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We generalise a result of Hardy, which asserts the impossibility of a function and its Fourier transform to be simultaneously "very rapidly decreasing", to: (i) all noncompact, semi-simple Lie groups with one conjugacy class of Cartan subgroups; (ii) SL(2, R); and (iii) all symmetric spaces of the noncompact type.

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A theorem of Hardy states that, if f is a function on R such that |f (x)| ≤ C e −α|x| 2 for all x in R and |f (ξ)| ≤ C e −β|ξ| 2 for all ξ in R, where α > 0, β > 0, and αβ > 1/4, then f = 0. Sitaram and Sundari generalised this theorem to semisimple groups with one conjugacy class of Cartan subgroups and to the K-invariant case for general semisimple groups. We extend the theorem to all semisimple groups.

No abstract