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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

  • LupaƟ, 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, Ä°., 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, Ä°. 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.
  • Ä°spir, 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

  • LupaƟ, 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, Ä°., 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, Ä°. 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.
  • Ä°spir, 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

Ä°smet YĂŒksel

ÜlkĂŒ Dinlemez

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

Cite

APA YĂŒksel, Ä°., & Dinlemez, Ü. (2015). Weighted Approximation by the 𝒒 −SzĂĄsz−Schurer−Beta Type Operators. Gazi University Journal of Science, 28(2), 231-238.
AMA YĂŒksel Ä°, 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, Ä°smet, 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 Ä°, 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 Ä°. 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, Ä°smet - 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 Ä°, Dinlemez Ü. Weighted Approximation by the 𝒒 −SzĂĄsz−Schurer−Beta Type Operators. Gazi University Journal of Science. 2015;28:231–238.
MLA YĂŒksel, Ä°smet 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 Ä°, Dinlemez Ü. Weighted Approximation by the 𝒒 −SzĂĄsz−Schurer−Beta Type Operators. Gazi University Journal of Science. 2015;28(2):231-8.