Random sampling and reconstruction of concentrated signals in a reproducing kernel space

Abstract

In this paper, we consider (random) sampling of signals concentrated on a bounded Corkscrew domain $\Omega$ of a metric measure space, and reconstructing concentrated signals approximately from their (un)corrupted sampling data taken on a sampling set contained in $\Omega$. We establish a weighted stability of bi-Lipschitz type for a (random) sampling scheme on the set of concentrated signals in a reproducing kernel space. The weighted stability of bi-Lipschitz type provides a weak robustness to the sampling scheme, however due to the nonconvexity of the set of concentrated signals, it does not imply the unique signal reconstruction. From (un)corrupted samples taken on a finite sampling set contained in $\Omega$, we propose an algorithm to find approximations to signals concentrated on a bounded Corkscrew domain $\Omega$. Random sampling is a sampling scheme where sampling positions are randomly taken according to a probability distribution. Next we show that, with high probability, signals concentrated on a bounded Corkscrew domain $\Omega$ can be reconstructed approximately from their uncorrupted (or randomly corrupted) samples taken at i.i.d. random positions drawn on $\Omega$, provided that the sampling size is at least of the order $\mu(\Omega) \ln (\mu(\Omega))$, where $\mu(\Omega)$ is the measure of the concentrated domain $\Omega$. Finally, we demonstrate the performance of proposed approximations to the original concentrated signal when the sampling procedure is taken either with small density or randomly with large size.