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A result by G. H. Hardy ([11]) says that if f and its Fourier transform f̂ are $O\left(\right|x{|}^m{e}^{-\alpha x²})$ and $O\left(\right|x\left|ⁿe^{-x²/\left(4\alpha \right)}\right)$ respectively for some m,n ≥ 0 and α > 0, then f and f̂ are $P\left(x\right)e^{-\alpha x²}$ and $P^{\text{'}}\left(x\right)e^{-x²/\left(4\alpha \right)}$ respectively for some polynomials P and P’. If in particular f is as above, but f̂ is $o\left(e^{-x²/\left(4\alpha \right)}\right)$, 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.