Mat\'{e}rn Class Tensor-Valued Random Fields and Beyond

We construct classes of homogeneous random fields on a three-dimensional Euclidean space that take values in linear spaces of tensors of a fixed rank and are isotropic with respect to a fixed orthogonal representation of the group of $3\times 3$ orthogonal matrices. The constructed classes depend on finitely many isotropic spectral densities. We say that such a field belong to either the Mat'{e}rn or the dual Mat'{e}rn class if all of the above densities are Mat'{e}rn or dual Mat'{e}rn. Several examples are considered.