Hardy’s theorem for the 𝑛-dimensional Euclidean motion group

Abstract

An uncertainty principle, due to Hardy, for Fourier transform pairs on R \Bbb R says that if the function f f is “very rapidly decreasing”, then the Fourier transform cannot also be “very rapidly decreasing” unless f f is identically zero. In this paper we state and prove an analogue of Hardy’s theorem for the n n -dimensional Euclidean motion group.