Hardy's Theorem for Gabor transform

TL;DR

This paper extends Hardy's theorem to Gabor transforms across various mathematical groups, including locally compact abelian groups and several classes of nilpotent Lie groups, broadening the theorem's applicability.

Contribution

It introduces Hardy's theorem analogues for Gabor transforms on diverse groups such as abelian, Euclidean motion, and multiple nilpotent Lie groups, expanding the theoretical framework.

Findings

Hardy's theorem analogues established for Gabor transforms on abelian groups.

Extension of results to Euclidean motion and nilpotent Lie groups.

Broadens understanding of Gabor analysis in non-commutative settings.

Abstract

We establish analogues of Hardy's theorem for Gabor transform on locally compact abelian groups, Euclidean motion group and several general classes of nilpotent Lie groups which include Heisenberg groups, thread-like nilpotent Lie groups, $2$-NPC nilpotent Lie groups and low-dimensional nilpotent Lie groups.