Advertisement

Pointwise Convergence Analysis for Nonlinear Double m-Singular Integral Operators

  • Gümrah Uysal
  • Hemen Dutta
Chapter

Abstract

In this chapter, m-singularity notion is discussed for double singular integral operators. In this direction, several results concerning pointwise convergence of nonlinear double m-singular integral operators are presented. This chapter is divided into six sections. In the first section, the reasons giving birth to m-singularity notion are explained and related theoretical background is mentioned. Also, the motivations giving inspiration to this note are presented. In the second section, the well-definiteness of the operators which are under the spotlights is shown on their domain. In the third section, an auxiliary result, pointwise convergence theorem, is proved. In the fourth section, main theorem, Fatou type convergence theorem, is proved. In the fifth section, corresponding rates of convergences are evaluated. In the last section, some concluding remarks are given.

Keywords

Pointwise convergence Fatou-type convergence Nonlinear bivariate integral operators m-Singularity 

References

  1. 1.
    S.E. Almali, A.D. Gadjiev, On approximation properties of certain multidimensional nonlinear integrals. J. Nonlinear Sci. Appl. 9(5), 3090–3097 (2016)MathSciNetCrossRefGoogle Scholar
  2. 2.
    A. Aral, V. Gupta, R.P. Agarwal, Applications of q-Calculus in Operator Theory (Springer, New York, 2013)CrossRefGoogle Scholar
  3. 3.
    C. Bardaro, On approximation properties for some classes of linear operators of convolution type. Atti Sem. Mat. Fis. Univ. Modena 33(2), 329–356 (1984)MathSciNetzbMATHGoogle Scholar
  4. 4.
    C. Bardaro, J. Musielak, G. Vinti, Nonlinear Integral Operators and Applications. De Gruyter Series in Nonlinear Analysis and Applications, vol. 9 (Walter de Gruyter & Co., Berlin, 2003)Google Scholar
  5. 5.
    C. Bardaro, H. Karsli, G. Vinti, On pointwise convergence of Mellin type nonlinear m-singular integral operators. Comm. Appl. Nonlinear Anal. 20(2), 25–39 (2013)MathSciNetzbMATHGoogle Scholar
  6. 6.
    R. Bracewell, The Fourier Transform and its Applications, 3rd edn. (McGraw-Hill Science, New York, 1999)zbMATHGoogle Scholar
  7. 7.
    P.L. Butzer, R.J. Nessel, Fourier Analysis and Approximation, vol. I (Academic, New York, 1971)CrossRefGoogle Scholar
  8. 8.
    A.P. Calderón, A. Zygmund, Singular integral operators and differential equations. Am. J. Math. 79, 901–921 (1957)MathSciNetCrossRefGoogle Scholar
  9. 9.
    P. Fatou, Series trigonometriques et series de Taylor. Acta Math. 30(1), 335–400 (1906)MathSciNetCrossRefGoogle Scholar
  10. 10.
    A.D. Gadjiev, On nearness to zero of a family of nonlinear integral operators of Hammerstein. Izv. Akad. Nauk Azerbaı̆džan, SSR Ser. Fiz.-Tehn. Mat. Nauk 2, 32–34 (1966)Google Scholar
  11. 11.
    A.D. Gadjiev, 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
  12. 12.
    E.G. Ibrahimov, On approximation order of functions by m-singular Gegenbauer integrals. Applied problems of mathematics and mechanics (Baku 2002). Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb. 17(205), 71–77 (2002)Google Scholar
  13. 13.
    E.J. Ibrahimov, S.A. Jafarova, On convergence and convergence order of Gegenbauer’s m-singular integrals. Proc. A. Razmadze Math. Inst. 159, 21–42 (2012)MathSciNetzbMATHGoogle Scholar
  14. 14.
    H. Karsli, Fatou type convergence of nonlinear m-singular integral operators. Appl. Math. Comput. 246, 221–228 (2014)MathSciNetzbMATHGoogle Scholar
  15. 15.
    H. Lebesgue, Sur les intégrales singulières. Annales de la faculté des sciences de Toulouse Sér. 3(1), 25–117 (1909)CrossRefGoogle Scholar
  16. 16.
