Nonuniform Average Sampling and Reconstruction of Signals with Finite Rate of Innovation

From an average (ideal) sampling/reconstruction process, the question arises whether and how the original signal can be recovered from its average (ideal) samples. We consider the above question under the assumption that the original signal comes from a prototypical space modelling signals with finite rate of innovation, which includes finitely-generated shift-invariant spaces, twisted shift-invariant spaces associated with Gabor frames and Wilson bases, and spaces of polynomial splines with non-uniform knots as its special cases. We show that the displayer associated with an average (ideal) sampling/reconstruction process, that has well-localized average sampler, can be found to be well-localized. We prove that the reconstruction process associated with an average (ideal) sampling process is robust, locally behaved, and finitely implementable, and thus we conclude that the original signal can be approximately recovered from its incomplete average (ideal) samples with noise in real time. Most of our results in this paper are new even for the special case that the original signal comes from a finitely-generated shift-invariant space.