Taiwanese Journal of Mathematics

A NOTE ON THE APPROXIMATION BY THE $q$-HYBRID SUMMATION INTEGRAL TYPE OPERATORS

Ülkü Dinlemez, İsmet Yüksel, and Birol Altın

Full-text: Open access

Abstract

In this study, it is introduced a $q-$Stancu type of hybrid summation integral type operators. It is investigated their approximation properties. It is given a weighted approximation theorem and obtained rates of convergence of these operators for continuous functions.

Article information

Source
Taiwanese J. Math., Volume 18, Number 3 (2014), 781-792.

Dates
First available in Project Euclid: 10 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499706440

Digital Object Identifier
doi:10.11650/tjm.18.2014.3801

Mathematical Reviews number (MathSciNet)
MR3213386

Zentralblatt MATH identifier
1357.41021

Subjects
Primary: 41A25: Rate of convergence, degree of approximation 41A36: Approximation by positive operators

Keywords
$q$-Stancu type operators $q$-Hybrid operators weighted approximation rates of approximation

Citation

Dinlemez, Ülkü; Yüksel, İsmet; Altın, Birol. A NOTE ON THE APPROXIMATION BY THE $q$-HYBRID SUMMATION INTEGRAL TYPE OPERATORS. Taiwanese J. Math. 18 (2014), no. 3, 781--792. doi:10.11650/tjm.18.2014.3801. https://projecteuclid.org/euclid.twjm/1499706440


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References

  • V. Gupta and E. Erkuş, On a hybrid family of summation integral type operators, J. Inequal. Pure Appl. Math., 7(1) (2006), Article 23.
  • J. Sinha and V. K. Singh, Rate of convergence on the mixed summation integral type operators, Gen. Math., 14(4) (2006), 29-36.
  • A. Lupaş, A $q-$analogue of the Bernstein operator, Seminar on Numerical and Statistical Calculus, University of Cluj-Napoca, 9 (1987), 85-92.
  • G. M. Phillips, Bernstein polynomials based on the $q$-integers, Ann. Numer. Math., 4 (1997), 511-518.
  • O. Do\ptmrs ğru and V. Gupta, Monotonicity and the asymptotic estimate of Bleimann Butzer and Hahn operators based on $q$-integers, Georgian Math. J., 12(3) (2005), 415-422.
  • O. Do\ptmrs ğru and V. Gupta, Korovkin-type approximation properties of bivariate $q$-Meyer-König and Zeller operators, Calcolo, 43(1) (2006), 51-63.
  • V. Gupta and W. Heping, The rate of convergence of $q-$Durrmeyer operators for $0<q<1$, Math. Methods Appl. Sci., 31(16) (2008), 1946-1955.
  • V. Gupta and A. Aral, Convergence of the $q$-analogue of Szá sz-beta operators, Appl. Math. Comput., 216(2) (2010), 374-380.
  • V. Gupta and H. Karsli, Some approximation properties by $q$ -Sz ász-Mirakyan-Baskakov-Stancu operators, Lobachevskii J. Math., 33(2) (2012), 175-182.
  • \ipI. Yüksel, Direct results on the q-mixed summation integral type operators, J. Appl. Funct. Anal., 8(2) (2013), 235-245.
  • F. H. Jackson, On $q$-definite integrals, quart. J. Pure Appl. Math., 41(15) (1910), 193-203.
  • H. T. Koelink and T. H. Koornwinder, $q-$special functions, a tutorial, Deformation Theory and Quantum Groups with Applications to Mathematical Physics, (Amherst, MA), 1990, pp. 141-142, Contemp. Math., 134, Amer. Math. Soc., Providence, RI, 1992.
  • V. G. Kac and P. Cheung, Quantum Calculus, Universitext, Springer-Verlag, New York, 2002.
  • A. De Sole and V. G. Kac, On integral representations of $q$-gamma and $q$-beta functions, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei, (9), Mat. Appl., 16(1) (2005), 11-29.
  • A. Aral, V. Gupta and R. P. Agarwal, Applications of $q$-Calculus in Operator Theory, Springer, New York, 2013.
  • R. A. De Vore and G. G. Lorentz, Constructive Approximation, Springer, Berlin, 1993.
  • A. D. Gadzhiev, Theorems of the type of P. P. Korovkin type theorems, Math. Zametki, 20(5) (1976), 781-786; English Translation, Math. Notes, 20(5/6) (1976), 996-998.
  • V. Gupta, G. S. Srivastava and A. Sahai, On simultaneous approximation by Szász-beta operators, Soochow J. Math., 21(1) (1995), 1-11.