Missing Frequencies and the Diameter of the Support. The Second Beurling-Malliavin Theorem and the Fabry Theorem

Abstract

The first five paragraphs of this chapter are devoted to the following form of the UP:

$$T\; \in \;\mathcal{D}'\left( \mathbb{R} \right),\quad \operatorname{supp} T \subset \left( { - a,a} \right),\;\hat{T}|\;\Lambda = 0 \Rightarrow \;T = 0 $$

where a is a positive number, Λ is a sufficiently “dense” set of real numbers (“frequencies”). The second Beurling-Malliavin theorem stated in §3 and proved in §§4–5 yields very precise conditions to be imposed onto the pair (a, Λ) to ensure this variant of the UP. The setting of the problem is discussed in 81; Sect. 5.8 contains some concrete examples.