论文标题
$ \ mathbb {r}^n $和$ \ mathbb {t}^n $的指数和beurling定理的完整性
Completeness of Exponentials and Beurling's Theorem on $\mathbb{R}^n$ and $\mathbb{T}^n$
论文作者
论文摘要
阿恩·贝林（Arne Beurling）的经典结果指出，如果$μ$有一定的衰减，则无零复合物孔测量$μ$的傅立叶变换在一组正面的Lebesgue度量上就不会消失。我们通过探索与众所周知的问题有关在连续函数的特定加权范数线性空间中的线性跨度密度的联系，证明了Beurling定理的几个可变类似物。在此过程中，我们还证明了这种类型的一些新结果，并在上述两个问题之间建立了等效性。我们还获得了Beurling定理的概括，并在$ n $ dimensional torus $ \ mathbb {t}^n。$上证明了这些结果。
A classical result of Arne Beurling states that the Fourier transform of a nonzero complex Borel measure $μ$ on the real line cannot vanish on a set of positive Lebesgue measure if $μ$ has certain decay. We prove a several variable analogue of Beurling's theorem by exploring its connection with the well-known problem concerning the density of linear span of exponentials in a certain weighted normed linear space of continuous functions. In the process, we also prove some new results of this genre and establish an equivalence between the above two problems. We also obtain a generalisation of Beurling's theorem and prove these results on the $n$-dimensional torus $\mathbb{T}^n.$
