Abstract
We first find simple characterizations of $\frac{1}{N} \mathbb{Z}$-invariance of arbitrary principal shift-invariant space $V(\phi)$. Then we find several equivalent conditions for $V(\phi)$ to admit periodic oversampling for a class of continuous frame generators $\phi$. In particular, when $\phi$ is band-limited and $\hat{\phi}$ is piecewise continuous, we find very simple and general sufficient conditions for $V(\phi)$ to admit periodic oversampling, which involve the extra invariance of $V(\phi)$, together with an illustrating example.
