Support of functions on compact groups, and their Fourier transforms 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.
 A: Certainly you can recover $f$ for a circle, because the group is abelian. In general, $f$ can be written as a linear combination of matrix coefficients, and for the Fourier transform of a general compact Lie group you take a trace, so you will loose information about $f$. 
For example, there are nonzero functions with vanishing Fourier transform in the non-commutative situation, e.g. take two orthogonal vectors in an irreducible subspace $V_\rho$ and consider the function given by the matrix coefficient $f:k \mapsto \langle v_1 , \rho(k) v_2 \rangle$. We have $\hat{f}( \rho')=0$ for all irreducible representations $\rho'$. Sketch of proof: Use that $tr\rho'(f) = tr\rho'(f^G)$ where $$f^G (x) = \int\limits_G f(g^{-1}xg) d g$$ and use the orthogonality relations to show that $f^G=0$.
But you can always check the support at the identity with the Plancherel formula
$$ f(1) = \sum\limits_{\rho}  C_\rho \hat{f}(\rho).$$
The fudge factor $C_\rho$ depend only on the dimension. Unfortuantely, I don't recall how. 
