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Abstract

A result by G. H. Hardy ([11]) says that if $f$ and its Fourier transform $\widehat {f}$ are $O(|x|^m e^{-\alpha x^2})$ and $O(|x|^n e^{-x^2/{(4\alpha )}})$ respectively for some $m,n\ge 0$ and $\alpha >0$, then $f$ and $\widehat {f}$ are $P(x)e^{-\alpha x^2}$ and $P'(x)e^{-x^2/{(4\alpha )}}$ respectively for some polynomials $P$ and $P'$. If in particular $f$ is as above, but $\widehat {f}$ is $o(e^{-x^2/{(4\alpha )}})$, then $f= 0$. In this article we will prove a complete analogue of this result for connected noncompact semisimple Lie groups with finite center. Our proof can be carried over to the real reductive groups of the Harish-Chandra class.