  In this note, we solve the dynamical sampling problem for a class of
shift-preserving operators $L:V\to V$ acting on a finitely generated
shift-invariant space $V$. We find conditions on $L$ and a finite set of
functions of $V$ so that the iterations of the operator $L$ on the functions
produce a frame generator set of $V$. This means that the integer translations
of the generators form a frame of $V$.
