Random average sampling and reconstruction in a reproducing kernel space

Abstract

Suppose that signals of interest reside in a reproducing kernel space defined on a metric measure space. We consider the scenario that the sampling positions are distributed on a bounded domain of a metric measure space, and the sampling data are local averages of the original signals in a reproducing kernel space. For signals concentrated on in that reproducing kernel space, we study the stability of this sampling procedure by establishing a weighted sampling inequality of bi‐Lipschitz type. This type of stability implies a weak version of conventional sampling inequality. We propose an iterative algorithm that reconstruct these concentrated signals from finite sampling data. The reconstruction error is characterized through the concentration ratio and the Hausdorff distance between the set of sampling positions and . We also consider the random sampling scheme where the sampling positions are i.i.d. randomly drawn on , and the sampling data are local averages of concentrated signals. We demonstrate that these concentrated signals can be approximated from the random sampling data with high probability when the sampling size is at least of the order with being the measure of .