Completeness of exponentials and Beurling's theorem regarding Fourier transform on $\mathbb{R}^n$ and $\mathbb{T}^n$

Abstract

A classical result of A. Beurling gives a relation between the decay of a complex Borel measure on R and the vanishing set of its Fourier transform. We prove several variable analogues of this result on the Euclidean space R n and the n-dimensional torus T n . We also prove some results on the well known weighted approximation problem of exponentials on R n and T n by establishing an equivalence with Beurling's theorem.