Premium
Hardy's Uncertainty Principle on Certain Lie Groups
Author(s) -
Astengo F.,
Cowling M.,
Di Blasio B.,
Sundari M.
Publication year - 2000
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610700001186
Subject(s) - mathematics , lie group , nilpotent , pure mathematics , rank (graph theory) , fourier transform , hardy space , lie theory , symmetric space , harmonic function , transformation group , group (periodic table) , simple lie group , transformation (genetics) , lie algebra , mathematical analysis , combinatorics , physics , adjoint representation of a lie algebra , quantum mechanics , lie conformal algebra , biochemistry , chemistry , gene
A theorem due to Hardy states that, if f is a function on R such that| f( x ) | ⩽ C e − α | x | 2for all x in R and| f ^ ( ξ ) | ⩽ C e ‐ β| ( ξ ) | 2for 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.