Optimal sampling points in reproducing kernel Hilbert spaces

Abstract

The recent development of compressed sensing seeks to extract information from as few samples as possible. In such applications, since the number of samples is restricted, one should deploy the sampling points wisely. We are motivated to study the optimal distribution of finite sampling points in reproducing kernel Hilbert spaces, the natural background function spaces for sampling. Formulation under the framework of optimal reconstruction yields a minimization problem. In the discrete measure case, we estimate the distance between the optimal subspace resulting from a general Karhunen–Loève transform and the kernel space to obtain another algorithm that is computationally favorable. Numerical experiments are then presented to illustrate the effectiveness of the algorithms for the searching of optimal sampling points.