Convolutions on the Sphere: Commutation with Differential Operators

TL;DR

This paper introduces a generalized convolution operation on the 2-sphere that commutes with differential operators and preserves vector/tensor normality, enabling advanced analysis of spherical systems in physics.

Contribution

It extends convolution definitions to vectors and tensors on the sphere, proving their commutation with differential operators and equivalence to Helmholtz-based scalar convolutions.

Findings

Convolution commutes with differential operators on the sphere.

Normal/tangent vector and tensor properties are preserved after convolution.

The generalized convolution aligns with traditional scalar convolutions of Helmholtz components.

Abstract

We generalize the definition of convolution of vectors and tensors on the 2-sphere, and prove that it commutes with differential operators. Moreover, vectors and tensors that are normal/tangent to the spherical surface remain so after the convolution. These properties make the new filtering operation particularly useful to analyzing and modeling nonlinear dynamics in spherical systems, such as in geophysics, astrophysics, and in inertial confinement fusion applications. An essential tool we use is the theory of scalar, vector, and tensor spherical harmonics. We then show that our generalized filtering operation is equivalent to the (traditional) convolution of scalar fields of the Helmholtz decomposition of vectors and tensors.