Sampling with positive definite kernels and an associated dichotomy

We study classes of reproducing kernels $K$ on general domains; these are kernels which arise commonly in machine learning models; models based on certain families of reproducing kernel Hilbert spaces. They are the positive definite kernels $K$ with the property that there are countable discrete sample-subsets $S$; i.e., proper subsets $S$ having the property that every function in $\mathcal{H}\left(K\right)$ admits an $S$-sample representation. We give a characterizations of kernels which admit such non-trivial countable discrete sample-sets. A number of applications and concrete kernels are given in the second half of the paper.