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Published in:
A Brief History of the Favard Operator and Its Variants Read chapter Read first chapter

2016 | OriginalPaper | Chapter

Bivariate Extension of Linear Positive Operators

Authors : P. N. Agrawal, Meenu Goyal

Published in: Mathematical Analysis, Approximation Theory and Their Applications

Publisher: Springer International Publishing

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Abstract

The goal of this chapter is to present a survey of the literature on approximation of functions of two variables by linear positive operators. We study the approximation properties of these operators in the space of functions of two variables, continuous on a compact set. We also discuss the convergence of the operators in a weighted space of functions of two variables and find the rate of this convergence by means of modulus of continuity.

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previous chapter A Brief History of the Favard Operator and Its Variants
next chapter Positive Green’s Functions for Boundary Value Problems with Conformable Derivatives
Literature
1.
Abel, U.: On the asymptotic approximation with bivariate operators of Bleimann, Butzer and Hahn. J. Approx. Theory 97(1), 181–198 (1999)MathSciNetCrossRefMATH
2.
Adell, J.A., de la Cal, J., Miguel, M.S.: On the property of monotonic convergence for multivariate Bernstein-type operators. J. Approx. Theory 80(1), 132–137 (1995)MathSciNetCrossRefMATH
3.
Agratini, O.: Bivariate positive operators in polynomial weighted spaces. Abstr. Appl. Anal. 2013, Art. ID 850760, 8 pp. (2013)
4.
Agrawal, P.N., Goyal, M.: Generalized Baskakov Kantorovich operators FILOMAT. Accepted
5.
Agrawal, P.N., Finta, Z., Sathish Kumar, A.: Bivariate q-Bernstein-Schurer-Kantorovich operators. Results Math. 67(3–4), 365–380 (2015)MathSciNetCrossRefMATH
6.
Altomare, F., Campiti, M.: Korovkin-Type Approximation Theory and Its Applications. Walter de Gruyter, Berlin (1994)CrossRefMATH
7.
Andrews, G.E., Askey, R., Roy, R.: Special Functions. Cambridge University Press, Cambridge (1999)CrossRefMATH
8.
Aral, A., Gupta, V., Agrawal, R.P.: Applications of q-Calculus in Operator Theory. Springer, New York (2013)CrossRefMATH
9.
Atakut, Ç., İspir, N.: On Bernstein type rational functions of two variables. Math. Slovaca 54(3), 291–301 (2004)MathSciNetMATH
10.
Bărbosu, D.: Aproximarea funcţiilor de mai multe variabile prin sume booleene de operatori liniari de tip interpolator. RISOPRINT, Cluj-Napoca (2002) (Romanian)
11.
Bărbosu, D.: Bivariate operators of Schurer-Stancu type. An. Ştiinţ. Univ. Ovidius Constanţa Ser Mat. 11(1), 1–8 (2003)MathSciNetMATH
12.
Baskakov, V.A.: An instance of a sequence of linear positive operators in the space of continuous functions. Dokl. Akad. Nauk. SSSR 113, 249–251 (1957)MathSciNetMATH
13.
Becker, M., Kucharski, D., Nessel, R.J.: Global approximation theorems for the Szàsz-Mirakyan operators in exponential weight spaces. In: Linear Spaces and Approximation (Proceedings of the Conference, Oberwolfach, 1977), vol. 40, pp. 319–333. Birkhäuser, Basel (1978)
14.
15.
Butzer, P.L., Berens, H.: Semi-Groups of Operators and Approximation, xi+318 pp. Springer, New York (1967)CrossRefMATH
16.
Căbulea, L.A., Aldea, M.: On some properties of Kantorovich bivariate operators. Proceedings of the International Conference on Theory and Applications of Mathematics and Informatics - ICTAMI. Alba Iulia (2003)MATH
17.
Doğru, O., Gupta, V.: Korovkin-type approximation properties of bivariate q-Meyer-König and Zeller operators. Calcolo 43(1), 51–63 (2006)MathSciNetCrossRefMATH
18.
Erencin, A.: Durrmeyer type modification of generalized Baskakov operators. Appl. Math. Comput. 218(8), 4384–4390 (2011)MathSciNetMATH
19.
Ernst, T.: The history of q-calculus and a new method, U.U.D.M. Report 2000:16. Department of Mathematics, Uppsala University, Uppsala (2000)
20.
Ersan, S.: Approximation properties of bivariate generalization of Bleimann, Butzer and Hahn operators Based on the q-integers. International Conference on Applied Mathematics, Cairo, Egypt, pp. 29–31 (2007)
21.
