  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 $\mathscr{H}\left(K\right)$ 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.
