Positive kernels in Lie group representations and Fourier interpolation

For a given function $f$ on a discrete subspace $\mathrm{\Gamma }$ of a noncompact Riemannian symmetric space $G\mathrm{/}K$, we construct analytic, noncompactly supported functions $W$ on $G\mathrm{/}K$ with $W{\mathrm{\mid }}_{\mathrm{\Gamma }}=f$ that, in addition, satisfy an invariant differential equation on $G\mathrm{/}K$. The key step is to regard $W$ as an orthogonal projection onto a space of coherent states in a representation of the Lie group $G$ in a Hilbert space that admits a strictly positive definite reproducing kernel. The techniques are implemented for the heat kernel on $G\mathrm{/}K$, and $W$ is expressed in terms of certain closed formulas for linearly independent coherent states.