Skip to main content
SpringerLink
Log in

Bivariate Extension of Linear Positive Operators

  • Chapter
  • First Online:
Mathematical Analysis, Approximation Theory and Their Applications

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 111))

  • 1493 Accesses

Abstract

The goal of this chapter is to present a survey of the literature on approximation of functions of two variables by linear positive operators. We study the approximation properties of these operators in the space of functions of two variables, continuous on a compact set. We also discuss the convergence of the operators in a weighted space of functions of two variables and find the rate of this convergence by means of modulus of continuity.

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. Abel, U.: On the asymptotic approximation with bivariate operators of Bleimann, Butzer and Hahn. J. Approx. Theory 97(1), 181–198 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  2. Adell, J.A., de la Cal, J., Miguel, M.S.: On the property of monotonic convergence for multivariate Bernstein-type operators. J. Approx. Theory 80(1), 132–137 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  3. Agratini, O.: Bivariate positive operators in polynomial weighted spaces. Abstr. Appl. Anal. 2013, Art. ID 850760, 8 pp. (2013)

    Google Scholar 

  4. Agrawal, P.N., Goyal, M.: Generalized Baskakov Kantorovich operators FILOMAT. Accepted

    Google Scholar 

  5. Agrawal, P.N., Finta, Z., Sathish Kumar, A.: Bivariate q-Bernstein-Schurer-Kantorovich operators. Results Math. 67(3–4), 365–380 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Altomare, F., Campiti, M.: Korovkin-Type Approximation Theory and Its Applications. Walter de Gruyter, Berlin (1994)

    Book  MATH  Google Scholar 

  7. Andrews, G.E., Askey, R., Roy, R.: Special Functions. Cambridge University Press, Cambridge (1999)

    Book  MATH  Google Scholar 

  8. Aral, A., Gupta, V., Agrawal, R.P.: Applications of q-Calculus in Operator Theory. Springer, New York (2013)

    Book  MATH  Google Scholar 

  9. Atakut, Ç., İspir, N.: On Bernstein type rational functions of two variables. Math. Slovaca 54(3), 291–301 (2004)

    MathSciNet  MATH  Google Scholar 

  10. Bărbosu, D.: Aproximarea funcţiilor de mai multe variabile prin sume booleene de operatori liniari de tip interpolator. RISOPRINT, Cluj-Napoca (2002) (Romanian)

    Google Scholar 

  11. Bărbosu, D.: Bivariate operators of Schurer-Stancu type. An. Ştiinţ. Univ. Ovidius Constanţa Ser Mat. 11(1), 1–8 (2003)

    MathSciNet  MATH  Google Scholar 

  12. Baskakov, V.A.: An instance of a sequence of linear positive operators in the space of continuous functions. Dokl. Akad. Nauk. SSSR 113, 249–251 (1957)

    MathSciNet  MATH  Google Scholar 

  13. Becker, M., Kucharski, D., Nessel, R.J.: Global approximation theorems for the Szàsz-Mirakyan operators in exponential weight spaces. In: Linear Spaces and Approximation (Proceedings of the Conference, Oberwolfach, 1977), vol. 40, pp. 319–333. Birkhäuser, Basel (1978)

    Google Scholar 

  14. Butzer, P.L.: On two dimensional Bernstein polynomials. Can. J. Math. 5, 107–113 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  15. Butzer, P.L., Berens, H.: Semi-Groups of Operators and Approximation, xi+318 pp. Springer, New York (1967)

    Book  MATH  Google Scholar 

  16. Căbulea, L.A., Aldea, M.: On some properties of Kantorovich bivariate operators. Proceedings of the International Conference on Theory and Applications of Mathematics and Informatics - ICTAMI. Alba Iulia (2003)

    MATH  Google Scholar 

  17. Doğru, O., Gupta, V.: Korovkin-type approximation properties of bivariate q-Meyer-König and Zeller operators. Calcolo 43(1), 51–63 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Erencin, A.: Durrmeyer type modification of generalized Baskakov operators. Appl. Math. Comput. 218(8), 4384–4390 (2011)

    MathSciNet  MATH  Google Scholar 

  19. Ernst, T.: The history of q-calculus and a new method, U.U.D.M. Report 2000:16. Department of Mathematics, Uppsala University, Uppsala (2000)

    Google Scholar 

  20. Ersan, S.: Approximation properties of bivariate generalization of Bleimann, Butzer and Hahn operators Based on the q-integers. International Conference on Applied Mathematics, Cairo, Egypt, pp. 29–31 (2007)

    Google Scholar 

  21. Gadjiev, A.D.: Positive linear operators in weighted spaces of functions of several variables. Izv. Akad. Nauk Azerb. SSR, Fiz.-Tekhh. Mat. Nauk 1(4), 32–37 (1980)

    Google Scholar 

  22. Gadjiev, A.D., Ghorbanalizadeh, A.M.: Approximation properties of a new type Bernstein-Stancu polynomials of one and two variables. Appl. Math. Comput. 216(3), 890–901 (2010)

    MathSciNet  MATH  Google Scholar 

  23. Gauchman, H.: Integral inequalities in q-calculus. Comput. Math. Appl. 47(2–3), 281–300 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  24. Goyal, M., Agrawal, P.N.: Bivariate generalized Baskakov-Kantorovich operators. (submitted)

    Google Scholar 

  25. Gurdek, M., Rempulska, L., Skorupka, M.: The Baskakov operators for functions of two variables. Collect. Math. 50(3), 289–302 (1999)

