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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Area of Bernstein-type polynomials


Author: Martin E. Price
Journal: Proc. Amer. Math. Soc. 42 (1974), 222-227
MSC: Primary 41A10; Secondary 26A63
DOI: https://doi.org/10.1090/S0002-9939-1974-0326236-0
MathSciNet review: 0326236
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Abstract: Bernstein polynomials in one variable are known to be total-variation diminishing when compared to the approximated function f. Here we consider the two variable case and give a counterexample to show they are not area-diminishing. Sufficient conditions are then given on a continuous function f to insure convergence in area. A similar theorem is proved for Kantorovitch polynomials in the case f is summable.


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DOI: https://doi.org/10.1090/S0002-9939-1974-0326236-0
Keywords: Generalized area, generalized absolute continuity, Kantorovitch polynomials, Bernstein polynomials
Article copyright: © Copyright 1974 American Mathematical Society