Positive kernels in Lie group representations and Fourier interpolation

Abstract

For a given function $f$ on a discrete subspace $\Gamma$ of a noncompact Riemannian symmetric space $G/K$, we construct analytic, noncompactly supported functions $W$ on $G/K$ with $W|_{\Gamma}=f$ that, in addition, satisfy an invariant differential equation on $G/K$. 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 $G/K$, and $W$ is expressed in terms of certain closed formulas for linearly independent coherent states.