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

## TL;DR

This paper develops a framework for constructing isotropic tensor-valued random fields in three-dimensional space, characterized by spectral densities that are either Matérn or dual Matérn, with applications to modeling complex tensor data.

## Contribution

It introduces a new class of homogeneous, isotropic tensor-valued random fields based on spectral densities, extending the Matérn class to tensor fields in 3D.

## Key findings

- Defined classes of tensor-valued random fields with Matérn spectral densities
- Provided examples illustrating the construction of these fields
- Extended the Matérn class concept to tensor-valued random fields

## Abstract

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.