= Strictly supercritical sampling alias counterexample
{title2=$g(t)=\sin(\pi t)h(t),\qquad g(n)=0$}
For every $\epsilon>0$, choose a nonzero smooth spectral bump supported in $(-\epsilon_0,\epsilon_0)$ with $\epsilon_0<\min(\epsilon,\pi)$, and let $h$ be its inverse Fourier transform. Then $g$ is a nonzero <Schwartz function>, vanishes on all integer samples, and its transform is the difference of bumps shifted to $\pi$ and $-\pi$. Its support lies strictly within $[-\pi-\epsilon,\pi+\epsilon]$. Thus even a small bandwidth increase destroys uniqueness from unit-spaced samples.
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