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Proceedings of the American Mathematical Society

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

 

 

Variation norm convergence of function sequences


Author: Randolph Constantine
Journal: Proc. Amer. Math. Soc. 45 (1974), 339-345
MSC: Primary 41A30; Secondary 26A51
DOI: https://doi.org/10.1090/S0002-9939-1974-0352808-3
MathSciNet review: 0352808
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Abstract: We prove that a pointwise convergent sequence of convex functions with a continuous limit converges with respect to the total variation norm. This yields a theorem on convexity-preserving operators which has as a corollary the result that a complex function $ f$ is absolutely continuous on $ [0,1]$ if and only if the sequence $ B.(f)$ of Bernstein polynomials of $ f$ converges to $ f$ with respect to the total variation norm.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1974-0352808-3
Keywords: Total variation norm, convex function, absolutely continuous function, Bernstein polynomial
Article copyright: © Copyright 1974 American Mathematical Society