  Let $(I,+)$ be a finite abelian group and $\mathbf{A}$ be a circular
convolution operator on $\ell^2(I)$. The problem under consideration is how to
construct minimal $\Omega \subset I$ and $l_i$ such that $Y=\{\mathbf{e}_i,
\mathbf{A}\mathbf{e}_i, \cdots, \mathbf{A}^{l_i}\mathbf{e}_i: i\in \Omega\}$ is
a frame for $\ell^2(I)$, where $\{\mathbf{e}_i: i\in I\}$ is the canonical
basis of $\ell^2(I)$. This problem is motivated by the spatiotemporal sampling
problem in discrete spatially invariant evolution systems. We will show that
the cardinality of $\Omega $ should be at least equal to the largest geometric
multiplicity of eigenvalues of $\mathbf{A}$, and we consider the universal
spatiotemporal sampling sets $(\Omega, l_i)$ for convolution operators
$\mathbf{A}$ with eigenvalues subject to the same largest geometric
multiplicity. We will give an algebraic characterization for such sampling sets
and show how this problem is linked with sparse signal processing theory and
polynomial interpolation theory.
