Discrete Approximation Processes of King’s Type

  • Octavian Agratini
  • Tudor Andrica
Part of the Springer Optimization and Its Applications book series (SOIA, volume 35)


This survey paper is focused on linear positive operators having the degree of exactness null and fixing the monomial of the second degree. The starting point is represented by J.P. King’s paper appearing in 2003. Our first aim is to sum up results obtained in the past five years. The second aim is to present a general class of discretizations following the features of the operators introduced by King.


Bernstein Polynomial Linear Positive Operator Bernstein Operator Weighted Modulus Baskakov Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    O. Agratini, Linear operators that preserve some test functions, International Journal of Mathematics and Mathematical Sciences, Vol. 2006, Article ID 94136, pp. 11, DOI 10.1155/IJMMS.Google Scholar
  2. 2.
    O. Agratini, On a class of linear positive bivariate operators of King type, Studia Univ. “Babeş-Bolyai”, Mathematica, 51(2006), f. 4, 13–22.MATHMathSciNetGoogle Scholar
  3. 3.
    O. Agratini, On the iterates of a class of summation-type linear positive operators, Computers Mathematics with Applications, 55(2008), 1178–1180.MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    D. Cárdenas-Morales, P. Garrancho, F.J. Mu noz-Delgado, Shape preserving approximation by Bernstein-type operators which fix polynomials, Applied Mathematics and Computation, 182(2006), 1615–1622.MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    E. Censor, Quantitative results for positive linear approximation operators, J. Approx. Theory, 4(1971), 442–450.MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    O. Duman, C. Orhan, An abstract version of the Korovkin approximation theorem, Publ. Math. Debrecen, 69(2006), f. 1-2, 33–46.MATHMathSciNetGoogle Scholar
  7. 7.
    O. Duman, M.A. Özarslan, Szász-Mirakjan type operators providing a better error estimation, Applied Math. Letters, 20(2007), 1184–1188.MATHCrossRefGoogle Scholar
  8. 8.
    A.D. Gadjiev, C. Orhan, Some approximation theorems via statistical convergence, Rocky Mountain J. Math., 32(2002), 129–138.MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    H. Gonska, P. Piţul, Remarks on an article of J.P. King, Schriftenreihe des Fachbereichs Mathematik, SM-DU-596, 2005, Universität Duisburg-Essen, 1–8.Google Scholar
  10. 10.
    J.P. King, Positive linear operators which preserve x 2, Acta Math. Hungar., 99(2003), f. 3, 203–208.MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    E. Kolk, Matrix summability of statistical convergent sequences, Analysis, 13(1993), 77–83.MATHMathSciNetGoogle Scholar
  12. 12.
    S. Ostrovska, The first decade of the q-Bernstein polynomials: results and perspectives, Journal of Mathematical Analysis and Approximation Theory, 2(2007), Number 1, 35–51.MATHMathSciNetGoogle Scholar
  13. 13.
    G.M. Phillips, Bernstein polynomials based on the q-integers, Ann. Numer. Math., 4(1997), 511–518.MATHMathSciNetGoogle Scholar
  14. 14.
    L. Rempulska, K. Tomczak, Approximation by certain linear operators preserving x 2, Turk. J. Math., 32(2008), 1–11.Google Scholar
  15. 15.
    V.I. Volkov, On the convergence of sequences of linear positive operators in the space of continuous functions of two variables (in Russian), Dokl. Akad. Nauk SSSR (N.S.), 115(1957), 17–19.MATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceBabeş-Bolyai UniversityCluj-NapocaRomania

Personalised recommendations