Let $f\in L^1(\mathbb{R})$ and let $V_f$ be the closed linear subspace of $L^1(\mathbb{R})$ generated by the translates $f(\cdot - y)$ of $f$. If $V_f=L^1(\mathbb{R})$, I want to show that $\hat{f}$ never vanishes.
We have $\hat{f}(\xi_0)=0$ iff $\hat{h}(\xi_0)=0$ for $h\in V_f$, but I'm not sure how to proceed beyond this. The Riemann-Lebesgue Lemma gives us that $L^1(\mathbb{R})$ is sent to $C_0(\mathbb{R})$ under the Fourier transform, $C_0(\mathbb{R})$ seems to contain functions that vanish...
I got this problem from 3.3 here: http://www.math.ucdavis.edu/~jlirion/course_notes/Prelim_Solutions.pdf.
EDITED: I originally neglected to say that Vf is the closed subspace generated by the translates, not the translates themselves. I don't know if it is reasonable to suppose that this subspace is L1 itself or just a subset...