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 | References | Similar Articles | Additional Information
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.
- Anil K. Bose, Functions satisfying a weighted average property, Trans. Amer. Math. Soc. 118 (1965), 472β487. MR 177128, DOI https://doi.org/10.1090/S0002-9947-1965-0177128-0
- Anil Kumar Bose, Functions satisfying a weighted average property. II, Trans. Amer. Math. Soc. 124 (1966), 540β551. MR 204679, DOI https://doi.org/10.1090/S0002-9947-1966-0204679-3
- Anil Kumar Bose, Generalized eigenfunctions of the Laplace operator and weighted average property, Proc. Amer. Math. Soc. 19 (1968), 55β62. MR 221121, DOI https://doi.org/10.1090/S0002-9939-1968-0221121-4
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Additional Information
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
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© Copyright 1970
American Mathematical Society