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
_α
(φ) by considering the generator φ as a totally positive function (TPF) of finite type. A function φ is called the TPF of finite type N if ∧φ̂(w)=Π
_l=1
^N
(1+2πiδ
_l
w)
^-1
for 0 ≠ δ
_l
∈ R and N ≥ 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 ∈ V
_α
(φ) by using its average sample values is also provided.