Reconstruction of signals in quasi shift-invariant spaces from local averages

This article is concerned with the average sampling in quasi shift-invariant spaces $V_{\alpha}(\varphi)$ by considering the generator $\varphi$ as a totally positive function (TPF) of finite type. A function $\varphi$ is called the TPF of finite type $N$ if $\hat{\varphi}(w)=\prod_{l=1}^{N}(1+2\pi i\delta_{l}w)^{-1}$ for $0\neq\delta_{l}\in \mathbb{R}$ and $N\geq 2$. We prove that if sampling points are close enough, then all signals belonging to the quasi shift-invariant space are reconstructed stably and uniquely by using its average sample values. An efficient iterative frame reconstruction algorithm for reconstruction of a signal $g\in V_{\alpha}(\varphi)$ by using its average sample values is also provided.