Some approximation properties of Urysohn type nonlinear operators

Harun Karsli

Abstract


The central issue of this paper is to continue the investigation
of convergence properties of Urysohn type operators. By using Urysohn
type operators we will extend the theory of interpolation to functionals
and operators. In details, the present paper centers around Urysohn type
nonlinear counterpart of the two dimensional Stancu operators dened
on a triangle. We construct our nonlinear operators by using a nonlinear
forms of the kernels together with the two dimensional Urysohn type operator
values instead of the sampling values of the function. Afterwards,
we investigate the convergence problem for these nonlinear operators.


Keywords


Urysohn integral operators; Stancu operator; two dimensional nonlinear Stancu operators; Urysohn type nonlinear Stancu operators.

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References


Altomare, F. and Campiti, M. (1994), Korovkin-Type Approximation Theory and its Applications, De Gruyter Studies in Mathematics, 17, Walter de Gruyter and Co., Berlin.

Bardaro, C., Mantellini, I., On the reconstruction of functions by means of nonlinear discrete operators. J. Concr. Appl. Math. 1 (2003), no. 4, 273-285.

Bardaro, C., Mantellini, I., Approximation properties in abstract modular spaces for a class of general sampling-type operators. Appl. Anal. 85 (2006), no. 4, 383-413.

Bardaro, C., Vinti, G., Urysohn integral operators with homogeneous kernel: approximation properties in modular spaces. Comment. Math. (Prace Mat.) 42 (2002), no. 2, 145-182.

Bardaro, C., Karsli, H., Vinti, G., Nonlinear integral operators with homogeneous kernels: pointwise approximation theorems, Appl. Anal., Vol. 90, Nos. 3-4, March-April (2011), 463-474.

Bardaro, C., Musielak, J., Vinti, G., Nonlinear Integral Operators and Applications, De Gruyter Series in Nonlinear Analysis and Applications, Vol. 9, xii + 201 pp., 2003.

Bernstein S. N., Demonstration du Theoreme de Weierstrass fondee sur le calcul des probabilites, Comm. Soc. Math. Kharkow 13, (1912/13), 1-2.

Butzer, P. L., On Bernstein Polynomials, Ph.D. Thesis, University of Toronto in November, (1951).

Butzer, P. L., On two dimensional Bernstein polynomials, Canad. J. Math. 5 (1953) 107-113.

Butzer, P. L. and Nessel, R. J., Fourier Analysis and Approximation, V.1, Academic Press, New York, London, 1971.

Demkiv, I. I., On Approximation of the Urysohn operator by Bernstein type operator polynomials, Visn. L'viv. Univ., Ser. Prykl. Mat. Inform., (2000), Issue 2, 26 - 30.

Karsli, H., Approximation by Urysohn type Meyer-Konig and Zeller operators to Urysohn integral operators. Results Math. 72 (2017), no. 3, 1571-1583.

Karsli, H., Approximation results for Urysohn type nonlinear Bernstein operators, Advances in Summability and Approximation Theory, Book Chapter, Springer-Verlag, (2018), accepted.

Karsli, H., Voronovskaya-type theorems for Urysohn type nonlinear Bernstein operators, Mathematical Methods in the Applied Sciences, (2018), accepted.

Karsli, H., Approximation Results for Urysohn Type Two Dimensional Nonlinear Bernstein Operators, Const. Math. Anal., 1 (2018), No. 1, pp. 45-57.

Karsli, H., Altin, H. E., A Voronovskaya-type theorem for a certain nonlinear Bernstein operators. Stud. Univ. Babes-Bolyai Math. 60 (2015), no. 2, 249-258.

Karsli, H., Tiryaki, I. U.; Altin, H. E., On convergence of certain nonlinear Bernstein operators. Filomat 30 (2016), no. 1, 141-155.

Karsli H., Tiryaki I. U., Altin H. E., Some approximation properties of a certain nonlinear Bernstein operators, Filomat 28:6 (2014), 1295-1305.

Lorentz G.G., Bernstein Polynomials, University of Toronto Press,Toronto (1953).

Lupaş L., Lupaş A., Polynomials of binomial type and approximation operators, Studia Univ. Babes-Bolyai, Mathematica, 32, 1987, 61-69.

Makarov, V. L., Demkiv, I. I., Approximation of the Urysohn operator by operator polynomials of Stancu type, Ukrainian Math Journal, (2012), 64(3), 356 - 386.

Musielak J., 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.

Stancu, D.D. { Approximation of functions by a new class of linear polynomial operators, Rev. Roumaine Math. Pures Appl., 13 (1968), 1173-1194.

Stancu, D.D., A new class of uniform approximating polynomial operators in two and several variables.Proceedings of the Conference on the Constructive Theory of Functions (Approximation Theory) (Budapest, 1969), 443-455. Akademiai Kiado, Budapest, 1972.

Urysohn, P., On a type of nonlinear integral equation. Mat. Sb., (1924), 31, 236-255.

Zabreiko P. P., Koshelev A. I., Krasnosel'skii M. A., Mikhlin S. G., Rakovscik L. S.and Stetsenko V. Ja., Integral Equations: A Reference Text, Noordhoff Int. Publ., Leyden, 1975.




DOI: http://dx.doi.org/10.24193/subbmath.2019.2.05

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