On pointwise convergence of bivariate nonlinear singular integral operators
Keywords:
Lipschitz condition, pointwise convergence, rate of convergence, nonlinear bivariate integral operator, generalized Lebesgue pointAbstract
In this paper, we present some theorems on pointwise convergence and the rate of pointwise convergence for the family of nonlinear bivariate singular integral operators of the following form:
whereis a real valued and integrable function on a bounded arbitrary closed, semi-closed or open region in or and is the set of non-negative indices with accumulation point .
References
Almali, S.E. (2016). On approximation properties of nonconvolution type of integral operators for non-integrable function, Advances in Mathematics, 5(2): 219-230.
Bardaro, C., Musielak, J. & Vinti, G. (2003). Nonlinear integral operators and applications. DeGruyter Series in Nonlinear Analysis and Applications 9. Walter de Gruyter & Co., Berlin. Pp. 201.
Bardaro, C., Vinti, G. & Karsli, H. (2011). Nonlinear integral operators with homogeneous kernels: pointwise approximation theorems. Applicable Analysis, 90(3-4):463-474.
Bardaro, C., Vinti, G. & Karsli, H. (2008). On pointwise convergence of linear integral operators with homogenous kernels. Integral Transforms and Special Functions, 19(5-6):429-439.
Bardaro, C., Karsli, H. & Vinti, G. (2013). On pointwise convergence of Mellin type nonlinear
m-singular integral operators. Communications on Applied Nonlinear Analysis, 20(2):25-39.
Bardaro, C. & Mantellini, I. (2006). Pointwise convergence theorems for nonlinear Mellin convolution operators. Int. J. Pure Appl. Math., 27(4):431-447.
Butzer, P.L. & Nessel, R.J. (1971). Fourier analysis and approximation. Volume I: One-Dimensional Theory. Academic Press, New York-London. Pp. 553.
Gadjiev, A.D. (1968). On the order of convergence of singular integrals which depend on two parameters. In: Special Problems of Functional Analysis and their Appl. to the Theory of Diff. Eq. and the Theory of Func. Pp. 40-44. Izdat. Akad. Nauk Azerba?daan. SSR, Baku.
Ghorpade, S.R. & Limaye, B.V. (2010). A Course in Multivariable Calculus and Analysis. Undergraduate Texts in Mathematics. Springer, New York. Pp. 475.
Jawarneh, Y. & Noorani, M.S.M. (2011). Inequalities of Ostrowski and Simpson type for mappings of two variables with bounded variation and applications. Transylvanian Journal of Mathematics and Mechanics, 3(2):8194.
Karsli, H. (2006). Convergence and rate of convergence by nonlinear singular integral operators depending on two parameters. Applicable Analysis, 85(6-7):781-791.
Karsli, H. (2008). On the approximation properties of a class of convolution type nonlinear singular integral operators. Georgian Mathematical Journal, 15(1):7786.
Karsli, H. (2014). Fatou type convergence of nonlinear m-singular integral operators. Applied Mathematics and Computation, 246:221-228.
Karsli, H. (2015). On Fatou type convergence of convolution type double singular integral operators, Analysis and Theory in Applications, 31(3):307320.
Karsli, H. & Ibikli, E. (2007). On convergence of convolution type singular integral operators depending on two parameters. Fasciculi Mathematici, 38:25-39.
Musielak, J. (1983). On some approximation problems in modular spaces. In: Constructive Function Theory 1981, (Proc. Int. Conf., Varna, June 1-5, 1981). Pp. 455-461. Publ. House Bulgarian Acad. Sci., Sofia.
Musielak, J. ( 2000). Nonlinear integral operators and summability in . Atti del Seminario Matematico e Fisico dell'Universit di Modena, 48(1):249-257.
Rydzewska, B. (1974). Approximation des fonctions de deux variables par des intgrales singulires doubles. Fasciculi Mathematici, 8:35-45.
Rydzewska, B. (1973). Approximation des fonctions par des intgrales singulires ordinaires. Fasciculi Mathematici, 7:71-81.
Siudut, S. (1988). On the convergence of double singular integrals. Commentationes Mathematicae, 28(1):143-146.
Sremr, J. (2010). Absolutely continuous functions of two variables in the sense of Carathodory.
Electronic Journal of Differential Equations, 2010:154.
Swiderski, T. & Wachnicki, E. (2000). Nonlinear singular integrals depending on two parameters. Commentationes Mathematicae, 40:181-189.
Taberski, R. (1964). On double integrals and Fourier Series. Annales Polonici Mathematici, 15: 97-115.
Taberski, R. (1962). Singular integrals depending on two parameters. Prace Matematyczne, 7:173-179.
Uysal, G., Yilmaz, M.M. & Ibikli, E. (2015). A study on pointwise approximation by double singular integral operators. Journal of Inequalities and Applications, 2015:94.
Uysal, G., (2016). A new approach to nonlinear singular integral operators depending on three parameters. Open Mathematics, 14:897907.
Vinti, G. & Zampogni, L. (2011). A unifying approach to convergence of linear sampling type operators in Orlicz spaces. Advances in Differential Equations, 16(5-6):573-600.
Yilmaz, M.M, Uysal, G. & Ibikli, E. (2014). A note on rate of convergence of double singular integral operators. Advances in Difference Equations, 2014:287.
Yilmaz, M.M, Mishra, L.N. & Uysal, G. (2017). On the approximation by convolution type
double singular integral operators, arXiv:1701.07186v1
