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Proceedings of the American Mathematical Society

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

 
 

 

Boundedness and dimension for weighted average functions


Author: David P. Stanford
Journal: Proc. Amer. Math. Soc. 24 (1970), 82-84
MSC: Primary 31.11
DOI: https://doi.org/10.1090/S0002-9939-1970-0252666-8
MathSciNet review: 0252666
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Abstract: The paper considers a weighted average property of the type $u({x_o}) = ({\smallint _B}uwdx)/({\smallint _B}wdx)$, $B$ a ball in ${E^n}$ with center ${x_o}$. A lemma constructing such functions is presented from which it follows that if $n = 1$ and the weight function $w$ is continuously differentiable but is not an eigenfunction of the $1$-dimensional Laplace operator, then $u$ is constant. It is also shown that if $w$ is integrable on ${E^n}$ and $u$ is bounded above or below, $u$ is constant.


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Keywords: Eigenfunction of Laplace operator, dimension of space of functions satisfying a weighted average property, weight function integrable over <IMG WIDTH="33" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="${E^n}$">, bounded function satisfying a weighted average property
Article copyright: © Copyright 1970 American Mathematical Society