Let $f\in L^1(\mathbb{R})$. s.t $\{f(x-t)\}_{t\in\mathbb{R}}$ is complete. Prove that $\hat{f}(\omega)\neq 0$ for all $\omega\in\mathbb{R}$
Suppose the system is complete for any $g\in L^1(\mathbb{R})$, $$\Vert g-\sum_{t}f(x-t)\Vert\le \epsilon$$ which means $$\int |g-\sum_tf(x-t)|\to 0$$ and now I thought applying the trasnform and getting contradiction. But where is the contradiction?