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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 $\int _Q {|f - {a_Q}| \leq K|Q|}$, where $|Q|$ 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 $\int _Q {|f - {p_Q}| \leq K|Q|}$, then $f$ is of bounded mean oscillation.


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Keywords: Bounded mean oscillation, polynomial approximation
Article copyright: © Copyright 1975 American Mathematical Society