    MathSciNet  MATH  Google Scholar 

  26. Ipatov, A.F.: Estimation of error and order of approximation of functions of two variables by Bernstein polynomials, (in Russian). Uč Zap. Petrozavodsk Univ. 4(4), 31–48 (1955)

    MathSciNet  Google Scholar 

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

    Book  MATH  Google Scholar 

  28. Kantorovich, L.V.: Sur certains développements suivant les polynômials de la formede S. Bernstein I, II. C. R. Acad. Sci. USSR 20, 563–568, 595–600 (1930)

    Google Scholar 

  29. Khan, R.A.: A bivariate extension of Bleimann-Butzer-Hahn operator. Approx. Theory Appl. 18(1), 90–100 (2002)

    MathSciNet  MATH  Google Scholar 

  30. Kingsley, E.H.: Bernstein polynomials for functions of two variables of class C (k). Proc. Am. Math. Soc. 2(1), 64–71 (1951)

    MathSciNet  MATH  Google Scholar 

  31. Lupas, A.: A q-analogue of the Bernstein operator. In: Seminar on Numerical and Statistical Calculus, University of Cluj-Napoca, vol. 9, pp. 85–92 (1987)

    MathSciNet  MATH  Google Scholar 

  32. Marinkovic, S., Rajkovic, P., Stankovic, M.: The inequalities for some types of q-integrals. Comput. Math. Appl. 56(10), 2490–2498 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  33. Martinez, F.L.: Some properties of two dimensional Bernstein polynomials. J. Approx. Theory 59(3), 300–306 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  34. Mihesan, V.: Uniform approximation with positive linear operators generated by generalized Baskakov method. Autom. Comput. Appl. Math. 7(1), 34–37 (1998)

    MathSciNet  Google Scholar 

  35. Örkcü, M.: Approximation properties of bivariate extension of q-Szász-Mirakjan-Kantorovich operators. J. Inequal. Appl. 324, 1–10 (2013)

    MATH  Google Scholar 

  36. Örkcü, M.: q-Szász-Mirakjan-Kantorovich operators of Functions of two variables in polynomial weighted spaces. Hindawi Publ. Corp. Abstr. Appl. Anal. 2013 Art. ID 823803, 9 pp. (2013)

    Google Scholar 

  37. Phillips, G.M.: Bernstein polynomials based on the q-integers. The heritage of P. L. Chebyshev: a Festschrift in honour of the 70th-birthday of Prof. T. J. Rivlin.. Ann. Numer. Math. 4, 511–518 (1997)

    Google Scholar 

  38. Pop, O.T.: The generalization of Voronovskaja’s theorem for a class of bivariate operators. Stud. Univ. Babes-Bolyai Math. 53(2), 85–107 (2008)

    MathSciNet  MATH  Google Scholar 

  39. Pop, O.T., Fărcaş, M.D.: About the bivariate operators of Kantorovich type. Acta Math. Univ. Comen. 78(1), 43–52 (2009)

    MathSciNet  MATH  Google Scholar 

  40. Rempulska, L., Graczyk, S.: On certain class of Szàsz-Mirakyan operators in exponential weight spaces. Int. J. Pure Appl. Math. 60(3), 259–267 (2010)

    MathSciNet  MATH  Google Scholar 

  41. Rempulska, L., Graczyk, S.: On Generalized Sz\(\acute{a}\) sz-Mirakyan operators of functions of two variables. Math. Slovaca 62(1), 87–98 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  42. Stancu, D.D.: A method for obtaining polynomials of Bernstein type of two variables. Am. Math. Mon. 70(3), 260–264 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  43. Stancu, D.D.: Approximation of functions by a new class of linear polynomial operators. Rev. Roum. Math. Pures Appl. 13(8), 1173–1194 (1968)

    MathSciNet  MATH  Google Scholar 

  44. Stancu, F.: Aproximarea funcţiilor de două şi mai multe variabile cu ajutorul operatorilor liniari şi pozitivi. Ph.D. Thesis, Cluj-Napoca (1984) (Romanian)

    Google Scholar 

  45. Volkov, V.I.: On the convergence sequences of linear positive operators in the space of continuous functions of two variables. Dokl. Akad. Nauk SSSR 115, 17–19 (1957) (Russian)

    MathSciNet  MATH  Google Scholar 

  46. Wafi, A., Khatoon, S.: Approximation by generalized Baskakov operators for functions of one and two variables in exponential and polynomial weight spaces. Thai. J. Math. 2, 53–66 (2004)

    MATH  Google Scholar 

  47. Wafi, A., Khatoon, S.: Convergence and Voronovskaja-type theorems for derivatives of generalized Baskakov operators. Cen. Eur. J. Math. 6(2), 325–334 (2008)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The second author is thankful to the “Council of Scientific and Industrial Research” (Grant code: 09/143(0836)/2013-EMR-1) India for financial support to carry out the above research work.

Author information

Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, 247667, India

    P. N. Agrawal & Meenu Goyal

Authors
  1. P. N. Agrawal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Meenu Goyal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Meenu Goyal .

Editor information

Editors and Affiliations

  1. Department of Mathematics, National Technical Unversity of Athens, Athens, Greece

    Themistocles M. Rassias

  2. Department of Mathematics, Netaji Subhas Instit. of Tech., New Delhi, India

    Vijay Gupta

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Agrawal, P.N., Goyal, M. (2016). Bivariate Extension of Linear Positive Operators. In: Rassias, T., Gupta, V. (eds) Mathematical Analysis, Approximation Theory and Their Applications. Springer Optimization and Its Applications, vol 111. Springer, Cham. https://doi.org/10.1007/978-3-319-31281-1_2

Download citation

Publish with us

Policies and ethics

Search

Navigation