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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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