Strictly supercritical sampling alias counterexample ($g(t)=\sin (\pi t)h(t),g(n)=0$)

For every $ϵ>0$, choose a nonzero smooth spectral bump supported in $(-{ϵ}_0,{ϵ}_0)$ with ${ϵ}_0<\min (ϵ,\pi )$, 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 $[-\pi -ϵ,\pi +ϵ]$. Thus even a small bandwidth increase destroys uniqueness from unit-spaced samples.