  We consider conditions on a given system $\mathcal{F}$ of vectors in Hilbert
space $\mathcal{H}$, forming a frame, which turn $\mathcal{H}$ into a
reproducing kernel Hilbert space. It is assumed that the vectors in
$\mathcal{F}$ are functions on some set $\Omega$. We then identify conditions
on these functions which automatically give $\mathcal{H}$ the structure of a
reproducing kernel Hilbert space of functions on $\Omega$. We further give an
explicit formula for the kernel, and for the corresponding isometric
isomorphism. Applications are given to Hilbert spaces associated to families of
Gaussian processes.
