Measures with locally finite support and spectrum

Abstract

The goal of this paper is the construction of measures μ on R n enjoying three conflicting but fortunately compatible properties: (i) μ is a sum of weighted Dirac masses on a locally finite set, (ii) the Fourier transformμ of μ is also a sum of weighted Dirac masses on a locally finite set, and (iii) μ is not a generalized Dirac comb. We give surprisingly simple examples of such measures. These unexpected patterns strongly differ from quasicrystals, they provide us with unusual Poisson's formulas, and they might give us an unconventional insight into aperiodic order.Poisson formula | spectrum | almost periodic T he Dirac mass at a ∈ R n is denoted by δ a or δ a ðxÞ. A purely atomic measure is a linear combination μ = P λ∈Λ cðλÞδ λ of Dirac masses where the coefficients cðλÞ are real or complex numbers and P jλj≤R jcðλÞj is finite for every R > 0. Then Λ is a countable set of points of R n . If Λ is closed and if cðλÞ ≠ 0, ∀λ ∈ Λ, then Λ is the support of μ. A subset Λ ⊂ R n is locally finite if Λ ∩ B is finite for every bounded set B. Equivalently Λ can be ordered as a sequence fλ j , j = 1,2, . . .g and λ j tends to infinity with j. A measure μ is a tempered distribution if it has a polynomial growth at infinity in the sense given by Laurent Schwartz in ref. 1. For instance, the measureThe distributional Fourier transformμ of μ is defined by the following condition: hμ, ϕi = hμ,φi shall hold for every test function ϕ belonging to the Schwartz class SðR n Þ. The spectrum S of μ is the (closed) support ofμ. Definition 1: A purely atomic measure μ on R n is a crystalline measure if i) the support Λ of μ is a locally finite set, ii) μ is a tempered distribution, and iii) the distributional Fourier transformμ of μ is also a purely atomic measure that is supported by a locally finite set S.If μ is a crystalline measure, its Fourier transform is also a crystalline measure.Definition 2: A measure μ on R is odd if for every compactly supported continuous function f we haveFor every set E of real numbers, let E + = E ∩ fx > 0g and E − = E ∩ fx < 0g. Let us denote by Q the field of rational numbers. Theorem 1 is proved in this article: Theorem 1. There exists an odd crystalline measure μ on R such that its support Λ and its spectrum S have the following properties: (i) Each finite subset of Λ + is linearly independent over Q and (ii) each finite subset of S + is linearly independent over Q.The spectrum S of μ is an increasing sequence s k , k ∈ Z, of real numbers such that (i) s −k = −s k , ∀k ∈ Z, and (ii) s 1 , s 2 , . . . , s N are linearly independent over Q for each integer N. Theorem 1 implies that P ∞ −∞ bðkÞexpð2πis k xÞ is a sum of Dirac masses on a locally finite set Λ. It is counterintuitive that these incoherent waves bðkÞexpð2πis k xÞ can be piled up in harmony so that their sum yields Dirac masses.Theorem 1 is valid in any dimension as the following proposition shows.Theorem 2. There exists a crystalline measure μ on R n such that i) μ is odd in the last variable x n , ii) the support Λ of μ is the union of ...

Spectral notions of aperiodic order

“…It then also holds for generic elements in the corresponding hull, equipped with a natural patch frequency measure. Moreover, nonperiodic measures with locally finite support and spectrum, as recently constructed in [57], are further examples with well-defined amplitudes. So, there is room for generalisation, and hence work to be done to clarify the situation.…”

Fourier quasicrystals and discreteness of the diffraction spectrum

“…Open problems 11.1. Very recently, Y. Meyer has found [Mey16] an interesting version of Theorem 1.2. Namely, he constructed measures µ whose supports and spectra are discrete closed sets, which can be described by simple effective formulas.…”

“…This is a stronger "non-periodicity" condition than in Theorem 1.2. However, such a measure cannot be translation-bounded, see [Mey16,Lemma 5].…”

Crystalline measures in two dimensions

“…We consider the Dirichlet series ζ(µ, s) = {λ∈Λ, λ =0} c(λ)|λ| −s in the complex variable s. Let γ n = π −n/2 2 1−n Γ(n/2) , where Γ denotes the Euler Gamma function. Then ζ(µ, s) is an entire function in the complex plane if 0 / ∈ S, while ζ(µ, s) − γ n a(0) s−n can be extended as an entire function of s ∈ C if 0 ∈ S [3], [7]. The connection between the Poisson summation formula and the properties of the Riemann zeta function is developed in Titchmarsh's treatise and can be traced back to Riemann.…”