Elastic Herglotz Functions

Solutions of the spectral Navier equation in the linearized theory of elasticity that satisfy the Herglotz boundness condition at infinity are introduced. The leading asymptotic terms in a neighborhood of infinity provide the far-field patterns, and the Herglotz norm is expressed as the sum of the $L^2 $-norms of these patterns over the unit sphere. Basic integral representations that connect the solid spherical Navier eigenvectors to the vector spherical harmonics over the unit sphere are utilized to prove the fundamental representation theorem for the elastic Herglotz solutions in full space. It is shown that the longitudinal and the transverse Herglotz kernels are exactly the corresponding far-field patterns of the irrotational and the solenoidal parts of the displacement field. Particular methods to obtain the displacement field from the far-field patterns, and vice versa, are also described.