Generalized Bernstein Operators on the Classical Polynomial Spaces

  • J. M. AldazEmail author
  • H. Render


We study generalizations of the classical Bernstein operators on the polynomial spaces \(\mathbb {P}_{n}[a,b]\), where instead of fixing \(\mathbf {1}\) and x, we reproduce exactly \(\mathbf {1}\) and a polynomial \(f_1\), strictly increasing on [ab]. We prove that for sufficiently large n, there always exist generalized Bernstein operators fixing \(\mathbf {1}\) and \(f_1\). These operators are defined by non-decreasing sequences of nodes precisely when \(f_1^\prime > 0\) on (ab), but even if \(f_1^\prime \) vanishes somewhere inside (ab), they converge to the identity.


Bernstein polynomial Bernstein operator 

Mathematics Subject Classification

Primary 41A10 


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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM) and Departamento de MatemáticasUniversidad Autónoma de MadridMadridSpain
  2. 2.School of Mathematical SciencesUniversity College DublinDublin 4Ireland

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