Skip to main content
SpringerLink
Log in

Korovkin-Type Approximation Theorem for Functions of Two Variables Via Statistical Summability (C, 1, 1)

  • Chapter
  • First Online:
Analytic Number Theory, Approximation Theory, and Special Functions

Abstract

The concept of statistical summability (C, 1, 1) has recently been introduced by Moricz (J. Math. Anal. Appl. 286:340–350, 2003). In this paper, we use this notion of summability to prove the Korovkin-type approximation theorem for functions of two variables.

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
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover 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. Anastassiou, G.A., Mursaleen, M., Mohiuddine, S.A.: Some approximation theorems for functions of two variables through almost convergence of double sequences. J. Comput. Anal. Appl. 13, 37–40 (2011)

    MATH  MathSciNet  Google Scholar 

  2. Belen, C., Mursaleen, M., Yildirim, M.: Statistical A-summability of double sequences and a Korovkin type approximation theorem. Bull. Korean Math. Soc. 49, 851–861 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  3. Demirci, K., Karakuş, S.: Korovkin-type approximation theorem for double sequences of positive linear operators via statistical A-summability. Results Math. doi:10.1007/s00025-011-0140-y

    Google Scholar 

  4. Dirik, F., Demirci, K.: A Korovkin type approximation theorem for double sequences of positive linesr operators of two variables in A-statistical sense. Bull. Korean Math. Soc. 47, 825–837 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  5. Dirik, F., Demirci, K.: Korovkin type approximation theorem for functions of two variables in statistical sense. Turk. J. Math. 34, 73–83 (2010)

    MATH  MathSciNet  Google Scholar 

  6. Edely, O.H.H., Mursaleen, M.: Tauberian theorems for statistically convergent double sequences. Inform. Sci. 176, 875–886 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  7. Edely, O.H.H., Mohiuddine, S.A., Noman, A.K.: Korovkin type approximation theorems obtained through generalized statistical convergence. Appl. Math. Lett. 23, 1382–1387 (2010)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  10. Korovkin, P.P.: Linear operators and approximation theory. Hindustan Publishing Co., Delhi (1960)

    Google Scholar 

  11. Kumar, V., Mursaleen, M.: On (λ, μ)-statistical convergence of double sequences on intuitionistic fuzzy normed spaces. Filomat 25, 109–120 (2011)

    Article  MathSciNet  Google Scholar 

  12. Mohiuddine, S.A.: An application of almost convergence in approximation theorems. Appl. Math. Lett. 24,1856–1860 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  13. Mohiuddine, S.A., Alotaibi, A., Mursaleen, M.: Statistical summability (C, 1) and a Korovkin type approximation theorem. J. Inequal. Appl. 2012, 172 (2012)

    Article  MathSciNet  Google Scholar 

  14. Móricz, F.: Tauberian theorems for double sequences that are statistically summability (C, 1, 1). J. Math. Anal. Appl. 286, 340–350 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  15. Mursaleen, M., Alotaibi, A.: Statistical summability and approximation by de la Vallée-Pousin mean. Appl. Math. Lett. 24, 320–324 (2011) [Erratum: Appl. Math. Lett. 25, 665 (2012)]

    Google Scholar 

  16. Mursaleen, M., Edely, O.H.H.: Statistical convergence of double sequences. J. Math. Anal. Appl. 288, 223–231 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  17. Mursaleen, M., Mohiuddine, S.A.: Statistical convergence of double sequences in intuitionistic fuzzy normed spaces. Chaos Solitons Fract. 41, 2414–2421 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  18. Mursaleen, M., Çakan, C., Mohiuddine, S.A., Savaş, E.: Generalized statistical convergence and statitical core of double sequences. Acta Math. Sinica 26, 2131–2144 (2010)

    Article  MATH  Google Scholar 

  19. Mursaleen, M., Karakaya, V., Ertürk, M., Gürsoy, F.: Weighted statistical convergence and its application to Korovkin type approximation theorem. Appl. Math. Comput. 218, 9132–9137 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  20. Pringsheim, A.: Zur theorie der zweifach unendlichen Zahlenfolgen. Math. Z. 53, 289–321 (1900)

    MATH  MathSciNet  Google Scholar 

  21. Srivastava, H.M., Mursaleen, M., Khan, A.: Generalized equi-statistical convergence of positive linear operators and associated approximation theorems. Math. Comput. Model. 55, 2040–2051 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  22. Taşdelen, F., Erençin, A.: The generalization of bivariate MKZ operators by multiple generating functions. J. Math. Anal. Appl. 331, 727–735 (2007)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Aligarh Muslim University, Aligarh, 202002, India

    M. Mursaleen

  2. Department of Mathematics, King Abdulaziz University, 80203, Jeddah, 21589, Saudi Arabia

    S. A. Mohiuddine

Authors
  1. M. Mursaleen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. A. Mohiuddine
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Mursaleen .

Editor information

Editors and Affiliations

  1. Mathematical Institute, Serbian Academy of Sciences and Arts, Belgrade, Serbia

    Gradimir V. Milovanović

  2. Department of Mathematics, ETH Zürich, Zürich, Switzerland

    Michael Th. Rassias

Additional information

Dedicated to Professor Hari M. Srivastava

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer Science+Business Media New York

About this chapter

Cite this chapter

Mursaleen, M., Mohiuddine, S.A. (2014). Korovkin-Type Approximation Theorem for Functions of Two Variables Via Statistical Summability (C, 1, 1). In: Milovanović, G., Rassias, M. (eds) Analytic Number Theory, Approximation Theory, and Special Functions. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0258-3_20

Download citation

Publish with us

Policies and ethics

Search

Navigation