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Proceedings of the American Mathematical Society

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

 
 

 

Linear projections which implement balayage in Fourier transforms


Author: George S. Shapiro
Journal: Proc. Amer. Math. Soc. 61 (1976), 295-299
MSC: Primary 43A25
DOI: https://doi.org/10.1090/S0002-9939-1976-0427957-3
MathSciNet review: 0427957
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Abstract: Let $\Lambda$ be a closed and discrete or compact subset of a second countable ${\text {LCA}}$ group $G$ and $E$ a subset of the dual group. Balayage is said to be possible for $(\Lambda ,\;E)$ if for every finite measure $\mu$ on $G$ there is some measure $\nu$ on $\Lambda$ whose Fourier transform, $\hat \nu$, agrees on $E$ with $\hat \mu$. If balayage is assumed possible just when $\mu$ is a point measure (with the norms of all the measures $\nu$ bounded by some constant), then there is a bounded linear projection, ${B_\Lambda }$, from the measures on $G$ onto those on $\Lambda$ with ${({B_\Lambda }\mu )^ \wedge } = \hat \mu$ on $E$. An application is made to balayage in product groups.


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Keywords: Balayage in Fourier transforms, point measures, measurable choice, analytic set, weak- <IMG WIDTH="15" HEIGHT="21" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$^{\ast }$"> integral, bounded linear projection, balayage in product groups
Article copyright: © Copyright 1976 American Mathematical Society