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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
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- $ ^{\ast}$ integral, bounded linear projection, balayage in product groups
Article copyright: © Copyright 1976 American Mathematical Society