Reproducing kernel for elastic Herglotz functions

We study the elastic Herglotz wave functions, which are entire solutions of the spectral Navier equation appearing in linearized elasticity theory with L^2 -far-field patterns. We characterize in three-dimensions the set of these functions 𝒲, as a closed subspace of a Hilbert space 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 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.