Theorem:
Suppose that the signal length $N$ is a prime integer. Let $\Omega$ be a subset of $\{ 0, \ldots , N-1 \}$ and let $f$ be a vector supported on $T$ such that $|T| < \dfrac{1}{2}|\Omega|$ then $f$ can be constructed uniquely from the set $\Omega$ and the Fourier transform of $f$ on the set $\Omega$.
Converse:
If $\Omega$ is not the set of all $N$ frequencies, then there exist distinct vectors $f$; $g$ such that $|\operatorname{supp}(f)|,|\operatorname{supp}(g)| < (1/2)|\Omega| + 1$ and such that the Fourier transform of $f$ on $\Omega$ and that of $g$ on $\Omega$ are equal.
I did not understand how for the main theorem the converse is the above statement?
