Korean J. Math.  Vol 26, No 3 (2018)  pp.483-501
DOI: https://doi.org/10.11568/kjm.2018.26.3.483

Some weighted approximation properties of nonlinear double integral operators

Gumrah Uysal, Vishnu Narayan Mishra, Sevilay Kirci Serenbay


In this paper, we present some recent results on weighted pointwise convergence and the rate of pointwise convergence for the family of nonlinear double singular integral operators in the following form:
T_{\eta }\left( f;x,y\right) =\underset{\mathbb{R}^{2}}{\iint }K_{\eta
}\left( t-x,s-y,f\left( t,s\right) \right) dsdt,\text{ }\left( x,y\right)
\in \mathbb{R}^{2},\text{ }\eta \in \Lambda ,
where the function $f: \mathbb{R}^{2}\rightarrow \mathbb{R}$ is Lebesgue measurable on $\mathbb{R}^{2}$ and $\Lambda $ is a non-empty set of indices. Further, we provide an example to support these theoretical results.


Generalized Lipschitz condition; Weighted pointwise convergence; Rate of convergence.

Subject classification

41A35, 41A25, 28C10.


Full Text:



G. Alexits, Convergence Problems of Orthogonal Series, Translated from the German by I. F ̈older. International Series of Monographs in Pure and Applied Mathematics, Vol. 20 Pergamon Press, New York, Oxford, Paris, 1961. (Google Scholar)

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

C. Bardaro, J. Musielak, G. Vinti, Nonlinear Integral Operators and Applications, De Gruyter Ser. Nonlinear Anal. Appl. 9, Walter de Gruyter, Berlin, 2003. (Google Scholar)

C. Bardaro, G. Vinti and H. Karsli, Nonlinear integral operators with homogeneous kernels: pointwise approximation theorems, Appl. Anal. 90 (3-4) (2011), 463–474. (Google Scholar)

P. L. Butzer and R. J. Nessel, Fourier Analysis and Approximation, Vol. I. Academic Press, New York, London, 1971. (Google Scholar)

A. D. Gadjiev, The order of convergence of singular integrals which depend on two parameters, Special problems of Functional Analysis and Their Appl. to the Theory of Diff. Eq. and the Theory of Func., Izdat. Akad. Nauk Azerba ̆ıdaˇzan. SSR. (1968), 40–44. (Google Scholar)

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

H. Karsli, Convergence and rate of convergence by nonlinear singular integral operators depending on two parameters, Appl. Anal. 85 (6-7) (2006), 781–791. (Google Scholar)

R. G. Mamedov, On the order of convergence of m-singular integrals at generalized Lebesgue points and in the space Lp (−∞, ∞), Izv. Akad. Nauk SSSR Ser. Mat. 27 (2) (1963), 287–304. (Google Scholar)

J. Musielak, On some approximation problems in modular spaces, Constructive Function Theory 1981, (Proc. Int. Conf., Varna, June 1-5, 1981), Sofia, Publ. House Bulgarian Acad. Sci. (1983), 455–461. (Google Scholar)

W. Rudin, Real and Complex Analysis, Mc-Graw Hill Book Co., London, 1987. (Google Scholar)

B. Rydzewska, Approximation des fonctions par des int ́egrales singuli`eres ordinaires, Fasc. Math. 7 (1973), 71–81. (Google Scholar)

B. Rydzewska, Approximation des fonctions de deux variables par des int ́egrales singuli`eres doubles, Fasc. Math. 8 (1974), 35–45. (Google Scholar)

S. Siudut, On the convergence of double singular integrals, Comment. Math. 28 (1) (1988), 143–146. (Google Scholar)

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, New Jersey, 1970. (Google Scholar)

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

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

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

R. Taberski, On double singular integrals, Prace Mat. 19 (1976), 155–160. (Google Scholar)

G. Uysal, A new approach to nonlinear singular integral operators depending on three parameters. Open Math. 14 (2016), 897–907. (Google Scholar)

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


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