Improved Ingham-type result on $\mathbb R^d$ and on connected, simply connected nilpotent Lie Groups

TL;DR

This paper extends Ingham-type uncertainty principles, linking the decay of Fourier transforms to the vanishing of functions on open sets, from Euclidean spaces to connected, simply connected nilpotent Lie groups.

Contribution

It improves previous results on  spaces and establishes analogous theorems on nilpotent Lie groups, broadening the scope of uncertainty principles.

Findings

Enhanced Ingham-type theorems on  Euclidean spaces.

New results on Fourier decay and function vanishing on nilpotent Lie groups.

Broadened understanding of uncertainty principles in non-commutative settings.

Abstract

In \cite{BRS} we have characterized the existance of a non zero function vanishing on an open set in terms of the decay of it's Fourier transform on the $d$-dimensional Euclidean space, the $d$-dimensional torus and on connected, simply connected two step nilpotent Lie groups. In this paper we improved these results on $\mathbb R^d$ and prove analogus results on connected, simply connected nilpotent Lie groups.