A Nice Representation for a Link Between Baskakov- and Szász–Mirakjan–Durrmeyer Operators and Their Kantorovich Variants

  • Margareta HeilmannEmail author
  • Ioan Raşa


In this paper we consider a link between Baskakov–Durrmeyer type operators and corresponding Kantorovich type modifications of their classical variants. We prove a useful representation for Kantorovich variants of arbitrary order for integer values of the linking parameter which leads to a simple proof of convexity properties for the linking operators. This also solves an open problem mentioned in Baumann et al. (Results Math. 69(3):297–315, 2016). Another open problem is presented at the end of the paper.


Linking operators Baskakov–Durrmeyer type operators Kantorovich modifications of operators 

Mathematics Subject Classification

41A36 41A10 41A28 



  1. 1.
    Baumann, K., Heilmann, M., Raşa, I.: Further results for \(k\)th order Kantorovich modification of linking Baskakov type operators. Results Math. 69(3), 297–315 (2016)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Gonska, H., Păltănea, R.: Quantitative convergence theorems for a class of Bernstein–Durrmeyer operators preserving linear functions. Ukr. Math. J. 62(7), 1061–1072 (2010)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Gonska, H., Păltănea, R.: Simultaneous approximation by a class of Bernstein–Durrmeyer operators preserving linear functions. Czechoslov. Math. J. 60(135), 783–799 (2010)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Gonska, H., Raşa, I., Stanila, E.-D.: The eigenstructure of operators linking the Bernstein and the genuine Bernstein–Durrmeyer operators. Mediterr. J. Math. 11, 561–576 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Gonska, H., Raşa, I., Stanila, E.-D.: Lagrange-type operators associated with \(U_n^\rho \). Publ. Inst. Math. (Beograd) (N.S.) 96(110), 159168 (2014)CrossRefGoogle Scholar
  6. 6.
    Heilmann, M.: Direct and converse results for operators of Baskakov–Durrmeyer operators. Approx. Theory Appl. 5(1), 105–127 (1989)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Heilmann, M.: Erhöhung der Konvergenzgeschwindigkeit bei der Approximation von Funktionen mit Hilfe von Linearkombinationen spezieller positiver linearer Operatoren. Habilitationschrift Universität Dortmund (1992)Google Scholar
  8. 8.
    Heilmann, M., Raşa, I.: \(k\)-th order Kantorovich modification of linking Baskakov type operators. In: Agrawal, P.N. et al. (eds.) Recent Trends in Mathematical Analysis and its Applications, Proceedings of the Conference ICRTMAA 2014, Rorkee, India, December 2014, Proceedings in Mathematics and Statistics 143, pp. 229–242 (2015)Google Scholar
  9. 9.
    Heilmann, M., Raşa, I.: A nice representation for a link between Bernstein-Durrmeyer and Kantorovich operators. In: Giri, D. et al. (eds.) Proceedings of ICMC 2017, Haldia, India, January 2017, Springer Proceedings in Mathematics and Computing, pp. 312–320 (2017)Google Scholar
  10. 10.
    Mazhar, S.M., Totik, V.: Approximation by modified Szász operators. Acta Sci. Math. 49, 257–269 (1985)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Păltănea, R.: A class of Durrmeyer type operators preserving linear functions. Ann. Tiberiu Popoviciu Sem. Funct. Eq. Approx. Conv. (Cluj-Napoca) 5, 109–117 (2007)zbMATHGoogle Scholar
  12. 12.
    Păltănea, R.: Modified Szász–Mirakjan operators of integral form. Carpath. J. Math. 24(3), 378–385 (2008)zbMATHGoogle Scholar
  13. 13.
    Păltănea, R.: Simultaneous approximation by a class of Szász–Mirakjan operators. J. Appl. Funct. Anal. 9(3–4), 356–368 (2014)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Phillips, R.S.: An inversion formula for Laplace transforms and semi-groups of linear operators. Ann. Math. 59(2), 325–356 (1954)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Schumaker, L.L.: Spline Functions: Basic Theory. Cambridge University Press, Cambridge (2007)CrossRefGoogle Scholar
  16. 16.
    Wagner, M.: Quasi-Interpolanten zu genuinen Baskakov–Durrmeyer–Typ Operatoren. Disssertation Universität Wuppertal (2013)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.School of Mathematics and Natural SciencesUniversity of WuppertalWuppertalGermany
  2. 2.Department of MathematicsTechnical UniversityCluj-NapocaRomania

Personalised recommendations