  A sharp version of the Balian-Low theorem is proven for the generators of
finitely generated shift-invariant spaces. If generators $\{f_k\}_{k=1}^K
\subset L^2(\mathbb{R}^d)$ are translated along a lattice to form a frame or
Riesz basis for a shift-invariant space $V$, and if $V$ has extra invariance by
a suitable finer lattice, then one of the generators $f_k$ must satisfy
$\int_{\mathbb{R}^d} |x| |f_k(x)|^2 dx = \infty$, namely, $\widehat{f_k} \notin
H^{1/2}(\mathbb{R}^d)$. Similar results are proven for frames of translates
that are not Riesz bases without the assumption of extra lattice invariance.
The best previously existing results in the literature give a notably weaker
conclusion using the Sobolev space $H^{d/2+\epsilon}(\mathbb{R}^d)$; our
results provide an absolutely sharp improvement with $H^{1/2}(\mathbb{R}^d)$.
Our results are sharp in the sense that $H^{1/2}(\mathbb{R}^d)$ cannot be
replaced by $H^s(\mathbb{R}^d)$ for any $s<1/2$.
