Uncertainty principles on certain Lie groups

Abstract

There are several ways of formulating the uncertainty principle for the Fourier transform on Rⁿ. Roughly speaking, the uncertainty principle says that if a function f is 'concentrated' then its Fourier transform f^~ cannot be 'concentrated' unless f is identically zero. Of course, in the above, we should be precise about what we mean by 'concentration'. There are several ways of measuring 'concentration' and depending on the definition we get a host of uncertainty principles. As several authors have shown, some of these uncertainty principles seem to be a general feature of harmonic analysis on connected locally compact groups. In this paper, we show how various uncertainty principles take form in the case of some locally compact groups including Rⁿ, the Heisenberg group, the reduced Heisenberg group and the Euclidean motion group of the plane.