Gadjiev, A.D.: Positive linear operators in weighted spaces of functions of several variables. Izv. Akad. Nauk Azerb. SSR, Fiz.-Tekhh. Mat. Nauk 1(4), 32–37 (1980)
22.
Gadjiev, A.D., Ghorbanalizadeh, A.M.: Approximation properties of a new type Bernstein-Stancu polynomials of one and two variables. Appl. Math. Comput. 216(3), 890–901 (2010)MathSciNetMATH
23.
24.
Goyal, M., Agrawal, P.N.: Bivariate generalized Baskakov-Kantorovich operators. (submitted)
25.
Gurdek, M., Rempulska, L., Skorupka, M.: The Baskakov operators for functions of two variables. Collect. Math. 50(3), 289–302 (1999)MathSciNetMATH
26.
Ipatov, A.F.: Estimation of error and order of approximation of functions of two variables by Bernstein polynomials, (in Russian). Uč Zap. Petrozavodsk Univ. 4(4), 31–48 (1955)MathSciNet
27.
28.
Kantorovich, L.V.: Sur certains développements suivant les polynômials de la formede S. Bernstein I, II. C. R. Acad. Sci. USSR 20, 563–568, 595–600 (1930)
29.
Khan, R.A.: A bivariate extension of Bleimann-Butzer-Hahn operator. Approx. Theory Appl. 18(1), 90–100 (2002)MathSciNetMATH
30.
Kingsley, E.H.: Bernstein polynomials for functions of two variables of class C (k). Proc. Am. Math. Soc. 2(1), 64–71 (1951)MathSciNetMATH
31.
Lupas, A.: A q-analogue of the Bernstein operator. In: Seminar on Numerical and Statistical Calculus, University of Cluj-Napoca, vol. 9, pp. 85–92 (1987)MathSciNetMATH
32.
Marinkovic, S., Rajkovic, P., Stankovic, M.: The inequalities for some types of q-integrals. Comput. Math. Appl. 56(10), 2490–2498 (2008)MathSciNetCrossRefMATH
33.
34.
Mihesan, V.: Uniform approximation with positive linear operators generated by generalized Baskakov method. Autom. Comput. Appl. Math. 7(1), 34–37 (1998)MathSciNet
35.
Örkcü, M.: Approximation properties of bivariate extension of q-Szász-Mirakjan-Kantorovich operators. J. Inequal. Appl. 324, 1–10 (2013)MATH
36.
Örkcü, M.: q-Szász-Mirakjan-Kantorovich operators of Functions of two variables in polynomial weighted spaces. Hindawi Publ. Corp. Abstr. Appl. Anal. 2013 Art. ID 823803, 9 pp. (2013)
37.
Phillips, G.M.: Bernstein polynomials based on the q-integers. The heritage of P. L. Chebyshev: a Festschrift in honour of the 70th-birthday of Prof. T. J. Rivlin.. Ann. Numer. Math. 4, 511–518 (1997)
38.
Pop, O.T.: The generalization of Voronovskaja’s theorem for a class of bivariate operators. Stud. Univ. Babes-Bolyai Math. 53(2), 85–107 (2008)MathSciNetMATH
39.
Pop, O.T., Fărcaş, M.D.: About the bivariate operators of Kantorovich type. Acta Math. Univ. Comen. 78(1), 43–52 (2009)MathSciNetMATH
40.
Rempulska, L., Graczyk, S.: On certain class of Szàsz-Mirakyan operators in exponential weight spaces. Int. J. Pure Appl. Math. 60(3), 259–267 (2010)MathSciNetMATH
41.
Rempulska, L., Graczyk, S.: On Generalized Sz\(\acute{a}\) sz-Mirakyan operators of functions of two variables. Math. Slovaca 62(1), 87–98 (2012)MathSciNetCrossRefMATH
42.
43.
Stancu, D.D.: Approximation of functions by a new class of linear polynomial operators. Rev. Roum. Math. Pures Appl. 13(8), 1173–1194 (1968)MathSciNetMATH
44.
Stancu, F.: Aproximarea funcţiilor de două şi mai multe variabile cu ajutorul operatorilor liniari şi pozitivi. Ph.D. Thesis, Cluj-Napoca (1984) (Romanian)
45.
Volkov, V.I.: On the convergence sequences of linear positive operators in the space of continuous functions of two variables. Dokl. Akad. Nauk SSSR 115, 17–19 (1957) (Russian)MathSciNetMATH
46.
Wafi, A., Khatoon, S.: Approximation by generalized Baskakov operators for functions of one and two variables in exponential and polynomial weight spaces. Thai. J. Math. 2, 53–66 (2004)MATH
47.
Wafi, A., Khatoon, S.: Convergence and Voronovskaja-type theorems for derivatives of generalized Baskakov operators. Cen. Eur. J. Math. 6(2), 325–334 (2008)MathSciNetCrossRefMATH
Metadata
Title
Bivariate Extension of Linear Positive Operators
Authors
P. N. Agrawal
Meenu Goyal
Copyright Year
2016
Book
Mathematical Analysis, Approximation Theory and Their Applications

Print ISBN: 978-3-319-31279-8

Electronic ISBN: 978-3-319-31281-1

Copyright Year: 2016

DOI
https://doi.org/10.1007/978-3-319-31281-1_2

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