Weighted Approximation by the 𝒒 −Szász−Schurer−Beta Type Operators

İsmet Yüksel; Ülkü Dinlemez

Research Article

Weighted Approximation by the 𝒒 −Szász−Schurer−Beta Type Operators

Year 2015, Volume: 28 Issue: 2, 231 - 238, 22.06.2015

İsmet Yüksel Ülkü Dinlemez

Abstract

In this study, we investigate approximation properties of a Schurer type generalization of q-Szász-beta type operators. We estimate the rate of weighted approximation of these operators for functions of polynomial growth on the interval [0,∞).

Keywords

q-Szász-Schurer-beta operators, q-mixed operators, weighted approximation, rates of approximation.

References

Lupaş, A., A −analogue of the Bernstein operator, Seminar on numerical and statistical calculus, University of Cluj-Napoca 9 (1987) 85-92.
Phillips, G. M.,Bernstein polynomials based on the q-integers, Ann. Numer. Math. 4 (1997) 511-518.
Doğru, O. and Gupta, V., Monotonicity and the asymptotic estimate of Bleimann Butzer and Hahn operators based on q -integers, Georgian Math. J. 12 (2005) (3) 415-422.
Doğru, O. and Gupta, V., Korovkin-type approximation properties of bivariate −Meyer- König and Zeller operators, Calcolo 43 (1) (2006) 51-63.
Gupta, V. and Aral, A., Convergence of the −analogue of Szász-beta operators, Appl. Math. Comput., 216 (2) (2010) 374-380.
Gupta, V. and Karslı, H., Some approximation properties by Szász -Mirakyan-Baskakov- Stancu operators, Lobachevskii J. Math. 33(2) (2012) 175-182.
Yüksel, İ., Approximation by −Phillips operators, Hacet. J. Math. Stat. 40 (2011) no. 2, 191-201.
Yüksel,·İ., Direct results on the −mixed summation integral type operators, J. Appl. Funct. Anal. (2) (2013) 235-245.
Dinlemez, Ü., Yüksel ·İ. and Altın, B., A note on the approximation by the −hybrid summation integral type operators, Taiwanese J. Math. 18(3) (2014) 781
Gupta, V. and Mahmudov, N. I., Approximation properties of the −Szasz-Mirakjan-Beta operators, Indian J. Industrial and Appl. Math. 3(2) (20012) 41-53.
Yüksel, İ. and Dinlemez, Ü., Voronovskaja type approximation theorem for −Szász-beta operators. Appl. Math. Comput. 235 (2014) 555-559.
Govil, N. K. and Gupta, V., −Beta-Szász-Stancu operators. Adv. Stud. Contemp. Math. 22(1) (2012) 123
Mahmudov, N. I., −Szász operators which preserve x2 . Slovaca 63(5) (2013) 1059-1072
Dinlemez, Ü., Convergence of the −Stancu- Szasz-beta type operators, J. Inequal. Appl. 2014, :354, 8 pp.
Jackson, F. H., On −definite integrals, quart. J. Pure Appl. Math., 41(15) (1910) 193-203.
Koelink, H. T. and Koornwinder, T. H., −special functions, a tutorial, Deformation theory and quantum groups with applications to mathematical physics (Amherst, MA, 1990) 141,142, Contemp. Math., 134,
Amer. Math. Soc., Providence, RI, 1992.
Kac, V. G. and Cheung, P., Quantum calculus, Universitext. Springer-Verlag, New York, 2002.
De Sole, A. and Kac, V. G., On integral representations of −gamma and −beta functions, Atti. Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 16(1) (2005) 11-29.
Aral, A., Gupta, V. and Agarwal, R. P., Applications of q-calculus in operator theory, Springer, New York, 2013.
Gupta, V., Srivastava, G. S. and Sahai, A., On simultaneous approximation by Szász-beta operators, Soochow J. Math. 21(1) (1995) 1-11.
Gupta V. and Agarwal, R. P., Convergence estimates in approximation theory. Springer, Cham, ISBN: 978-3-319-02764-7 2014.
Deo, N., Direct result on the Durrmeyer variant of Beta operators. Southeast Asian Bull. Math. 32(2) (2008) 283-290.
Deo, N., Direct result on exponential-type operators. Appl. Math. Comput. 204(1) (2008) 109-115
De Vore R. A. and Lorentz, G. G., Constructive Approximation, Springer, Berlin 1993.
Gadzhiev, A. D., Theorems of the type of P. P. Korovkin type theorems, Math. Zametki 20(5) (1976) 786; English Translation, Math. Notes, 20(5/6) (1976) 996-998.
İspir, N., On modified Baskakov operators on weighted spaces, Turkish J. Math. 25(3) (2001) 355

