We consider convolution sampling and reconstruction of signals in certain reproducing kernel subspaces of L p , 1 ≤ p ≤ ∞. We show that signals in those subspaces could be stably reconstructed from their convolution samples taken on a relatively separated set with small gap. Exponential convergence and error estimates are established for the iterative approximationprojection reconstruction algorithm.

We consider random sampling in finitely generated shift-invariant spaces V (Φ) ⊂ L 2 (R n ) generated by a vector Φ = (ϕ 1 , . . . , ϕ r ) ∈ L 2 (R n ) r . Following the approach introduced by Bass and Gröchenig, we consider certain relatively compact subsets V R,δ (Φ) of such a space, defined in terms of a concentration inequality with respect to a cube with side lengths R. Under very mild assumptions on the generators, we show that for R sufficiently large, taking O(R n log(R n 2 /α ′ )) many random samples (taken independently uniformly distributed within C R ) yields a sampling set for V R,δ (Φ) with high probability. Here α ′ ≤ n is a suitable constant. We give explicit estimates of all involved constants in terms of the generators ϕ 1 , . . . , ϕ r .

In this paper, we investigate the non-uniform weighted average sampling for the weighted multiply generated shift-invariant space. The conditions for sampling point set to be a set of sampling for the weighted multiply generated shift-invariant space are obtained. Furthermore, the explicit bound expressions of a set of sampling inequalities for this shift-invariant spaces also are obtained.

Background: Periodogram analysis of time-series is widespread in biology. A new challenge for analyzing the microarray time series data is to identify genes that are periodically expressed. Such challenge occurs due to the fact that the observed time series usually exhibit non-idealities, such as noise, short length, and unevenly sampled time points. Most methods used in the literature operate on evenly sampled time series and are not suitable for unevenly sampled time series.