HARDY’S THEOREM FOR GABOR TRANSFORM

Hardy’s uncertainty principle for the Gabor transform is proved for locally compact abelian groups having noncompact identity component and groups of the form $\mathbb{R}^{n}\times K$ , where $K$ is a compact group having irreducible representations of bounded dimension. We also show that Hardy’s theorem fails for a connected nilpotent Lie group $G$ which admits a square integrable irreducible representation. Further, a similar conclusion is made for groups of the form $G\times D$ , where $D$ is a discrete group.