Year 2015, Volume: 28 Issue: 2, 231 - 238, 22.06.2015

İsmet Yüksel Ülkü Dinlemez

Abstract

References

Lupaş, A., A −analogue of the Bernstein operator, Seminar on numerical and statistical calculus, University of Cluj-Napoca 9 (1987) 85-92.
Phillips, G. M.,Bernstein polynomials based on the q-integers, Ann. Numer. Math. 4 (1997) 511-518.
Doğru, O. and Gupta, V., Monotonicity and the asymptotic estimate of Bleimann Butzer and Hahn operators based on q -integers, Georgian Math. J. 12 (2005) (3) 415-422.
Doğru, O. and Gupta, V., Korovkin-type approximation properties of bivariate −Meyer- König and Zeller operators, Calcolo 43 (1) (2006) 51-63.
Gupta, V. and Aral, A., Convergence of the −analogue of Szász-beta operators, Appl. Math. Comput., 216 (2) (2010) 374-380.
Gupta, V. and Karslı, H., Some approximation properties by Szász -Mirakyan-Baskakov- Stancu operators, Lobachevskii J. Math. 33(2) (2012) 175-182.
Yüksel, İ., Approximation by −Phillips operators, Hacet. J. Math. Stat. 40 (2011) no. 2, 191-201.
Yüksel,·İ., Direct results on the −mixed summation integral type operators, J. Appl. Funct. Anal. (2) (2013) 235-245.
Dinlemez, Ü., Yüksel ·İ. and Altın, B., A note on the approximation by the −hybrid summation integral type operators, Taiwanese J. Math. 18(3) (2014) 781
Gupta, V. and Mahmudov, N. I., Approximation properties of the −Szasz-Mirakjan-Beta operators, Indian J. Industrial and Appl. Math. 3(2) (20012) 41-53.
Yüksel, İ. and Dinlemez, Ü., Voronovskaja type approximation theorem for −Szász-beta operators. Appl. Math. Comput. 235 (2014) 555-559.
Govil, N. K. and Gupta, V., −Beta-Szász-Stancu operators. Adv. Stud. Contemp. Math. 22(1) (2012) 123
Mahmudov, N. I., −Szász operators which preserve x2 . Slovaca 63(5) (2013) 1059-1072
Dinlemez, Ü., Convergence of the −Stancu- Szasz-beta type operators, J. Inequal. Appl. 2014, :354, 8 pp.
Jackson, F. H., On −definite integrals, quart. J. Pure Appl. Math., 41(15) (1910) 193-203.
Koelink, H. T. and Koornwinder, T. H., −special functions, a tutorial, Deformation theory and quantum groups with applications to mathematical physics (Amherst, MA, 1990) 141,142, Contemp. Math., 134,
Amer. Math. Soc., Providence, RI, 1992.
Kac, V. G. and Cheung, P., Quantum calculus, Universitext. Springer-Verlag, New York, 2002.
De Sole, A. and Kac, V. G., On integral representations of −gamma and −beta functions, Atti. Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 16(1) (2005) 11-29.
Aral, A., Gupta, V. and Agarwal, R. P., Applications of q-calculus in operator theory, Springer, New York, 2013.
Gupta, V., Srivastava, G. S. and Sahai, A., On simultaneous approximation by Szász-beta operators, Soochow J. Math. 21(1) (1995) 1-11.
Gupta V. and Agarwal, R. P., Convergence estimates in approximation theory. Springer, Cham, ISBN: 978-3-319-02764-7 2014.
Deo, N., Direct result on the Durrmeyer variant of Beta operators. Southeast Asian Bull. Math. 32(2) (2008) 283-290.
Deo, N., Direct result on exponential-type operators. Appl. Math. Comput. 204(1) (2008) 109-115
De Vore R. A. and Lorentz, G. G., Constructive Approximation, Springer, Berlin 1993.
Gadzhiev, A. D., Theorems of the type of P. P. Korovkin type theorems, Math. Zametki 20(5) (1976) 786; English Translation, Math. Notes, 20(5/6) (1976) 996-998.
İspir, N., On modified Baskakov operators on weighted spaces, Turkish J. Math. 25(3) (2001) 355

There are 27 citations in total.

Details

Primary Language	English
Subjects	Engineering
Journal Section	Mathematics
Authors	İsmet Yüksel Ülkü Dinlemez
Publication Date	June 22, 2015
Published in Issue	Year 2015 Volume: 28 Issue: 2

Cite

APA	Yüksel, İ., & Dinlemez, Ü. (2015). Weighted Approximation by the 𝒒 −Szász−Schurer−Beta Type Operators. Gazi University Journal of Science, 28(2), 231-238.
AMA	Yüksel İ, Dinlemez Ü. Weighted Approximation by the 𝒒 −Szász−Schurer−Beta Type Operators. Gazi University Journal of Science. June 2015;28(2):231-238.
Chicago	Yüksel, İsmet, and Ülkü Dinlemez. “Weighted Approximation by the 𝒒 −Szász−Schurer−Beta Type Operators”. Gazi University Journal of Science 28, no. 2 (June 2015): 231-38.
EndNote	Yüksel İ, Dinlemez Ü (June 1, 2015) Weighted Approximation by the 𝒒 −Szász−Schurer−Beta Type Operators. Gazi University Journal of Science 28 2 231–238.
IEEE	İ. Yüksel and Ü. Dinlemez, “Weighted Approximation by the 𝒒 −Szász−Schurer−Beta Type Operators”, Gazi University Journal of Science, vol. 28, no. 2, pp. 231–238, 2015.
ISNAD	Yüksel, İsmet - Dinlemez, Ülkü. “Weighted Approximation by the 𝒒 −Szász−Schurer−Beta Type Operators”. Gazi University Journal of Science 28/2 (June 2015), 231-238.
JAMA	Yüksel İ, Dinlemez Ü. Weighted Approximation by the 𝒒 −Szász−Schurer−Beta Type Operators. Gazi University Journal of Science. 2015;28:231–238.
MLA	Yüksel, İsmet and Ülkü Dinlemez. “Weighted Approximation by the 𝒒 −Szász−Schurer−Beta Type Operators”. Gazi University Journal of Science, vol. 28, no. 2, 2015, pp. 231-8.
Vancouver	Yüksel İ, Dinlemez Ü. Weighted Approximation by the 𝒒 −Szász−Schurer−Beta Type Operators. Gazi University Journal of Science. 2015;28(2):231-8.

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