Reproducing kernel for elastic Herglotz functions

Abstract

We study the elastic Herglotz wave functions, which are entire solutions of the spectral Navier equation appearing in the linearized elasticity theory with $L^2-$far-field patterns. We characterize in three-dimensions the set of these functions $\mathcal{W},$ as a close subspace of a Hilbert space $\mathcal{H}$ of vector valued functions such that they and their spherical gradients belong to a certain weighted $L^2$ space. This allows us to prove that $\mathcal{W}$ is a reproducing kernel Hilbert space and to calculate the reproducing kernel. Finally, we outline the proof for the two-dimensional case and give the corresponding reproducing kernel.