It is known that sufficiently "well-behaved" functions admitting a Fourier transform are uniquely associated with it: two functions with identical Fourier transforms will be identical. But what happens if their Fourier transforms only coincide in some bounded set?
More in detail, let $f_1$, $f_2:\mathbb{R}\rightarrow\mathbb{C}$ continuous functions in $L^1(\mathbb{R})\cap L^2(\mathbb{R})$ whose Fourier transforms are equal in a bounded interval, say $[-\Omega,\Omega]$, i.e. \begin{equation} \forall\omega\in[-\Omega,\Omega]:\quad\int_\mathbb{R}f_1(\omega)e^{-i\omega x}\,\mathrm{d}x=\int_\mathbb{R}f_2(\omega)e^{-i\omega x}\,\mathrm{d}x, \end{equation} while possibly being different outside the interval. Can we say something about the relation between $f_1$ and $f_2$? Are they related somehow?