A generic research on nonlinear non-convolution type singular integral operators
0
조회
0
다운로드
1
인용
0
참고문헌
키워드
초록
In this paper, we present some general results on the pointwise convergence of the non-convolution type nonlinear singular integral operators in the following form: \begin{equation*} T_{\lambda }(f;x)=\underset{\Omega }{\int }K_{\lambda }\left( t,x,f\left(t\right) \right) dt,\text{ }x\in \Psi ,\text{ }\lambda \in \Lambda , \end{equation*} where $\Psi =\left \langle a,b\right \rangle $ and $\Omega =\left \langle A,B\right \rangle $ stand for arbitrary closed, semi-closed or open bounded intervals in $\mathbb{R}$ or these set notations denote $\mathbb{R} $, and $\Lambda $ is a set of non-negative numbers, to the function $f\in L_{p,w}\left( \Omega \right) $, where $L_{p,w}\left( \Omega \right) $ denotes the space of all measurable functions $f$ for which $\left \vert \frac{f}{w}\right \vert ^{p}$ $(1\leq p<\infty )$ is integrable on $\Omega ,$ and $w: \mathbb{R} \rightarrow \mathbb{R}^{+}$ is a weight function satisfying some conditions.
Abstract
In this paper, we present some general results on the pointwise convergence of the non-convolution type nonlinear singular integral operators in the following form: \begin{equation*} T_{\lambda }(f;x)=\underset{\Omega }{\int }K_{\lambda }\left( t,x,f\left(t\right) \right) dt,\text{ }x\in \Psi ,\text{ }\lambda \in \Lambda , \end{equation*} where $\Psi =\left \langle a,b\right \rangle $ and $\Omega =\left \langle A,B\right \rangle $ stand for arbitrary closed, semi-closed or open bounded intervals in $\mathbb{R}$ or these set notations denote $\mathbb{R} $, and $\Lambda $ is a set of non-negative numbers, to the function $f\in L_{p,w}\left( \Omega \right) $, where $L_{p,w}\left( \Omega \right) $ denotes the space of all measurable functions $f$ for which $\left \vert \frac{f}{w}\right \vert ^{p}$ $(1\leq p<\infty )$ is integrable on $\Omega ,$ and $w: \mathbb{R} \rightarrow \mathbb{R}^{+}$ is a weight function satisfying some conditions.
참고문헌 (28)
- G. Alexits, "Konvergenzprobleme der Orthogonalreihen" (1960), Verlag der Un-garischen Akademie der Wissenschaften
- C. Bardaro, "On approximation properties for some classes of linear operators of convolution type", Atti Sem. Mat. Fis. Univ. Modena, 33(2), pp.329~356 (1984)
- C. Bardaro, "On approximation properties of certain noncon-volution integral operators", J. Approx. Theory, 62(3), pp.358~371 (1990)
- C. Bardaro, "Approximation by nonlinear integral operators in some modular function spaces", Ann. Polon. Math, 63, pp.173~182 (1996)
- C. Bardaro, "Nonlinear integral operators and applications" (2003), Walter de Gruyter, Publ
- C. Bardaro, "Pointwise convergence theorems for nonlinear Mellin convolution operators", Int. J. Pure Appl. Math, 27(4), pp.431~447 (2006)
- P. L. Butzer, "Fourier Analysis and Approximation: Vol. I" (1971), Academic Press
- Deepmala, "A study on some problems on existence of solutions for nonlinear functional-integral equations", Acta Mathematica Scientia, 33 B(5), pp.1305~1313 (2013)
- Deepmala, "A study on fixed point theorems for nonlinear contractions and its applications", Ph. D., pp.2014
- S. Esen, "Convergence and the order of convergence of family of non-convolution type integral operators at characteristic points", Ph. D., pp.2002
- S. Esen, "The order of approximation by the family of integral operators with positive kernel", Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb, 28, pp.117~122 (2008)
- A. D. Gadjiev, "Special Problems of Functional Analysis and their Appl. to the Theory of Diff. Eq. and the Theory of Func", pp.40~44 (1968), Izdat. Akad. Nauk Azerbaidzan. SSR
- O. O. Guller, "On weighted approximation by singular integral operators depending on two parameters", 4th Interna-tional Eurasian Conference on Mathematical Sciences and Applications (IECMSA 2015), pp.89~ (2015)
- H. Karsli, "On approximation properties of non-convolution type nonlinear integral operators", Anal. Theory Appl, 26(2), pp.140~152 (2010)
- 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), pp.287~304 (1963)
- V. N. Mishra, "Trigonometric approximation of signals (Functions) in Lp(p >= 1)-norm", International Journal of Contempo-rary Mathematical Sciences, 7(19), pp.909~918 (2012)
- V. N. Mishra, "On Chlodowsky variant of (p; q)Kantorovich-Stancu-Schurer operators", Int. J. Analysis and Applications, 11(1), pp.28~39 (2016)
- L. N. Mishra, "On the concept of existence and local attractivity of solutions for some quadratic Volterra integral equation of fractional order", Appl. Math. Comput, 285, pp.174~183 (2016)
- L. N. Mishra, "On existence theorems for some generalized nonlinear functional-integral equations with appli-cations", Filomat (2016)
- L. N. Mishra, "Solvability and asymptotic behav-ior for some nonlinear quadratic integral equation involving Erdelyi-Kober fractional integrals on the unbounded interval", Progress in Fractional Differentiation and Applications, 2(3), pp.153~168 (2016)
- J. Musielak, "On some approximation problems in modular spaces", Constructive Function Theory, Proceedings of International Conference Varna, pp.181~189 (1983), Bulgarian Academic of Sciences
- J. Musielak, "Approximation by nonlinear singular integral operators in generalized Orlicz spaces", Comment. Math. Prace Mat, 31, pp.79~88 (1991)
- T. Swiderski, "Nonlinear singular integrals depending on two parameters", Comment. Math, 40, pp.181~189 (2000)
- H. K. Pathak, "Common fixed point theorems for PD-operator pairs under relaxed conditions with applications", J. Comp. Appl. Math, 239, pp.103~113 (2013)
- W. Rudin, "Real and Complex Analysis" (1987), Mc-Graw Hill Book Co
- B. Rydzewska, "Approximation des fonctions par des integrales singulieres ordinaires", Fasc. Math, 7, pp.71~81 (1973)
- R. Taberski, "Singular integrals depending on two parameters", Prace Mat, 7, pp.173~179 (1962)
- G. Uysal, "A note on nonlinear singular integral operators depending on two parameters", New Trends Math. Sci, 4(1), pp.104~114 (2016)