I need idea about my problem but if you have another type proof please give a hint not a whole solution.
Problem: Let $f\in L^{2}[-1,1], \; \hat{f}\in L^{2}(\mathbb{R})$ and $Q_{c}\hat{f}=0.$ i.e. $\int_{-c}^{c}f(x)e^{-2\pi ixt}dx=0.$ Prove that $f=0.$
What I have done:
the overall idea is to use Plancherel theorem, i.e. $\Vert f\Vert_{2}=\Vert \hat{f}\Vert_{2}.$ I want to prove that $\hat{f}(t)=0$ for all $t.$ For this, I see that $\vert\hat{f}(t)\vert\leq \int_{-1}^{1}\vert f(x)\vert dx<\infty.$ Now, I want to prove that $\hat{f}(t)$ is analytic so that I can use Liouville theorem and proving that $\hat{f}=0.$
For the analyticity, I see that $e^{2\pi ixt}$ is analytic. But I can't use Leibniz integral rule because of my integrand ($f(x)e^{-2\pi ixt}$) in $\hat{f}$ is not necessarily continuous.