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Let N be a H -type group and let S = NA be an one dimensional solvable extension of N . For the Helgason Fourier transform on S we prove the following analogue of Hardy’s theorem. Let \(\hat f\) (λ, Y , Z ) stand for the Helgason Fourier transform of f and let h α denote the heat kernel associated to the Laplace-Beltrami operator. Suppose a function f on S satisfies the conditions | f ( x )| ≤ ch α ( x ) and \({{\int\limits_N |\hat f (\lambda, Y,Z)|^2 (1+|Z|^2)^\gamma dY dZ \leq c e^{{-2\beta \lambda^2}}}}\) for all x S ,λℝ where \({{\gamma > \frac{{k-1}}{{2}}, k}}\) being the dimension of the centre of N . Then f =0 or f = ch α depending on whether α
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