1/9/2024 0 Comments Pure point measureLicensed under Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4. Transactions of the London Mathematical SocietyĬopyright 2020 The Authors. Item Type:Īperiodicity, pure point measures, Fourier transform We relate the decomposition over a, b of a measure d (on R ) into absolutely continuous, pure point, and singular continuous pieces to the. More generally, we analyse positive definite, doubly sparse measures in a natural cut and project setting, which results in a Poisson summation type formula. 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. 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. Their structure is reasonably well understood in Euclidean space, based on the use of tempered distributions. is called a pure point measure if (X) Ex ex u(x) for any Borel. The basic intuition is now that order in the original measure will show up as a (large) pure point component in its Fourier transform. The diffraction is then described by the Fourier transform of this measure. Fourier‐transformable Radon measures are called doubly sparse when both the measure and its transform are pure point measures with sparse support. Now we define the Lebesgue measures of sets in B, the Borel sets in R. In mathematical diffraction theory, the solid in question is modeled by a measure.
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