  The structure of certain types of quasi shift-invariant spaces, which take
the form
$V(\psi,\mathcal{X}):=\overline{\text{span}}^{L_2}\{\psi(\cdot-x_j):j\in\mathbb{Z}\}$
for a discrete set $\mathcal{X}=(x_j)\subset\mathbb{R}$ is investigated.
Additionally, the relation is explored between pairs $(\psi,\mathcal{X})$ and
$(\phi,\mathcal{Y})$ such that interpolation of functions in
$V(\psi,\mathcal{X})$ via interpolants in $V(\phi,\mathcal{Y})$ solely from the
samples of the original function is possible and stable. Some conditions are
given for which the sampling problem is stable, and for which recovery of
functions from their interpolants from a family of spaces
$V(\phi_\alpha,\mathcal{Y})$ is possible.
