How can we recover the support of a function on a compact Lie group G from its Fourier transform?

I make my question more precisley :

Let $f$ be a conjugation invariant function on a compact semi-simple Lie group $G$. So $f$ can be regarded as a function on the maximal tours ( or Lie algebra of the maximal torus) but perhaps $f$ supported in a convex subset A of the lie algebra of the maximal torus. Assume that we have the fourier transform of this function (on the dual of the lie algebra of the maximal torus). Now from this data we want to describe the support $A$ of the function $f$.

From the fourier expansion (rather than fourier transform) I think is a hard problem to find the support $A$. Any comment/answer are welcome.