). International Series of Numerical Mathematics, vol. 90, pp. 225–232. Birkhäuser, Basel (1989)

    Chapter  Google Scholar 

  24. Mamedov, R.G.: On the order of convergence of m-singular integrals at generalized Lebesgue points and in the space \(L_{p}\left( -\infty ,\infty \right)\). Izv. Akad. Nauk SSSR Ser. Mat. 27(2), 287–304 (1963)

    Google Scholar 

  25. Musielak, J.: On some approximation problems in modular spaces. In: Constructive Function Theory, Proceedings of International Conference Varna, 1–5 June 1981. Publication House of Bulgarian Academic of Sciences, Sofia, pp. 455–461 (1983)

    Google Scholar 

  26. Musielak, J.: Approximation by nonlinear singular integral operators in generalized Orlicz spaces. Comment. Math. Prace Mat. 31, 79–88 (1991)

    MathSciNet  MATH  Google Scholar 

  27. Musielak, J.: Nonlinear integral operators and summability in \({\mathbb{R}}^{2}\). Atti Sem. Mat. Fis. Univ. Modena 48(1), 249–257 (2000)

    Google Scholar 

  28. Natanson, I.P.: Theory of Functions of a Real Variable, vol. 2. Translated from the Russian by Leo F. Boron. Frederick Ungar Publishing Co., New York (1960)

    Google Scholar 

  29. Rydzewska, B.: Approximation des fonctions par des intégrales singulières ordinaires. Fasc. Math. 7, 71–81 (1973)

    MathSciNet  MATH  Google Scholar 

  30. Rydzewska, B.: Approximation des fonctions de deux variables par des intégrales singulières doubles. Fasc. Math. 8, 35–45 (1974)

    MathSciNet  MATH  Google Scholar 

  31. Rydzewska, B.: Point-approximation des fonctions par des certaines intégrales singuliéres. Fasc. Math. 10, 13–24 (1978)

    MathSciNet  MATH  Google Scholar 

  32. Saks, S.: Theory of the Integral. G.E. Stechert & Co., New York (1937)

    MATH  Google Scholar 

  33. Siudut, S.: On the convergence of double singular integrals. Comment. Math. Prace Mat. 28(1), 143–146 (1988)

    MathSciNet  MATH  Google Scholar 

  34. Siudut, S.: A theorem of Romanovski type for double singular integrals. Comment. Math. Prace Mat. 29, 277–289 (1989)

    MathSciNet  MATH  Google Scholar 

  35. Siudut, S.: On the Fatou type convergence of abstract singular integrals. Comment. Math. Prace Mat. 30(1), 171–176 (1990)

    MathSciNet  MATH  Google Scholar 

  36. Spivak, M.D.: Calculus, 3rd edn. Publish or Perish Inc., Houston (1994)

    MATH  Google Scholar 

  37. Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, New Jersey (1970)

    MATH  Google Scholar 

  38. Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, New Jersey (1971)

    MATH  Google Scholar 

  39. Swiderski, T., Wachnicki, E.: Nonlinear singular integrals depending on two parameters. Comment. Math. 40, 181–189 (2000)

    MathSciNet  MATH  Google Scholar 

  40. Taberski, R.: Singular integrals depending on two parameters. Prace Mat. 7, 173–179 (1962)

    MathSciNet  MATH  Google Scholar 

  41. Taberski, R.: On double integrals and Fourier series. Ann. Polon. Math. 15, 97–115 (1964)

    Article  MathSciNet  Google Scholar 

  42. Taberski, R.: On double singular integrals. Comment. Math. Prace Mat. 19(1), 155–160 (1976)

    MathSciNet  MATH  Google Scholar 

  43. Uysal, G., Ibikli, E.: Weighted approximation by double singular integral operators with radially defined kernels. Math. Sci. (Springer) 10(4), 149–157 (2016)

    Article  MathSciNet  Google Scholar 

  44. Uysal, G., Yilmaz, M.M., Ibikli, E.: On pointwise convergence of bivariate nonlinear singular integral operators. Kuwait J. Sci. 44(2), 46–57 (2017)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Division of Technology of Information Security, Department of Computer Technologies, Karabuk University, 78050, Karabuk, Turkey

    Gümrah Uysal

  2. Department of Mathematics, Gauhati University, Guwahati, 781014, India

    Hemen Dutta

Authors
  1. Gümrah Uysal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hemen Dutta
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gümrah Uysal .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Gauhati University, Guwahati, India

    Hemen Dutta

  2. Computational Intelligence Laboratory, Department of Electrical & Computer Engineering, University of Manitoba, Winnipeg, MB, Canada

    James F. Peters

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Uysal, G., Dutta, H. (2020). On Weighted Convergence of Double Singular Integral Operators Involving Summation. In: Dutta, H., Peters, J. (eds) Applied Mathematical Analysis: Theory, Methods, and Applications. Studies in Systems, Decision and Control, vol 177. Springer, Cham. https://doi.org/10.1007/978-3-319-99918-0_18

Download citation

Publish with us

Policies and ethics

Search

Navigation