Approximation Results for Urysohn Type Two Dimensional Nonlinear Bernstein Operators

Harun Karslı

doi:10.33205/cma.453027

Research Article

Approximation Results for Urysohn Type Two Dimensional Nonlinear Bernstein Operators

Year 2018, Volume: 1 Issue: 1, 45 - 57, 15.09.2018

Harun Karslı

https://doi.org/10.33205/cma.453027

Cited By: 7

Abstract

In the present work, our aim of this study is generalization and extension of the theory of interpolation of two dimensional 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 in two dimensional case. 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 form of the kernels together with the Urysohn type operator values instead of the sampling values of the function.

Keywords

Urysohn integral operators, Urysohn type two dimensional nonlinear Bernstein operators, Nonlinear Bernstein operators, Urysohn type two dimensional nonlinear Bernstein operators

References

[1] Bardaro, C., Mantellini, I., A Voronovskaya-type theorem for a general classof discrete operators, Rocky Mountain J. Math. 39 (5) (2009) 1411-1442.
[2] Bardaro, C., Mantellini, I., Pointwise convergence theorems for nonlinearMellin convolution operators. Int. J. Pure Appl. Math. 27 (2006), no. 4,431-447.
[3] Bardaro, C., Mantellini, I., On the reconstruction of functions by means ofnonlinear discrete operators. J. Concr. Appl. Math. 1 (2003), no. 4, 273-285.
[4] Bardaro, C., Mantellini, I., Approximation properties in abstract modularspaces for a class of general sampling-type operators. Appl. Anal. 85 (2006),no. 4, 383-413.
[5] Bardaro, C., Vinti, G., Urysohn integral operators with homogeneous kernel:approximation properties in modular spaces. Comment. Math. (PraceMat.) 42 (2002), no. 2, 145-182.
[6] Bardaro, C., Musielak, J., Vinti, G., Nonlinear Integral Operators andApplications, De Gruyter Series in Nonlinear Analysis and Applications,Vol. 9, xii + 201 pp., 2003.
[7] Bardaro, C., Karsli, H. and Vinti, G., Nonlinear integral operators withhomogeneous kernels: pointwise approximation theorems, Applicable Analysis,Vol. 90, Nos. 3{4, March{April (2011), 463-474.
[8] Bardaro, C., Karsli, H. and Vinti, G., On pointwise convergence of linearintegral operators with homogeneous kernels , Integral Transforms andSpecial Functions, 19(6), (2008), 429-439.
[9] S. N Bernstein, Demonstration du Th ̆eoreme de Weierstrass fond ̆ee sur le calcul des probabilit ̆es, Comm. Soc. Math. Kharkow 13, (1912/13), 1
[10] Bojanic R. and Cheng F., Rate of convergence of Bernstein polynomials forfunctions with derivatives of bounded variation, J. Math. Anal.and Appl.,141, (1989), 136-151.
[11] P. L. Butzer, On two dimensional Bernstein polynomials, Canad. J. Math.5 (1953) 107-113.
[12] P. L. Butzer, On Bernstein Polynomials, Ph.D. Thesis, University of Toronto in November, (1951).
[13] Costarelli, D., Vinti, G., Degree of approximation for nonlinear multivariatesampling Kantorovich operators on some functions spaces. Numer. Funct.Anal. Optim. 36 (2015), no. 8, 964-990.
[14] Costarelli, D., Vinti, G., Approximation by nonlinear multivariate samplingKantorovich type operators and applications to image processing. Numer.Funct. Anal. Optim. 34 (2013), no. 8, 819-844.
[15] Demkiv I. On approximation of Urison operator with operator polynomialsof Bernstein type // Visn. Lviv univ. Ser. appl. math. and inform. 2000.Vol. 2. p.26-30.
[16] Demkiv I. On approximation of Urison operator with operator polynomialsof Bernstein type in the case two variables // Visn. Lviv polytechnicNational University. Ser. appl. math. 2000. p.111-115.
[17] DeVore R.A., Lorentz G.G., Constructive Approximation, Springer, NewYork, 1993.
[18] Karsli, H., Some convergence results for nonlinear singular integral operators,Demonstratio. Math., Vol. XLVI No 4, 729-740 (2013).
[19] Karsli, H., Convergence and rate of convergence by nonlinear singular integraloperators depending on two parameters, Appl. Anal. 85(6,7), (2006),781-791.
[20] Karsli, H., Tiryaki, I. U.; Altin, H. E., On convergence of certain nonlinearBernstein operators. Filomat 30 (2016), no. 1, 141-155.
[21] H. Karsli, H.E. Altin, A Voronovskaya-type theorem for a certain nonlinear Bernstein operators. Stud. Univ. Babe ̧s- Bolyai Math. 60 (2015), no. 2, 249–258
[22] H. Karsli, H.E. Altin, Convergence of certain nonlinear counterpart of the Bernstein operators. Commun. Fac. Sci. Univ. Ank. Sér. A1 Math. Stat. 64 (2015), no. 1, 75–86
[23] Karsli H., Tiryaki I. U., Altin H. E., Some approximation properties of acertain nonlinear Bernstein operators, Filomat 28:6 (2014), 1295-1305.
[24] H. Karsli, Approximation by Urysohn type Meyer-König and Zeller operators to Urysohn integral operators. Results Math. 72 (2017), no. 3, 1571–15
[25] Karsli, H., Approximation results for Urysohn type nonlinear Bernstein operators,Advances in Summability and Approximation Theory, Book Chapter,Springer-Verlag, (2018), accepted.
[26] Karsli, H., Voronovskaya-type theorems for Urysohn type nonlinear Bernsteinoperators, Mathematical Methods in the Applied Sciences, (2018), accepted.
[27] Lorentz G.G., Bernstein Polynomials, University of Toronto Press,Toronto(1953).
[28] V. L. Makarov and V.V. Khlobystov, On the identification of Nonlinear operators and its application, BEM IX, (1987), No. 1, 43 - 58.
[29] 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)
[30] Urysohn, P. (1923). Sur une classe d'equations integrales non lineaires, Mat.Sb., 31, 236 - 255.
[31] Urysohn, P.: On a type of nonlinear integral equation. Mat. Sb., 31 236{355(1924)
[32] Wazwaz, A. M. (2011). Linear and Nonlinear Integral Equations: Methodsand Applications, Beijing and Springer - Verlag Berlin, Higher Educationpress.
[33] P. P. Zabreiko, A.I. Koshelev, M.A. Krasnosel’skii, S.G. Mikhlin, L.S. Rakovscik and V.Ja. Stetsenko, Integral Equa- tions: A Reference Text, Noordhoff Int. Publ., Leyden, (1975)

