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Let $G$ be a finite abelian group. Let $f: G \to {\mathbb C}$ be a signal(i.e. function). The classical uncertainty principle asserts that the productof the size of the support of $f$ and its Fourier transform $\hat f$,$\text{supp}(f)$ and $\text{supp}(\hat f)$ respectively, must satisfy thecondition: $$|\text{supp}(f)| \cdot |\text{supp}(\hat f)| \geq |G|.$$  In the first part of this paper, we improve the uncertainty principle forsignals with Fourier transform supported on generic sets. This improvement isachieved by employing {\it the restriction theory} and {\it the Salem set}mechanism from harmonic analysis. Then we investigate some applications ofuncertainty principles that were developed in the first part of this paper, tothe problem of unique recovery of finite sparse signals in the absence of somefrequencies.  Donoho and Stark (\cite{DS89}) showed that a signal of length $N$ can berecovered exactly, even if some of the frequencies are unobserved, providedthat the product of the size of the number of non-zero entries of the signaland the number of missing frequencies is not too large, leveraging theclassical uncertainty principle for vectors. Our results broaden the scope fora natural class of signals in higher-dimensional spaces. In the case when thesignal is binary, we provide a very simple exact recovery mechanism through theDRA algorithm.

Uncertainty Principles on Finite Abelian Groups, Restriction Theory, and Applications to Sparse Signal Recovery