A note on measures vanishing at infinity

Abstract

In this paper, we review the basic properties of measures vanishing at infinity and prove a version of the Riemann-Lebesgue lemma for Fourier transformable measures.2010 Mathematics Subject Classification. 43A25, 52C23.

Tempered distributions with translation bounded measure as Fourier transform and the generalized Eberlein decomposition

“…Proof (a)This follows from [34, Theorem 4.1] or [36, Theorem 5.7]. (b)This is a consequence of [29, Corollary 35]. (c)By Corollary 5.11, we have $$f \in \mathcal {WLS}_2({\mathbb {R}}^d)$$. The claim follows now from Theorem 5.13.$$\Box$$ …”

“…Let us start with necessary conditions. Here, we follow the approach of [29, Theorem 27]. Proposition Let $$\omega \in \mathcal {DTBM}_{\operatorname{0a}}({\mathbb {R}}^d)$$.…”

“…Remark (a)The converse of Proposition 5.3 is not true [29, Remark 28]. (b)Let $$\omega \in {\textsf {S}}^{\prime }({\mathbb {R}}^d)$$ be such that $$\widehat{\omega }$$ is absolutely continuous (but not necessarily translation bounded). Then, exactly as in Proposition 5.3, one can show that , for all $$\varphi \in C_{\mathsf {c}}^\infty ({\mathbb {R}}^d)$$. …”

On arithmetic progressions in model sets

“…First, since a n are linearly independent, we have n + a n ≠ m + a m for all n ≠ m. Indeed, if we assume by contradiction that there exists some n ≠ m such that n + a n = m + a m then (m − n) ⋅ 1 + 1 ⋅ a m + (−1) ⋅ a n = 0 , contradicting the linear independence. It follows that δ Λ − δ Z is a measure vanishing at infinity, and hence null weakly almost periodic [32].…”