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Proceedings of the American Mathematical Society

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

 
 

 

Characterizations of bounded mean oscillation


Author: Stephen Jay Berman
Journal: Proc. Amer. Math. Soc. 51 (1975), 117-122
MSC: Primary 42A92; Secondary 46E30
DOI: https://doi.org/10.1090/S0002-9939-1975-0374805-5
MathSciNet review: 0374805
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Abstract: Recall that an integrable function $ f$ on a cube $ {Q_0}$ in $ {{\mathbf{R}}^n}$ is said to be of bounded mean oscillation if there is a constant $ K$ such that for every parallel subcube $ Q$ of $ {Q_0}$ there exists a constant $ {a_Q}$ such that $ \int_Q {\vert f - {a_Q}\vert \leq K\vert Q\vert} $, where $ \vert Q\vert$ denotes the volume of $ Q$. We prove here that if there is an integer $ d$ and a constant $ K$ such that for every parallel subcube $ Q$ of $ {Q_0}$ there exists a polynomial $ {p_Q}$ of degree $ \leq d$ such that $ \int_Q {\vert f - {p_Q}\vert \leq K\vert Q\vert} $, then $ f$ is of bounded mean oscillation.


References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1975-0374805-5
Keywords: Bounded mean oscillation, polynomial approximation
Article copyright: © Copyright 1975 American Mathematical Society

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