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

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

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,∞).

References

  • Lupaß, 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 SzĂĄsz-beta operators, Appl. Math. Comput., 216 (2) (2010) 374-380.
  • Gupta, V. and Karslı, H., Some approximation properties by SzĂĄsz -Mirakyan-Baskakov- Stancu operators, Lobachevskii J. Math. 33(2) (2012) 175-182.
  • YĂŒksel, İ., Approximation by −Phillips operators, Hacet. J. Math. Stat. 40 (2011) no. 2, 191-201.
  • YĂŒksel,·İ., Direct results on the −mixed summation integral type operators, J. Appl. Funct. Anal. (2) (2013) 235-245.
  • Dinlemez, Ü., YĂŒksel ·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.
  • YĂŒksel, İ. and Dinlemez, Ü., Voronovskaja type approximation theorem for −SzĂĄsz-beta operators. Appl. Math. Comput. 235 (2014) 555-559.
  • Govil, N. K. and Gupta, V., −Beta-SzĂĄsz-Stancu operators. Adv. Stud. Contemp. Math. 22(1) (2012) 123
  • Mahmudov, N. I., −SzĂĄsz 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 SzĂĄsz-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

Abstract

References

  • Lupaß, 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 SzĂĄsz-beta operators, Appl. Math. Comput., 216 (2) (2010) 374-380.
  • Gupta, V. and Karslı, H., Some approximation properties by SzĂĄsz -Mirakyan-Baskakov- Stancu operators, Lobachevskii J. Math. 33(2) (2012) 175-182.
  • YĂŒksel, İ., Approximation by −Phillips operators, Hacet. J. Math. Stat. 40 (2011) no. 2, 191-201.
  • YĂŒksel,·İ., Direct results on the −mixed summation integral type operators, J. Appl. Funct. Anal. (2) (2013) 235-245.
  • Dinlemez, Ü., YĂŒksel ·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.
  • YĂŒksel, İ. and Dinlemez, Ü., Voronovskaja type approximation theorem for −SzĂĄsz-beta operators. Appl. Math. Comput. 235 (2014) 555-559.
  • Govil, N. K. and Gupta, V., −Beta-SzĂĄsz-Stancu operators. Adv. Stud. Contemp. Math. 22(1) (2012) 123
  • Mahmudov, N. I., −SzĂĄsz 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 SzĂĄsz-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 YĂŒksel

ÜlkĂŒ Dinlemez

Publication Date June 22, 2015
Published in Issue Year 2015 Volume: 28 Issue: 2

Cite

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