Doubly sparse measures on second countable locally compact Abelian groups are shown to be supported on finitely many translates of a lattice with trigonometric polynomial amplitudes.
On the Fourier Transformability of Strongly Almost Periodic Measures
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abstract
In this paper we characterize the Fourier transformability of a strongly almost periodic measure in terms of an integrability condition for its Fourier Bohr series. We also provide a necessary and sufficient condition for a strongly almost periodic measure to be a Fourier transform of a measure. We discuss the Fourier transformability of a measure on $\RR^d$ in terms of its Fourier transform as a tempered distribution. We conclude by looking at a large class of such measures coming from the cut and project formalism.
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2019 1verdicts
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Pure point measures with sparse support and sparse Fourier--Bohr support
Doubly sparse measures on second countable locally compact Abelian groups are shown to be supported on finitely many translates of a lattice with trigonometric polynomial amplitudes.