NONLINEARm SINGULAR INTEGRAL OPERATORS IN THE FRAMEWORK OF FATOU TYPE WEIGHTED CONVERGENCE

In the present paper, we prove some theorems concerning Fatoutype weighted pointwise convergence of nonlinear m singular integral operators of the form

___

  • 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).
  • Bardaro, C., On approximation properties for some classes of linear operators of convolution type. Atti Sem. Mat. Fis. Univ. Modena, 33 (1984), 329-356.
  • Bardaro, C., Musielak, J. and Vinti, G., Approximation by nonlinear singular integral oper- ators in some modular function spaces. Ann. Polon. Math. 63 (2) (1996), 173-182.
  • Bardaro, C., Musielak, J. and Vinti, G., Nonlinear Integral Operators and Applications. De Gruyter Ser. Nonlinear Anal. Appl. 9, Walter de Gruyter, Berlin (2003).
  • Bardaro, C., Karsli, H. and Vinti, G., On pointwise convergence of Mellin type nonlinear m-singular integral operators. Comm. Appl. Nonlinear Anal. 20 (2) (2013), 25–39.
  • Butzer, P. L. and Nessel, R. J., Fourier Analysis and Approximation vol. I. Academic Press, New York, London (1971).
  • Carlsson, M., Fatou-type theorems for general approximate identities. Math. Scand. 102 (2) (2008), 231–252.
  • Donoghue, W. F. J., A theorem of the Fatou type. Monatsh. Math. 67 (1963), 225–228.
  • Fatou, P., Séries trigonométriques et séries de Taylor. Acta Math. 30(1) (1906), 335–400.
  • Gadjiev, A. D., The order of convergence of singular integrals which depend on two parame- ters. In: Special Problems of Functional Analysis and their Appl. to the Theory of Diğ. Eq. and the Theory of Func., Izdat. Akad. Nauk Azerba¼ıdaµzan. SSR., (1968), 40–44.
  • Gadjiev, A. D., On nearness to zero of a family of nonlinear integral operators of Hammerstein. Izv. Akad. Nauk Azerba¼ıdµzan. SSR Ser. Fiz.-Tehn. Mat. Nauk, 2 (1966), 32-34.
  • Gripenberg, G., Londen, S. O. and Stağans, O., Volterra Integral and Functional Equations, Encyclopedia of Mathematics and its Applications, no. 34, Cambridge University Press, Cam- bridge (1990).
  • Ibrahimov, E. J. and Jafarova, S. A., On convergence and convergence order of Gegenbauer’s m-singular integrals. Proc. A. Razmadze Math. Inst. 159 (2012), 21–42.
  • Karsli, H. and Ibikli, E., On convergence of convolution type singular integral operators depending on two parameters. Fasc. Math. 38 (2007), 25–39.
  • Karsli, H., Fatou type convergence of nonlinear m-singular integral operators. Appl. Math. Comput. 246 (2014), 221–228.
  • Loomis, L. H., The converse of the Fatou theorem for positive harmonic functions. Trans. Amer. Math. Soc. 53 (1943), 239–250.
  • Mamedov, R. G., On the order of convergence of m-singular integrals at generalized Lebesgue points and in the space Lp( 1; 1). Izv. Akad. Nauk SSSR Ser. Mat. 27 (2) (1963), 287-304.
  • Musielak, J., On some approximation problems in modular spaces. In: Constructive Function Theory 1981, (Proc. Int. Conf., Varna, June 1-5, 1981). Publ. House Bulgarian Acad. Sci., So…a (1983), 455-461.
  • Musielak, J., Approximation by nonlinear singular integral operators in generalized Orlicz spaces. Comment. Math. Prace Mat. 31 (1991), 79–88.
  • Musielak, J., Nonlinear approximation in some modular function spaces: I. Math. Japonica, (1993), 83-90.
  • Rudin, W., Real and Complex Analysis. Mc-Graw Hill Book Co., London (1987).
  • Rydzewska, B., Approximation des fonctions par des intégrales singulières ordinaires. Fasc. Math. 7 (1973), 71–81.
  • Rydzewska, B., Point-approximation des fonctions par des certaines intégrales singulières. Fasc. Math. 10 (1978), 13–24.
  • Siudut, S., On the Fatou type convergence of abstract singular integrals. Comment. Math. Prace Mat. 30 (1) (1990), 171–176.
  • Stein, E. M., Singular Integrals and Diğerentiability Properties of Functions. Princeton Uni- versity Press, New Jersey (1970).
  • Swiderski, T. and Wachnicki, E., Nonlinear singular integrals depending on two parameters. Comment Math. 40 (2000), 181–189.
  • Taberski, R., Singular integrals depending on two parameters. Prace Mat. 7 (1962), 173-179.
  • Taberski, R., On double integrals and Fourier series. Ann. Polon. Math. 15 (1964), 97–115.
  • Taberski, R., On double singular integrals. Prace Mat. 19 (1976), 155–160.
  • Uysal, G. and Ibikli, E., Weighted approximation by double singular integral operators with radially de…ned kernels. Math. Sci. (Springer) 10 (4) (2016), 149–157.
  • Current address : Department of Computer Technologies, Division of Technology of Information Security, Karabuk University, Karabuk 78050, TURKEY