The Fourier transform of a tempered distribution is supported at the origin

Question

If the Fourier transform of a tempered distribution $G$ is supported at the origin, does this imply that $G$ is a constant? Can anyone give a reference or a short argument?

Answer 1

No, for instance if $S(x) = x$ then $\hat S = i\delta'$ (up to a multiplicative constant depending on how you define the fourier transform). More generaly if $S$ is a polynomial $S(x) = \sum a_k x^k $ then $\hat S = \sum a_k i^k\delta^{(k)} $