Uniqueness of the Fourier transform on the Euclidean motion group

TL;DR

This paper proves that integrable functions on the Euclidean motion group with finite rank Fourier transforms must be zero and explores a new form of the uncertainty principle related to Heisenberg uniqueness pairs on this group and its variants.

Contribution

It establishes a uniqueness result for functions with finite rank Fourier transforms and introduces a novel uncertainty principle on the Euclidean motion group.

Findings

Functions with finite rank Fourier transforms on the Euclidean motion group must vanish.

Introduces a new variance of the uncertainty principle involving Heisenberg uniqueness pairs.

Extends the uncertainty principle to product groups involving Euclidean space and compact groups.

Abstract

In this article, we prove that if the Fourier transform of a certain integrable function on the Euclidean motion group is of finite rank, then the function has to vanish identically. Further, we explore a new variance of the uncertainty principle, the Heisenberg uniqueness pairs on the Euclidean motion group as well as on the product group $\mathbb R^n\times K,$ where $K$ is a compact group.