Year 2018, Volume: 1 Issue: 1, 45 - 57, 15.09.2018

Harun Karslı

https://doi.org/10.33205/cma.453027

Cited By: 7

Abstract

References

[1] Bardaro, C., Mantellini, I., A Voronovskaya-type theorem for a general classof discrete operators, Rocky Mountain J. Math. 39 (5) (2009) 1411-1442.
[2] Bardaro, C., Mantellini, I., Pointwise convergence theorems for nonlinearMellin convolution operators. Int. J. Pure Appl. Math. 27 (2006), no. 4,431-447.
[3] Bardaro, C., Mantellini, I., On the reconstruction of functions by means ofnonlinear discrete operators. J. Concr. Appl. Math. 1 (2003), no. 4, 273-285.
[4] Bardaro, C., Mantellini, I., Approximation properties in abstract modularspaces for a class of general sampling-type operators. Appl. Anal. 85 (2006),no. 4, 383-413.
[5] Bardaro, C., Vinti, G., Urysohn integral operators with homogeneous kernel:approximation properties in modular spaces. Comment. Math. (PraceMat.) 42 (2002), no. 2, 145-182.
[6] Bardaro, C., Musielak, J., Vinti, G., Nonlinear Integral Operators andApplications, De Gruyter Series in Nonlinear Analysis and Applications,Vol. 9, xii + 201 pp., 2003.
[7] Bardaro, C., Karsli, H. and Vinti, G., Nonlinear integral operators withhomogeneous kernels: pointwise approximation theorems, Applicable Analysis,Vol. 90, Nos. 3{4, March{April (2011), 463-474.
[8] Bardaro, C., Karsli, H. and Vinti, G., On pointwise convergence of linearintegral operators with homogeneous kernels , Integral Transforms andSpecial Functions, 19(6), (2008), 429-439.
[9] S. N Bernstein, Demonstration du Th ̆eoreme de Weierstrass fond ̆ee sur le calcul des probabilit ̆es, Comm. Soc. Math. Kharkow 13, (1912/13), 1
[10] Bojanic R. and Cheng F., Rate of convergence of Bernstein polynomials forfunctions with derivatives of bounded variation, J. Math. Anal.and Appl.,141, (1989), 136-151.
[11] P. L. Butzer, On two dimensional Bernstein polynomials, Canad. J. Math.5 (1953) 107-113.
[12] P. L. Butzer, On Bernstein Polynomials, Ph.D. Thesis, University of Toronto in November, (1951).
[13] Costarelli, D., Vinti, G., Degree of approximation for nonlinear multivariatesampling Kantorovich operators on some functions spaces. Numer. Funct.Anal. Optim. 36 (2015), no. 8, 964-990.
[14] Costarelli, D., Vinti, G., Approximation by nonlinear multivariate samplingKantorovich type operators and applications to image processing. Numer.Funct. Anal. Optim. 34 (2013), no. 8, 819-844.
[15] Demkiv I. On approximation of Urison operator with operator polynomialsof Bernstein type // Visn. Lviv univ. Ser. appl. math. and inform. 2000.Vol. 2. p.26-30.
[16] Demkiv I. On approximation of Urison operator with operator polynomialsof Bernstein type in the case two variables // Visn. Lviv polytechnicNational University. Ser. appl. math. 2000. p.111-115.
[17] DeVore R.A., Lorentz G.G., Constructive Approximation, Springer, NewYork, 1993.
[18] Karsli, H., Some convergence results for nonlinear singular integral operators,Demonstratio. Math., Vol. XLVI No 4, 729-740 (2013).
[19] Karsli, H., Convergence and rate of convergence by nonlinear singular integraloperators depending on two parameters, Appl. Anal. 85(6,7), (2006),781-791.
[20] Karsli, H., Tiryaki, I. U.; Altin, H. E., On convergence of certain nonlinearBernstein operators. Filomat 30 (2016), no. 1, 141-155.
[21] H. Karsli, H.E. Altin, A Voronovskaya-type theorem for a certain nonlinear Bernstein operators. Stud. Univ. Babe ̧s- Bolyai Math. 60 (2015), no. 2, 249–258
[22] H. Karsli, H.E. Altin, Convergence of certain nonlinear counterpart of the Bernstein operators. Commun. Fac. Sci. Univ. Ank. Sér. A1 Math. Stat. 64 (2015), no. 1, 75–86
[23] Karsli H., Tiryaki I. U., Altin H. E., Some approximation properties of acertain nonlinear Bernstein operators, Filomat 28:6 (2014), 1295-1305.
[24] H. Karsli, Approximation by Urysohn type Meyer-König and Zeller operators to Urysohn integral operators. Results Math. 72 (2017), no. 3, 1571–15
[25] Karsli, H., Approximation results for Urysohn type nonlinear Bernstein operators,Advances in Summability and Approximation Theory, Book Chapter,Springer-Verlag, (2018), accepted.
[26] Karsli, H., Voronovskaya-type theorems for Urysohn type nonlinear Bernsteinoperators, Mathematical Methods in the Applied Sciences, (2018), accepted.
[27] Lorentz G.G., Bernstein Polynomials, University of Toronto Press,Toronto(1953).
[28] V. L. Makarov and V.V. Khlobystov, On the identification of Nonlinear operators and its application, BEM IX, (1987), No. 1, 43 - 58.
[29] 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)
[30] Urysohn, P. (1923). Sur une classe d'equations integrales non lineaires, Mat.Sb., 31, 236 - 255.
[31] Urysohn, P.: On a type of nonlinear integral equation. Mat. Sb., 31 236{355(1924)
[32] Wazwaz, A. M. (2011). Linear and Nonlinear Integral Equations: Methodsand Applications, Beijing and Springer - Verlag Berlin, Higher Educationpress.
[33] P. P. Zabreiko, A.I. Koshelev, M.A. Krasnosel’skii, S.G. Mikhlin, L.S. Rakovscik and V.Ja. Stetsenko, Integral Equa- tions: A Reference Text, Noordhoff Int. Publ., Leyden, (1975)

