  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 $\mathscr{H}_{F}$.
