An Uncertainty Principle of Paley and Wiener on Euclidean Motion Group

TL;DR

This paper extends Paley-Wiener type uncertainty principles to the Euclidean motion group, linking support properties of functions to Fourier decay and exploring implications for Schrödinger equation solutions.

Contribution

It establishes a Paley-Wiener analogue for functions on the Euclidean motion group and connects it to uniqueness in Schrödinger equation solutions.

Findings

Proves a Paley-Wiener type theorem on $M(n)$.

Shows a connection between support and Fourier decay on $M(n)$.

Demonstrates a uniqueness property for Schrödinger equation solutions.

Abstract

A classical result due to Paley and Wiener characterizes the existence of a non-zero function in $L^2(\mathbb{R})$, supported on a half line, in terms of the decay of its Fourier transform. In this paper we prove an analogue of this result for compactly supported continuous functions on the Euclidean motion group $M(n)$. We also relate this result to a uniqueness property of solutions to the initial value problem for time-dependent Schr\"odinger equation on $M(n)$.