The shift bound for abelian codes and generalizations of the Donoho-Stark uncertainty principle

Let $G$ be a finite abelian group. If $f:G\to \text{bC}$ is a nonzero function with Fourier transform $\text{hf}$, the Donoho-Stark uncertainty principle states that $\mathrm{\mid }\text{supp}(f)\mathrm{\mid }\mathrm{\mid }\text{supp}(\text{hf})\mathrm{\mid }\ge \mathrm{\mid }G\mathrm{\mid }$. The purpose of this paper is twofold. First, we present the shift bound for abelian codes with a streamlined proof. Second, we use the shifting technique to prove a generalization and a sharpening of the Donoho-Stark uncertainty principle. In particular, the sharpened uncertainty principle states, with notation above, that $\mathrm{\mid }\text{supp}(f)\mathrm{\mid }\mathrm{\mid }\text{supp}(\text{hf})\mathrm{\mid }\ge \mathrm{\mid }G\mathrm{\mid }+\mathrm{\mid }\text{supp}(f)\mathrm{\mid }-\mathrm{\mid }H(\text{supp}(f))\mathrm{\mid }$, where $H(\text{supp}(f))$ is the stabilizer of $\text{supp}(f)$ in $G$.