On certain multidimensional nonlinear integrals

Year 2020, Volume: 69 Issue: 2, 1356 - 1367, 31.12.2020

Özge Özalp Güller Gumrah Uysal

https://doi.org/10.31801/cfsuasmas.762646

Abstract

The aim of the paper is to obtain generalized convergence results for nonlinear multidimensional integrals of the form:

L_{η}(ω;x)=((ηⁿ)/(Ω_{n-1}))∫_{D}K(η|t-x|,ω(t))dt.

We will prove pointwise convergence of the family L_{η}(ω;x) as η→∞ at a fixed point x∈D which represents any generalized Lebesgue point of function ω∈L₁(D), where D is an open bounded subset of Rⁿ. Moreover, we will consider the case D=Rⁿ.

The aim of the paper is to obtain generalized convergence results for nonlinear multidimensional integrals of the form:

L_{η}(ω;x)=((ηⁿ)/(Ω_{n-1}))∫_{D}K(η|t-x|,ω(t))dt.

We will prove pointwise convergence of the family L_{η}(ω;x) as η→∞ at a fixed point x∈D which represents any generalized Lebesgue point of function ω∈L₁(D), where D is an open bounded subset of Rⁿ. Moreover, we will consider the case D=Rⁿ.

Keywords

Taylor expansion, Generalized Lebesgue point, Pointwise convergence

References

• Aliev, I. A., Gadjiev, A. D., Aral, A., On approximation properties of a family of linear operators at critical value of parameter, J. Approx. Theory, 138(2) (2006), 242--253.
• Almali, S. E., On pointwise convergence of the family of Urysohn-type integral operators, Math. Methods Appl. Sci., 42(16) (2019), 5346--5353.
• Almali, S. E., Gadjiev, A. D., On approximation properties of certain multidimensional nonlinear integrals, J. Nonlinear Sci. Appl., 9(5) (2016), 3090-3097.
• Altomare, F., Campiti, M., Korovkin-type Approximation Theory and Its Applications, Appendix A by Michael Pannenberg and Appendix B by Ferdinand Beckhoff, Berlin: De Gruyter Studies in Mathematics 17., Walter de Gruyter & Company, 1994.
• Altin, H. E., On pointwise approximation properties of certain nonlinear Bernstein operators, Tbilisi Math. J., 12(2) (2019), 47--58.
• Angeloni, L., Vinti, G., Convergence in variation and rate of approximation for nonlinear integral operators of convolution type, Results Math., 49(1-2) (2006), 1--23. (Erratum: Results Math. 57 (2010), no. 3-4, 387--391)
• Bardaro, C., Musielak, J., Vinti, G., Nonlinear Integral Operators and Applications, De Gruyter Series in Nonlinear Analysis and Applications, 9., Berlin: Walter de Gruyter & Company, 2003.
• Butzer, P. L., Nessel, R. J., Fourier analysis and approximation Vol. 1: One-dimensional Theory, Pure and Applied Mathematics, Vol. 40., New York-London: Academic Press, 1971.
• Costarelli, D., Vinti, G., Approximation by nonlinear multivariate sampling Kantorovich type operators and applications to image processing, Numer. Funct. Anal. Optim., 34(8) (2013), 819--844.
• Folland, G. B., Real Analysis: Modern Techniques and Their Applications, Pure and Applied Mathematics, New York: Wiley, 1999.
• Gadjiev, A. D., On nearness to zero of a family of nonlinear integral operators of Hammerstein, (Russian-Azerbaijani summary) Izv. Akad. Nauk Azerbaĭdžan. SSR Ser. Fiz.-Tehn. Mat. Nauk, 2 (1966), 32--34.
• Gadjiev, A. D., The order of convergence of singular integrals which depend on two parameters, (Russian), In: Special problems of functional analysis and their applications to the theory of differential equations and the theory of functions (Russian), (1968), 40--44.
• Karsli, H., On approximation properties of non-convolution type nonlinear integral operators, Anal. Theory Appl., 26(2) (2010), 140--152.
• Musielak, J., On some approximation problems in modular spaces, Constructive function theory '81 (Varna, 1981), Publ. House Bulgar. Acad. Sci., Sofia (1983), 455--461.
• Rudin, W., Real and Complex Analysis, Third edition., New York: McGraw-Hill Book Company, 1987.
• Serenbay, S. K., Yilmaz, M. M., Ibikli, E., The convergence of a family of integral operators (in Lp space) with a positive kernel, J. Comput. Appl. Math., 235(16) (2011), 4567--4575.
• Stein, E. M., Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30. Princeton-New Jersey: Princeton University Press, 1970.
• Stein, E. M., Weiss, G., Introduction to Fourier analysis on Euclidean Spaces, Princeton Mathematical Series, No. 32. Princeton-New Jersey: Princeton University Press, 1971.
• Świderski, T., Wachnicki, E., Nonlinear singular integrals depending on two parameters, Comment. Math. (Prace Mat.), 40 (2000), 181--189.
• Taberski, R., Singular integrals depending on two parameters, Prace Mat., 7 (1962), 173-179.
• Uysal, G., Mishra, V. N., Serenbay, S. K., Some weighted approximation properties of nonlinear double integral operators, Korean J. Math., 26(3) (2018), 483--501.
• Yilmaz, B., Jackson type generalization of nonlinear integral operators, J. Comput. Anal. Appl., 17(1) (2014), 77--83.
• Yilmaz, B., On some approximation properties of the Gauss-Weierstrass operators, Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat., 68(2) (2019), 2154--2160.
• Yilmaz, B., Ari, D. A., A note on the modified Picard integral operators, Math. Methods Appl. Sci., (2020), 1-6. https://doi.org/10.1002/mma.6360