The following problem arose in a question 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 $\hat f(1) \neq 0$ ?
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I didn't think about the non Abelian case, but for a commutative group say like $\mathbb{R}$ you can do the following. Take a function with support in $\frac{1}{2} U$ and symmetric with respect to the origin. Its convolution with itself answers your question. 

