TY  - JOUR
AB  - Fourier‐transformable Radon measures are called doubly sparse when both the measure and its transform are pure point measures with sparse support. Their structure is reasonably well understood in Euclidean space, based on the use of tempered distributions. Here, we extend the theory to second countable, locally compact Abelian groups, where we can employ general cut and project schemes and the structure of weighted model combs, along with the theory of almost periodic measures. In particular, for measures with Meyer set support, we characterise sparseness of the Fourier–Bohr spectrum via conditions of crystallographic type, and derive representations of the measures in terms of trigonometric polynomials. More generally, we analyse positive definite, doubly sparse measures in a natural cut and project setting, which results in a Poisson summation type formula.
DA  - 2020
DO  - 10.1112/tlm3.12020
LA  - eng
IS  - 1
M2  - 1
PY  - 2020
SP  - 1-32
T2  - Transactions of the London Mathematical Society
TI  - Pure point measures with sparse support and sparse Fourier–Bohr support
UR  - https://nbn-resolving.org/urn:nbn:de:0070-pub-29431088
Y2  - 2024-11-22T03:35:04
ER  -