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Proceedings of the American Mathematical Society

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

 

 

Best monotone approximation in $ L\sb 1[0,1]$


Authors: Robert Huotari and David Legg
Journal: Proc. Amer. Math. Soc. 94 (1985), 279-282
MSC: Primary 41A29; Secondary 26A48
DOI: https://doi.org/10.1090/S0002-9939-1985-0784179-X
MathSciNet review: 784179
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Abstract: If $ f$ is a bounded Lebesgue measurable function on [0,1] and $ 1 < p < \infty $, let $ {f_p}$ denote the best $ {L_p}$-approximation to $ f$ by nondecreasing functions. It is shown that $ {f_p}$ converges almost everywhere as $ p$ decreases to one to a best $ {L_1}$-approximation to $ f$ by nondecreasing functions. The set of best $ {L_1}$-approximations to $ f$ by nondecreasing functions is shown to include its supremum and infimum.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1985-0784179-X
Keywords: Best $ {L_1}$-approximation, nondecreasing function, convergence
Article copyright: © Copyright 1985 American Mathematical Society