Nonuniform sampling, reproducing kernels, and the associated Hilbert spaces

Abstract

In a general context of positive definite kernels $k$, we develop tools and algorithms for sampling in reproducing kernel Hilbert space $\mathscr{H}$ (RKHS). With reference to these RKHSs, our results allow inference from samples; more precisely, reconstruction of an "entire" (or global) signal, a function $f$ from $\mathscr{H}$, via generalized interpolation of $f$ from partial information obtained from carefully chosen distributions of sample points. We give necessary and sufficient conditions for configurations of point-masses $\delta_{x}$ of sample-points $x$ to have finite norm relative to the particular RKHS $\mathscr{H}$ considered. When this is the case, and the kernel $k$ is given, we obtain an induced positive definite kernel $\left\langle \delta_{x},\delta_{y}\right\rangle _{\mathscr{H}}$. We perform a comparison, and we study when this induced positive definite kernel has $l^{2}$ rows and columns. The latter task is accomplished with the use of certain symmetric pairs of operators in the two Hilbert spaces, $l^{2}$ on one side, and the RKHS $\mathscr{H}$ on the other. A number of applications are given, including to infinite network systems, to graph Laplacians, to resistance metrics, and to sampling of Gaussian fields.