von Neumann indices and classes of positive definite functions

With view to applications, we establish a correspondence between two problems: (i) the problem of finding continuous positive definite extensions of functions $F$ which are defined on open bounded domains $\Omega$ in $\mathbb{R}$, on the one hand; and (ii) spectral theory for elliptic differential operators acting on $\Omega$, (constant coefficients.) A novelty in our approach is the use of a reproducing kernel Hilbert space $\mathscr{H}_{F}$ computed directly from $\left(\Omega,F\right)$, as well as algorithms for computing relevant orthonormal bases in $\mathscr{H}_{F}$.