Skip to main content
SpringerLink
Log in

Statistical Approximation of the q-Bernstein-Durrmeyer Type Operators

  • Conference paper
  • First Online:
Fuzzy Systems & Operations Research and Management

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 367))

Abstract

In 2012, a kind of q-Bernstein-Durrmeyer type operators is introduced, and some approximate properties of these operators are studied by Ren. In this paper the statistical approximation properties of these operators are investigated. The Korovkin type statistical convergence theorem of these operators is established. Then the rates of statistical convergence of these operators are also studied by means of modulus of continuity and the help of functions of the Lipschitz class.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Phillips, G.M.: Bernstein polynomials based on the \(q\)-integers. Ann. Numer. Math. 4, 511–518 (1997)

    MathSciNet  MATH  Google Scholar 

  2. Gupta, V., Finta, Z.: On certain \(q\)-Durrmeyer type operators. Appl. Math. Comput. 209, 415–420 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Örkcü, M., Doğru, O.: \(q\)-Szász-Mirakyan-Kantorovich type operators preserving some test functions. Appl. Math. Lett. 24, 1588–1593 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Muraru, C.V.: Note on \(q\)-Bernstein-Schurer operators. Stud. Univ. Babeş-Bolyai Math. 56, 489–495 (2011)

    MathSciNet  Google Scholar 

  5. Doǧru, O., Gupta, V.: Korovkin-type approximation properties of bivariate q-Meyer-König and Zeller operators. Calcolo 43(1), 51–63 (2012)

    Google Scholar 

  6. Cai, Q.B., Zeng, X.M.: On the convergence of a modified q-Gamma operators. J. Comput. Anal. Appl. 15(5), 826–832 (2013)

    MathSciNet  MATH  Google Scholar 

  7. Dalmanoǧlu, Ö., Doǧru, O.: On statistical approximation properties of Kantorovich type q-Bernstein operators. Math. Comput. Model. 52(5–6), 760–771 (2010)

    Article  Google Scholar 

  8. Gupta, V., Radu, C.: Statistical approximation properties of q-Baskokov-Kantorovich operators. Cent. Eur. J. Math. 7(4), 809–818 (2009)

    MathSciNet  MATH  Google Scholar 

  9. Örkcü, M., Doǧru, O.: Weighted statistical approximation by Kantorovich type q-Szász-Mirakjan operators. Appl. Math. Comput. 217(20), 7913–7919 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ersan, S., Doǧru, O.: Statistical approximation properties of q-Bleimann, Butzer and Hahn operators. Math. Comput. Model. 49(7–8), 1595–1606 (2009)

    Article  MATH  Google Scholar 

  11. Mahmudov, N, Sabancigil, P.: A q-analogue of the Meyer-König and Zeller operators. Bull. Malays. Math. Sci. Soc. (2), 35(1), 39–51 (2012)

    Google Scholar 

  12. Ren, M.Y., Zeng, X.M.: On statistical approximation properties of modified q-Bernstein-Schurer operators. Bull. Korean Math. Soc. 50(4), 1145–1156 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ren, M Y: Approximation properties of the q-Bernstein-Durrmeyer type operators. Fuzzy Syst. Math. 26(5), 107–112 (2012) (Chinese)

    Google Scholar 

  14. Kac, V.G., Cheung, P.: Quantum Calculus. Universitext. Springer, New York (2002)

    Google Scholar 

  15. Gasper, G, Rahman, M.: Basic Hypergeometric Series. Encyclopedia of Mathematics and its Applications, vol. 35. Cambridge University Press, Cambridge (1990)

    Google Scholar 

  16. Fast, H.: Sur la convergence statistique. Collog. Math. 2, 241–244 (1951)

    MathSciNet  MATH  Google Scholar 

  17. Niven, I., Zuckerman, H.S., Montgomery, H.: An Introduction to the Theory of Numbers. Wiley, New york (1991)

    Google Scholar 

  18. Doǧru O.: On statistical approximation properties of Stancu type bivariate generalization of \(q\)-Balás-Szabados operators. In: Seminar on Numerical Analysis and Approximation Theory, Cluj-Napoca, Univ. Babeş-Bolya, pp. 179–194 (2006)

    Google Scholar 

  19. Gadjiev, A.D., Orhan, C.: Some approximation theorems via statistical convergence. Rocky Mt. J. Math. 32, 129–138 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  20. King, J.P.: Positive linear operators which preserve \(x^{2}\). Acta Math. Hungar. 99(3), 203–208 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. Mahmudov, N.I.: q-Szasz-Mirakjan operators which preserve \(x^{2}\). J. Comput. Appl. Math. 235, 4621–4628 (2011)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

This work is supported by the National Natural Science Foundation of China (No. 61170324), the Class A Science and Technology Project of Education Department of Fujian Province of China (No. JA12324), and the Natural Science Foundation of Fujian Province of China (No. 2013J01017 and No. 2014J01021).

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Wuyi University, Wuyishan, 354300, China

    Mei-Ying Ren

Authors
  1. Mei-Ying Ren
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mei-Ying Ren .

Editor information

Editors and Affiliations

  1. Guangzhou University, Guangzhou, China

    Bing-Yuan Cao

  2. Foreign Affairs Office, National Defense University, Beijing, China

    Zeng-Liang Liu

  3. Higher Education Mega Center, Guangzhou University, Guangzhou, China

    Yu-Bin Zhong

  4. Zhuhai Campus, Beijing Normal University, Zhuhai, China

    Hong-Hai Mi

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Ren, MY. (2016). Statistical Approximation of the q-Bernstein-Durrmeyer Type Operators. In: Cao, BY., Liu, ZL., Zhong, YB., Mi, HH. (eds) Fuzzy Systems & Operations Research and Management. Advances in Intelligent Systems and Computing, vol 367. Springer, Cham. https://doi.org/10.1007/978-3-319-19105-8_11

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-19105-8_11

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-19104-1

  • Online ISBN: 978-3-319-19105-8

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation