Hardy's Uncertainty Principle on Certain Lie Groups

By F. Astengo, M. Cowling, B. Di Blasio, M. Sundari. 2000

62 citations recorded by the source corpus.

A theorem due to Hardy states that, if f is a function on R such that |f(x)|⩽Ce−α|x|2 for all x in R and |f^(ξ)|⩽Ce‐β|(ξ)|2 for all ξ in R, where α > 0, β > 0, and αβ > 1/4, then f = 0. A version of this celebrated theorem is proved for two classes of Lie groups: two‐step nilpotent Lie groups and harmonic NA groups, the latter being a generalisation of noncompact rank‐1 symmetric spaces. In the first case the group Fourier transformation is considered; in the second case an analogue of the Helgason–Fourier transformation for symmetric spaces is considered.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/427bee5394e6fee7a5e536dd889458e4e223cff3