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

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- by George S. Shapiro PDF
- Trans. Amer. Math. Soc.
**225**(1977), 183-198 Request permission

## 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 $(\Lambda ,E)$.

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## Additional Information

- © Copyright 1977 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**225**(1977), 183-198 - MSC: Primary 43A25; Secondary 42A44
- DOI: https://doi.org/10.1090/S0002-9947-1977-0425510-4
- MathSciNet review: 0425510