The following problem arose in [a question][1] I recently asked : given a (possibly non abelian) compact group $G$ and a neighbourhood $U$ of the identity in $G$, can we always find a function $f : G \mapsto \mathbb{R}$, which vanishes outside $U$, whose Fourier transform is nonnegative, and which satisfies $\hat f(1) \neq 0$ ?

  [1]: http://mathoverflow.net/questions/110427/the-identity-element-of-a-compact-group-is-a-limit-point-of-any-polynomial-seque