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Power series of the operators \(U_n^{\varrho }\)

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Abstract

We study power series of members of a class of positive linear operators reproducing linear function constituting a link between genuine Bernstein-Durrmeyer and classical Bernstein operators. Using the eigenstructure of the operators we give a non-quantitative convergence result towards the inverse Voronovskaya operators. We include a quantitative statement via a smoothing approach.

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References

  1. Abel, U.: Geometric series of Bernstein-Durrmeyer operators. East J. Approx. 15, 439–450 (2009)

    MathSciNet  Google Scholar 

  2. Abel, U., Ivan, M., Păltănea, R.: Geometric series of Bernstein operators revisited. J. Math. Anal. Appl. 400, 22–24 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  3. Abel, U., Ivan, M., Păltănea, R.: Geometric series of positive linear operators and inverse Voronovskaya theorem, arXiv:1304.5721, 21 Apr. 2013

  4. Cooper, S., Waldron, S.: The eigenstructure of the Bernstein operator. J. Approx. Theory 105, 133–165 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. Gonska, H., Kovacheva, R.K.: The second order modulus revisited: remarks, applications, problems. Conferenze del seminario di matematica dell’universita di Bari, vol. 257 (1994)

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

    Article  MATH  MathSciNet  Google Scholar 

  7. Gonska, H., Păltănea, R.: Quantitative convergence theorems for a class of Bernstein-Durrmeyer operators preserving linear functions. Ukrainian Math. J. 62, 913–922 (2010)

    Article  MATH  Google Scholar 

  8. Gonska, H., Raşa, I., Stănilă, E.: The eigenstructure of operators linking the Bernstein and the genuine Bernstein-Durrmeyer operators. Mediterr. J. Math. (2013). doi:10.1007/s00009-013-0347-0

  9. Păltănea, R.: The power series of Bernstein operators. Autom. Comput. Appl. Math. 15, 247–253 (2006)

    Google Scholar 

  10. Păltănea, R.: A class of Durrmeyer type operators preserving linear functions. Ann. Tiberiu Popoviciu Sem. Funct. Equat. Approxim. Convex. (Cluj-Napoca) 5, 109–117 (2007)

    MATH  Google Scholar 

  11. Raşa, I.: Power series of Bernstein operators and approximation of resolvents. Mediterr. J. Math. 9, 635–644 (2012)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Faculty of Mathematics, University of Duisburg-Essen, Forsthausweg 2, 47057 , Duisburg, Germany

    Heiner Gonska & Elena Dorina Stănilă

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

    Ioan Raşa

Authors
  1. Heiner Gonska
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  2. Ioan Raşa
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  3. Elena Dorina Stănilă
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Correspondence to Heiner Gonska.

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Gonska, H., Raşa, I. & Stănilă, E.D. Power series of the operators \(U_n^{\varrho }\) . Positivity 19, 237–249 (2015). https://doi.org/10.1007/s11117-014-0293-5

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  • DOI: https://doi.org/10.1007/s11117-014-0293-5

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