If we have $f\in L^{2}(\mathbb{R})$. How can I prove that (if $f$ is not zero), $f$ and its Fourier transform $\cal{F}(f)$ can't be zero out of a bounded interval?
I think it involves the inversion formula, but how can we prove this in a rigurous way?
Thanks for any help.
One way to go about this is the Paley-Wiener theorem:
The Fourier transform of a distribution of compact support on $\mathbb{R}^n$ is an entire function on $\mathbb{C}^n$
So, assume that both $f$ and $\widehat f$ are compactly supported. Then they are both compactly-supported entire functions (as, more or less, the Fourier transforms of each other). How many functions are both entire and compactly supported?
Take some point $x_0$ outside the support of $f$, and expand in a Taylor series there (possible because $f$ is entire). You know that $f$ and all of its derivatives are zero there, so what is the resulting Taylor series?
the zero function.
