Existence of a function in L1(G) whose Fourier transform never vanish

I am reading Wiener Tauberian theorem on locally compact abelian group setting and there I have read that: The closed linear span of the translates of $ f $ is $ L^1(G) $ if and only if $ \hat{f}$ never vanishes, where G is a locally compact abelian group. Now my question is can we always find a function from $f\in L^1(G) $ whose Fourier transform ($ \hat{f}$ ) never vanishes. I only have example when $G=\mathbf R^n$ and the example is Gaussian Kernel. Is there any idea to go for general setting ? Thank you,