Discrete reproducing kernel Hilbert spaces: sampling and distribution of Dirac-masses

We study reproducing kernels, and associated reproducing kernel Hilbert spaces (RKHSs) H over infinite, discrete and countable sets V . In this setting we analyze in detail the distributions of the corresponding Dirac point-masses of V . Illustrations include certain models from neural networks: An Extreme Learning Machine (ELM) is a neural networkconfiguration in which a hidden layer of weights are randomly sampled, and where the object is then to compute resulting output. For RKHSs H of functions defined on a prescribed countable infinite discrete set V , we characterize those which contain the Dirac masses δx for all points x in V . Further examples and applications where this question plays an important role are: (i) discrete Brownian motion-Hilbert spaces, i.e., discrete versions of the Cameron-Martin Hilbert space; (ii) energy-Hilbert spaces corresponding to graph-Laplacians where the set V of vertices is then equipped with a resistance metric; and finally (iii) the study of Gaussian free fields.