Generalized Bernstein Operators on the Classical Polynomial Spaces

We study generalizations of the classical Bernstein operators on the polynomial spaces P n [ a , b ] , where instead of fixing 1 and x , we reproduce exactly 1 and a polynomial f 1 , strictly increasing on [ a ,  b ]. We prove that for sufficiently large n , there always exist generalized Bernstein...

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Bibliographic Details
Published in:Mediterranean journal of mathematics 2018-12, Vol.15 (6), p.1-22, Article 222
Main Authors: Aldaz, J. M., Render, H.
Format: Article
Language:English
Subjects:
Fixing
Mathematics
Mathematics and Statistics
Operators
Polynomials
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Summary:We study generalizations of the classical Bernstein operators on the polynomial spaces P n [ a , b ] , where instead of fixing 1 and x , we reproduce exactly 1 and a polynomial f 1 , strictly increasing on [ a ,  b ]. We prove that for sufficiently large n , there always exist generalized Bernstein operators fixing 1 and f 1 . These operators are defined by non-decreasing sequences of nodes precisely when f 1 ′ > 0 on ( a ,  b ), but even if f 1 ′ vanishes somewhere inside ( a ,  b ), they converge to the identity.
ISSN:1660-5446
1660-5454
DOI:10.1007/s00009-018-1266-x