There are 33 citations in total.

Details

Primary Language	English
Subjects	Mathematical Sciences
Journal Section	Articles
Authors	Harun Karslı
Publication Date	September 15, 2018
Published in Issue	Year 2018 Volume: 1 Issue: 1

Cite

APA	Karslı, H. (2018). Approximation Results for Urysohn Type Two Dimensional Nonlinear Bernstein Operators. Constructive Mathematical Analysis, 1(1), 45-57. https://doi.org/10.33205/cma.453027
AMA	Karslı H. Approximation Results for Urysohn Type Two Dimensional Nonlinear Bernstein Operators. CMA. September 2018;1(1):45-57. doi:10.33205/cma.453027
Chicago	Karslı, Harun. “Approximation Results for Urysohn Type Two Dimensional Nonlinear Bernstein Operators”. Constructive Mathematical Analysis 1, no. 1 (September 2018): 45-57. https://doi.org/10.33205/cma.453027.
EndNote	Karslı H (September 1, 2018) Approximation Results for Urysohn Type Two Dimensional Nonlinear Bernstein Operators. Constructive Mathematical Analysis 1 1 45–57.
IEEE	H. Karslı, “Approximation Results for Urysohn Type Two Dimensional Nonlinear Bernstein Operators”, CMA, vol. 1, no. 1, pp. 45–57, 2018, doi: 10.33205/cma.453027.
ISNAD	Karslı, Harun. “Approximation Results for Urysohn Type Two Dimensional Nonlinear Bernstein Operators”. Constructive Mathematical Analysis 1/1 (September 2018), 45-57. https://doi.org/10.33205/cma.453027.
JAMA	Karslı H. Approximation Results for Urysohn Type Two Dimensional Nonlinear Bernstein Operators. CMA. 2018;1:45–57.
MLA	Karslı, Harun. “Approximation Results for Urysohn Type Two Dimensional Nonlinear Bernstein Operators”. Constructive Mathematical Analysis, vol. 1, no. 1, 2018, pp. 45-57, doi:10.33205/cma.453027.
Vancouver	Karslı H. Approximation Results for Urysohn Type Two Dimensional Nonlinear Bernstein Operators. CMA. 2018;1(1):45-57.