    R.G. Mamedov, 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
  17. 17.
    S.G. Mikhlin, S. Prössdorf, Singular Integral Operators. Translated from the German by Albrecht Böttcher and Reinhard Lehmann (Springer, Berlin, 1986)CrossRefGoogle Scholar
  18. 18.
    J. Musielak, 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 455–461 (1983)Google Scholar
  19. 19.
    J. Musielak, Approximation by nonlinear singular integral operators in generalized Orlicz spaces. Comment. Math. Prace Mat. 31, 79–88 (1991)MathSciNetzbMATHGoogle Scholar
  20. 20.
    J. Musielak, Nonlinear integral operators and summability in \( \mathbb {R} ^{2}.\) Atti Sem. Mat. Fis. Univ. Modena 48(1), 249–257 (2000)Google Scholar
  21. 21.
    I.P. Natanson, Theory of Functions of a Real Variable, vol. 2. Translated from the Russian by Leo F. Boron. (Frederick Ungar Pub. Co., New York, 1960)Google Scholar
  22. 22.
    M. Papadopoulos, The use of singular integrals in wave propagation problems, with application to the point load source in a semi-infinite elastic medium. Proc. R. Soc. A276, 204–237 (1963)zbMATHGoogle Scholar
  23. 23.
    B. Rydzewska, Approximation des fonctions par des intégrales singulières ordinaires. Fasc. Math.7, 71–81 (1973)MathSciNetzbMATHGoogle Scholar
  24. 24.
    B. Rydzewska, Approximation des fonctions de deux variables par des intégrales singulières doubles. Fasc. Math. 8, 35–45 (1974)MathSciNetzbMATHGoogle Scholar
  25. 25.
    B. Rydzewska, Point-approximation des fonctions par des certaines intégrales singuliéres. Fasc. Math. 10, 13–24 (1978)MathSciNetzbMATHGoogle Scholar
  26. 26.
    S. Saks, Theory of the Integral (G.E. Stechert & Co., New York, 1937)zbMATHGoogle Scholar
  27. 27.
    S. Siudut, On the convergence of double singular integrals. Comment. Math. Prace Mat. 28(1), 143–146 (1988)MathSciNetzbMATHGoogle Scholar
  28. 28.
    S. Siudut, A theorem of Romanovski type for double singular integrals. Comment. Math. Prace Mat. 29, 277–289 (1989)MathSciNetzbMATHGoogle Scholar
  29. 29.
    S. Siudut, On the Fatou type convergence of abstract singular integrals. Comment. Math. Prace Mat. 30(1) 171–176 (1990)MathSciNetzbMATHGoogle Scholar
  30. 30.
    E.M. Stein, Singular Integrals and Differentiability Properties of Functions (Princeton Univ. Press, New Jersey, 1970)zbMATHGoogle Scholar
  31. 31.
    E.M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces (Princeton Univ. Press, New Jersey, 1971)zbMATHGoogle Scholar
  32. 32.
    T. Swiderski, E. Wachnicki, Nonlinear singular integrals depending on two parameters. Comment. Math. 40, 181–189 (2000)MathSciNetzbMATHGoogle Scholar
  33. 33.
    R. Taberski, Singular integrals depending on two parameters. Prace Mat. 7, 173–179 (1962)MathSciNetzbMATHGoogle Scholar
  34. 34.
    R. Taberski, On double integrals and Fourier series. Ann. Polon. Math. 15, 97–115 (1964)MathSciNetCrossRefGoogle Scholar
  35. 35.
    G. Uysal, Nonlinear m-singular integral operators in the framework of Fatou type weighted convergence. Commun. Fac. Sci. Univ. Ank. Sér. A1 Math. Stat. 67(1), 262–276 (2018)Google Scholar
  36. 36.
    G. Uysal, On nonlinear bivariate [m 1, m 2] −singular integral operators, Math. Methods Appl. Sci. 1–13 (2018).  https://doi.org/10.1002/mma.5425
  37. 37.
    G. Uysal, M.M. Yilmaz, E. Ibikli, On pointwise convergence of bivariate nonlinear singular integral operators. Kuwait J. Sci. 44(2), 46–57 (2017)MathSciNetGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Gümrah Uysal
    • 1
  • Hemen Dutta
    • 2
  1. 1.Department of Computer Technologies, Division of Technology of Information SecurityKarabuk UniversityKarabukTurkey
  2. 2.Department of MathematicsGauhati UniversityGuwahatiIndia

Personalised recommendations