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Proceedings of the American Mathematical Society

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

 

 

Best $ L\sb 1$-approximation of $ L\sb 1$-approximately continuous functions on $ (0,1)\sp n$ by nondecreasing functions


Authors: R. B. Darst and Shu Sheng Fu
Journal: Proc. Amer. Math. Soc. 97 (1986), 262-264
MSC: Primary 41A50
DOI: https://doi.org/10.1090/S0002-9939-1986-0835876-X
MathSciNet review: 835876
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Abstract: For $ n \geq 1$, let $ \Omega $ denote the open unit $ n$-cube, $ {(0,1)^n}$. Let $ \mu $ denote Lebesgue measure, let $ \Sigma $ consist of the Lebesgue measurable subsets of $ \Omega $, and let $ {L_1} = {L_1}(\Omega ,\Sigma ,\mu )$. Let $ {A_1}$ consist of the approximately continuous functions in $ {L_1}$ and let $ M$ consist of the equivalence classes in $ {L_1}$ which contain nondecreasing functions. Let $ f \in {A_1}$. It is shown that there is a unique best $ {L_1}$-approximation $ [g]$ to $ f$ in $ M$. If $ n = 1,g$ is continuous, but if $ n > 1,[g]$ may not consist of a continuous function.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1986-0835876-X
Keywords: Approximate continuity, best approximation, $ {L_1}$, nondecreasing functions
Article copyright: © Copyright 1986 American Mathematical Society