Spherical Basis Functions in Hardy Spaces with Localization Constraints

TL;DR

This paper introduces spherical basis functions within Hardy spaces on the sphere, providing approximation error bounds and insights into norm-minimizing vector fields under localization constraints, advancing inverse potential field problem solutions.

Contribution

It presents a novel set of spherical basis functions that belong to Hardy spaces, enabling improved approximation and analysis of localized vector fields in inverse problems.

Findings

Derived error bounds for spherical basis function approximations.

Analyzed norm-minimizing vector fields with localization constraints.

Demonstrated the basis functions' suitability within Hardy spaces.

Abstract

Subspaces obtained by the orthogonal projection of locally supported square-integrable vector fields onto the Hardy spaces $H_+(\mathbb{S})$ and $H_-(\mathbb{S})$, respectively, play a role in various inverse potential field problems since they characterize the uniquely recoverable components of the underlying sources. Here, we consider approximation in these subspaces by a particular set of spherical basis functions. Error bounds are provided along with further considerations on norm-minimizing vector fields that satisfy the underlying localization constraint. The new aspect here is that the used spherical basis functions are themselves members of the subspaces under consideration.