Abstract
In the present work, our aim of this study is generalization and extension of the theory of interpolation of functions to functionals or operators by means of Urysohn-type nonlinear operators. In accordance with this purpose, we introduce and study a new type of Urysohn-type nonlinear operators. In particular, we investigate the convergence problem for nonlinear operators that approximate the Urysohn-type operator. The starting point of this study is motivated by the important applications that approximation properties of certain families of nonlinear operators have in signal–image reconstruction and in other related fields. We construct our nonlinear operators by using a nonlinear forms of the kernels together with the Urysohn-type operator values instead of the sampling values of the function. As far as we know, this will be first use of such kind of operators in the theory of interpolation and approximation. Hence, the present study is a generalization and extension of some previous results.
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References
S.N. Bernstein, Demonstration du Thĕoreme de Weierstrass fondĕe sur le calcul des probabilitĕs. Commun. Soc. Math. Kharkow 13, 1–2 (1912/13)
G.G. Lorentz, Bernstein Polynomials (University of Toronto Press, Toronto, 1953)
J. Musielak, On some approximation problems in modular spaces, in Constructive Function Theory 1981, (Proc. Int. Conf., Varna, June 1-5, 1981) (Publ. House Bulgarian Acad. Sci., Sofia, 1983), pp. 455–461
C. Bardaro, J. Musielak, G. Vinti, Nonlinear Integral Operators and Applications, De Gruyter series in nonlinear analysis and applications, vol 9 (2003), p. xii + 201
C. Bardaro, I. Mantellini, A Voronovskaya-type theorem for a general class of discrete operators. Rocky Mountain J. Math. 39(5), 1411–1442 (2009)
C. Bardaro, I. Mantellini, Pointwise convergence theorems for nonlinear Mellin convolution operators. Int. J. Pure Appl. Math. 27(4), 431–447 (2006)
C. Bardaro, I. Mantellini, On the reconstruction of functions by means of nonlinear discrete operators. J. Concr. Appl. Math. 1(4), 273–285 (2003)
C. Bardaro, I. Mantellini, Approximation properties in abstract modular spaces for a class of general sampling-type operators. Appl. Anal. 85(4), 383–413 (2006)
C. Bardaro, G. Vinti, Urysohn integral operators with homogeneous kernel: approximation properties in modular spaces. Comment. Math. (Prace Mat.) 42(2), 145–182 (2002)
D. Costarelli, G. Vinti, Degree of approximation for nonlinear multivariate sampling Kantorovich operators on some functions spaces. Numer. Funct. Anal. Optim. 36(8), 964–990 (2015)
D. Costarelli, G. Vinti, Approximation by nonlinear multivariate sampling Kantorovich type operators and applications to image processing. Numer. Funct. Anal. Optim. 34(8), 819–844 (2013)
P.P. Zabreiko, A.I. Koshelev, M.A. Krasnosel’skii, S.G. Mikhlin, L.S. Rakovscik, VJa Stetsenko, Integral Equations: A Reference Text (Noordhoff Int. Publ., Leyden, 1975)
C. Bardaro, H. Karsli, G. Vinti, Nonlinear integral operators with homogeneous kernels: pointwise approximation theorems. Appl. Anal. 90(3–4), 463–474 (2011)
C. Bardaro, H. Karsli, G. Vinti, On pointwise convergence of linear integral operators with homogeneous kernels. Integral Transform. Special Funct. 19(6), 429–439 (2008)
H. Karsli, Some convergence results for nonlinear singular integral operators. Demonstr. Math. XLVI(4), 729–740 (2013)
H. Karsli, Convergence and rate of convergence by nonlinear singular integral operators depending on two parameters. Appl. Anal. 85(6,7), 781–791 (2006)
H. Karsli, I.U. Tiryaki, H.E. Altin, Some approximation properties of a certain nonlinear Bernstein operators. Filomat 28(6), 1295–1305 (2014)
H. Karsli, H.E. Altin, Convergence of certain nonlinear counterpart of the Bernstein operators. Commun. Fac. Sci. Univ. Ank. Sé r. A1 Math. Stat. 64(1), 75–86 (2015)
H. Karsli, H.E. Altin, A Voronovskaya-type theorem for a certain nonlinear Bernstein operators. Stud. Univ. Babeş-Bolyai Math. 60(2), 249–258 (2015)
H. Karsli, I.U. Tiryaki, H.E. Altin, On convergence of certain nonlinear Bernstein operators. Filomat 30(1), 141–155 (2016)
H. Karsli, Approximation by Urysohn type Meyer-König and Zeller operators to Urysohn integral operators. Results Math. 72(3), 1571–1583 (2017)
P. Urysohn, Sur une classe d’equations integrales non lineaires. Mat. Sb. 31, 236–255 (1923)
P. Urysohn, On a type of nonlinear integral equation. Mat. Sb. 31, 236–355 (1924)
A.M. Wazwaz, Linear and Nonlinear Integral Equations: Methods and Applications (Higher Education press, Beijing, 2011)
I.I. Demkiv, On Approximation of the Urysohn operator by Bernstein type operator polynomials. Visn. L’viv. Univ., Ser. Prykl. Mat. Inform. (2), 26–30 (2000)
V.L. Makarov, I.I. Demkiv, Approximation of the Urysohn operator by operator polynomials of Stancu type. Ukrainian Math. J. 64(3), 356–386 (2012)
R.A. DeVore, G.G. Lorentz, Constructive Approximation (Springer, New York, 1993)
R. Bojanic, F. Cheng, Rate of convergence of Bernstein polynomials for functions with derivatives of bounded variation. J. Math. Anal. Appl. 141, 136–151 (1989)
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Karsli, H. (2018). Approximation Results for Urysohn-Type Nonlinear Bernstein Operators. In: Mohiuddine, S., Acar, T. (eds) Advances in Summability and Approximation Theory. Springer, Singapore. https://doi.org/10.1007/978-981-13-3077-3_14
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