Some convergence results for nonlinear singular integral operators

• Autorzy: Karsli, H.
• Języki publikacji: EN

Abstrakty

EN

In this paper, we establish some pointwise convergence results for a family of certain nonlinear singular integral operators Tλf of the form (...), acting on functions with bounded (Jordan) variation on an interval [a, b] as λ→λ0. Here, the kernels (...) satisfy some suitable singularity assumptions. We remark that the present study is a continuation and extension of the study of pointwise approximation of the family of nonlinear singular integral operators (1) begun in [18].

Słowa kluczowe

PL

zmienność ograniczona; warunek Lipschitza; zbieżność punktowa

EN

nonlinear singular integrals; bounded variation; Lipschitz condition; pointwise convergence

• Czasopismo: Demonstratio Mathematica
• Rocznik: 2013
• Tom: Vol. 46, nr 4
• Strony: 729--740
• Opis fizyczny: Bibliogr. 23 poz.

Twórcy

autor

Karsli, H.

• Abant Izzet Baysal University, Faculty of Science and Arts, Department of Mathematics, 14280 Golkoy Bolu,

Bibliografia

• [1] L. Angeloni, G. Vinti, Convergence in variation and rate of approximation for nonlinear integral operators of convolution type, Results Math. 49 (2006), 1–23. Erratum: 57 (2010), 387–391.
• [2] L. Angeloni, G. Vinti, Approximation by means of nonlinear integral operators in the space of functions with bounded -variation, Differential Integral Equations 20 (2007), 339–360. Erratum: 23(7–8) (2010), 795–799.
• [3] L. Angeloni, G. Vinti, Convergence and rate of approximation for linear integral operators in BV-spaces in multidimensional setting, J. Math. Anal. Appl. 349 (2009), 317–334.
• [4] C. Bardaro, J. Musielak, G. Vinti, Nonlinear Integral Operators and Applications, de Gruyter Ser. Nonlinear Anal. Appl. Vol. 9, 2003.
• [5] C. Bardaro, P. L. Butzer, R. L. Stens, G. Vinti, Convergence in variation and rates of approximation for Bernstein-type polynomials and singular convolution integrals, Analysis (Munich) 23(4) (2003), 299–346.
• [6] C. Bardaro, H. Karsli, G. Vinti, On pointwise convergence of linear integral operators with homogeneous kernels, Integral Transforms Spec. Funct. 19(6) (2008), 429–439.
• [7] C. Bardaro, H. Karsli, G. Vinti, Nonlinear integral operators with homogeneous kernels: pointwise approximation theorems, Appl. Anal. 90(3,4) (2011), 463–474.
• [8] C. Bardaro, I. Mantellini, Pointwise convergence theorems for nonlinear Mellin convolution operators, Int. J. Pure Appl. Math. 27(4) (2006), 431–447.
• [9] C. Bardaro, S. Sciamannini, G. Vinti, Convergence in BV by nonlinear Mellin-type convolution operators, Funct. Approx. 29 (2001), 17–28.
• [10] R. Bojanic, F. Cheng, Estimates for the rate of approximation of functions of bounded variation by Hermite-Fejér polynomials, Second Edmonton Conference on Approximation Theory, Edmonton, Alta., 1982, 5–17.
• [11] R. Bojanic, M. Vuilleumier, On the rate of convergence of Fourier Legendre series of functions of bounded variation, J. Approx. Theory 31 (1981), 67–79.
• [12] P. L. Butzer, R. J. Nessel, Fourier Analysis and Approximation, Vol. 1, Academic Press, New York, London, 1971.
• [13] F. Cheng, On the rate of convergence of Bernstein polynomials of functions of bounded variation, J. Approx. Theory 39 (1983), 259–274.
• [14] V. Gupta, M. K. Gupta, Rate of convergence for certain families of summation integral type operators, J. Math. Anal. Appl. 296 (2004), 608–618.
• [15] Y. H. Hua, S. Y. Shaw, Rate of approximation for functions of bounded variation by integral operators, Period. Math. Hungar. 46(1) (2003), 41–60.
• [16] H. Karsli, Convergence and rate of convergence by nonlinear singular integral operators depending on two parameters, Appl. Anal. 85(6,7) (2006), 781–791.
• [17] H. Karsli, On approximation properties of a class of convolution type nonlinear singular integral operators, Georgian Math. J. 15(1) (2008), 77–86.
• [18] H. Karsli, V. Gupta, Rate of convergence by nonlinear integral operators for functions of bounded variation, Calcolo 45(2) (2008), 87–99.
• [19] H. Karsli, E. Ibikli, On convergence of convolution type singular integral operators depending on two parameters, Fasc. Math. 38 (2007), 25–39.
• [20] J. Musielak, 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, 1983.
• [21] S. Y. Shaw, W. C. Liaw, Y. L. Lin, Rates for approximation of functions in BV [a, b] and DBV [a, b] by positive linear operators, Chinese J. Math. 21(2) (1993), 171–193.
• [22] T. Świderski, E. Wachnicki, Nonlinear singular integrals depending on two parameters, Comment. Math. Prace Mat. 15 (2000), 181–189.
• [23] X. M. Zeng, W. Chen, On the rate of convergence of the generalized Durrmeyer type operators for functions of bounded variation, J. Approx. Theory 102 (2000), 1–12