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kth Order Kantorovich Modification of Linking Baskakov-Type Operators

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Mathematical Analysis and its Applications

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 143))

Abstract

In 1957 Baskakov introduced a general method for the construction of positive linear operators depending on a real parameter c. The so-called genuine Baskakov–Durrmeyer-type operators form a class of operators reproducing the linear functions, interpolating at (finite) endpoints of the interval, and having other nice properties. In this paper we consider a nontrivial link between Baskakov-type operators and genuine Baskakov–Durrmeyer-type operators. We establish explicit representations for the images of monomials and for the moments; they are useful, e.g., in studying asymptotic formulas.

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References

  1. Baskakov, V.A.: An instance of a sequence of positive linear operators in the space of continuous functions. Dokl. Akad. Nauk SSSR 113(2), 249–251 (1957)

    MathSciNet  MATH  Google Scholar 

  2. Durrmeyer, J.L.: Une formule d’inversion de la transformée de Laplace: applications à la théorie des moments, Thèse de 3e cycle, Faculté des Sciences de l’Université de Paris, 1967

    Google Scholar 

  3. Gonska, H., Păltănea, R.: Simultaneous approximation by a class of Bernstein-Durrmeyer operators preserving linear functions. Czechoslovak Math. J. 60(135), 783–799 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Gonska, H., Păltănea, R.: Quantitative convergence theorems for a class of Bernstein-Durrmeyer operators preserving linear functions. Ukrainian Math. J. 62(7), 1061–1072 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Gould, H.W.: Combinatorial Identities: A Standardized Set of Tables Listing 500 Binomial Coefficient Summations. Henry W. Gould, Morgantown (1972)

    Google Scholar 

  6. Heilmann, M.: Direct and converse results for operators of Baskakov-Durrmeyer operators. Approx. Theory Appl. 5(1), 105–127 (1989)

    MathSciNet  MATH  Google Scholar 

  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. Heilmann, M., Raşa, I.: \(k\)-th order Kantorovich type modification of the operators \(U_n^\rho \). J. Appl. Funct. Anal. 9(3–4), 320–334 (2014)

    MathSciNet  Google Scholar 

  9. Lupaş, A.: Die Folge der Betaoperatoren, Dissertation, Universität Stuttgart (1972)

    Google Scholar 

  10. Mazhar, S.M., Totik, V.: Approximation by modified Szász operators. Acta Sci. Math. 49, 257–269 (1985)

    MathSciNet  MATH  Google Scholar 

  11. Păltănea, R.: A class of Durrmeyer type operators preserving linear functions. Ann. Tiberiu Popoviciu Semin. Funct. Equ. Approx. Convexity (Cluj-Napoca) 5, 109–117 (2007)

    Google Scholar 

  12. Păltănea, R.: Modified Szász-Mirakjan operators of integral form. Carpathian J. Math. 24(3), 378–385 (2008)

    MATH  Google Scholar 

  13. Păltănea, R.: Simultaneous approximation by a class of Szász-Mirakjan operators. J. Appl. Funct. Anal. 9(3–4), 356–368 (2014)

    MathSciNet  Google Scholar 

  14. Phillips, R.S.: An inversion formula for Laplace transforms and semi-groups of linear operators. Ann. Math. 59(2), 325–356 (1954)

    Google Scholar 

  15. Phillips, G.M.: Interpolation and Approximation by Polynomials. Springer (2003)

    Google Scholar 

  16. Sahai, A., Prasad, G.: On simultaneous approximation by modified Lupas operators. J. Approx. Theory 45, 122–128 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  17. Stanila, E.D.: On Bernstein-Euler-Jacobi operators, PhD Thesis University of Duisburg-Essen (2014)

    Google Scholar 

  18. Wagner, M.: Quasi-Interpolanten zu genuinen Baskakov-Durrmeyer-Typ Operatoren, Disssertation Universität Wuppertal (2013)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Faculty of Mathematics and Natural Sciences, University of Wuppertal, Gaußstraße 20, Wuppertal, 42119, Germany

    Margareta Heilmann

  2. Department of Mathematics, Technical University, Str. Memorandumului 28, Cluj-napoca, 400114, Romania

    Ioan Raşa

Authors
  1. Margareta Heilmann
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  2. Ioan Raşa
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Corresponding author

Correspondence to Margareta Heilmann .

Editor information

Editors and Affiliations

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

    P. N. Agrawal

  2. Department of Mathematics, University of Central Florida, Orlando, Florida, USA

    R. N. Mohapatra

  3. Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand, India

    Uaday Singh

  4. Dept. of Mathematics and Statistics, University of Victoria, Victoria, British Columbia, Canada

    H. M. Srivastava

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Heilmann, M., Raşa, I. (2015). kth Order Kantorovich Modification of Linking Baskakov-Type Operators. In: Agrawal, P., Mohapatra, R., Singh, U., Srivastava, H. (eds) Mathematical Analysis and its Applications. Springer Proceedings in Mathematics & Statistics, vol 143. Springer, New Delhi. https://doi.org/10.1007/978-81-322-2485-3_18

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