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Transactions of the American Mathematical Society

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Balayage in Fourier transforms: general results, perturbation, and balayage with sparse frequencies

Author: George S. Shapiro
Journal: Trans. Amer. Math. Soc. 225 (1977), 183-198
MSC: Primary 43A25; Secondary 42A44
MathSciNet review: 0425510
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Abstract: Let $ \Lambda $ be a discrete subset of an LCA group and E a compact subset of the dual group. Balayage is said to be possible for $ (\Lambda ,E)$ if the Fourier transform of each measure on G is equal on E to the Fourier transform of some measure supported by $ \Lambda $. Following Beurling, we show that this condition is equivalent to the possibility of bounding certain functions with spectra in E by their bounds on $ \Lambda $. We derive consequences of this equivalence, among them a necessary condition on $ \Lambda $ for balayage when E is compact and open (a condition analogous to a density condition Beurling and Landau gave for balayage in Euclidean spaces).

We show that if balayage is possible for $ (\Lambda ,E)$ and if $ \Lambda '$ is close to $ \Lambda $, then balayage is possible for $ (\Lambda ',E)$. Explicit bounds for the needed closeness in R and $ {R^n}$ are given.

Using these perturbation techniques, we give examples of perfect sets $ E \subset R$ with the property that there are ``arbitrarily sparse'' sets $ \Lambda $ with balayage possible for $ (\Lambda ,E)$.

References [Enhancements On Off] (What's this?)

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Keywords: Balayage in Fourier transforms, convolution, set of sampling, dual Banach spaces, surjective linear operator, density, uniformly discrete, relatively dense, compact open subgroup, sparse set, Dirichlet set, homogeneous perfect set
Article copyright: © Copyright 1977 American Mathematical Society