On the Structure and Interpolation Properties of Quasi Shift-invariant Spaces

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.