Large Fourier Quasicrystals and Wiener’s Theorem

Abstract

Abstract. We find new simple conditions for support of a discrete measure on Euclidean space to be a finite union of translated lattices. The arguments are based on a local analog of Wiener's Theorem on absolutely convergent trigonometric series and theory of almost periodic functions.

Local Versions of the Wiener-Levi Theorem

“…For the case G = R d and holomorphic h on a neighborhood V ⊂ C of K the last theorem was proved in [2]. Its application allowed us to obtain in [2] some strengthening of one of the theorems in the theory of Fourier quasicrystals. Another application of Theorem 6 see [3].…”

Local Wiener's Theorem and Coherent Sets of Frequencies

“…with the norm f W = n |c n |. In [6] we proved Theorem 2. Let K ⊂ C be an arbitrary compact set, h(z) be a holomorphic function on a neighborhood of K, and f ∈ W .…”

Temperate distributions with locally finite support and spectrum on Euclidean spaces

“…Also, in Section 7 we will use the following local version of the classical Wiener-Levi Theorem: Proposition 3. [ [7]] Let K ⊂ C be an arbitrary compact set, h(z) be a holomorphic function on a neighborhood of K, and f ∈ W . Then there is a function…”

“…Remark that the periodic structure of a crystalline measure in several dimensions follows under extra assumptions on the masses ( [7], [8]):…”

“…with the same K as in (3). If H is a positive real number, but Γ is uniformly discrete, then (7) is valid for all m ([4], Proposition 3), In particular, if f is a measure with uniformly bounded masses and support of bounded density, then K = H = 0 and f is a measure with uniformly bounded masses as well.…”