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

## TL;DR

This paper investigates the structure of quasi shift-invariant spaces, explores conditions for stable interpolation and sampling, and examines the relationship between different such spaces for function recovery.

## Contribution

It provides new insights into the structure of quasi shift-invariant spaces and establishes conditions for stable interpolation and sampling within these spaces.

## Key findings

- Conditions for stable sampling are identified.
- Criteria for function recovery from interpolants are established.
- Relations between different quasi shift-invariant spaces are characterized.

## Abstract

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.