# On spectral representations of tensor random fields on the sphere

## TL;DR

This paper investigates spectral decompositions of tensor random fields on the sphere using rotation group representations, focusing on isotropy, continuity, and Gaussianity of the coefficients.

## Contribution

It introduces a spectral decomposition framework for tensor fields on the sphere based on group representation theory, extending previous scalar field analyses.

## Key findings

- Derived spectral decompositions for tensor fields on the sphere.
- Characterized properties and Gaussianity conditions of spectral coefficients.
- Provided conditions for weak isotropy and mean square continuity.

## Abstract

We study the representations of tensor random fields on the sphere basing on the theory of representations of the rotation group. Introducing specific components of a tensor field and imposing the conditions of weak isotropy and mean square continuity, we derive their spectral decompositions in terms of generalized spherical functions. The properties of random coefficients of the decompositions are characterized, including such an important question as conditions of Gaussianity.