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

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 $(\Lambda ,E)$.