  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.
