Nonuniform Sampling, Reproducing Kernels, and the Associated Hilbert Spaces

In a general context of positive definite kernels k, we develop tools and algorithms for sampling in reproducing kernel Hilbert space 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 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 δx of sample-points x to have finite norm relative to the particular RKHS ℋ considered. When this is the case, and the kernel k is given, we obtain an induced positive definite kernel < δx,δy>ℋ. We perform a comparison, and we study when this induced positive definite kernel has l2 rows and columns. The latter task is accomplished with the use of certain symmetric pairs of operators in the two Hilbert spaces, l2 on one side, and the RKHS 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.