An alternative to Slepian functions on the unit sphere - A space-frequency analysis based on localized spherical polynomials

TL;DR

This paper introduces a novel space-frequency analysis framework for spherical harmonics that combines ultraspherical polynomial theory and Slepian functions, enabling efficient localized function approximation on the sphere.

Contribution

It develops a new localized basis for spherical harmonics with proven localization and approximation properties, along with a fast computation scheme.

Findings

Proves localization and approximation properties of the new basis.

Provides a scheme for fast coefficient computation.

Analyzes the basis's efficiency in approximating localized functions.

Abstract

In this article, we present a space-frequency theory for spherical harmonics based on the spectral decomposition of a particular space-frequency operator. The presented theory is closely linked to the theory of ultraspherical polynomials on the one hand, and to the theory of Slepian functions on the 2-sphere on the other. Results from both theories are used to prove localization and approximation properties of the new band-limited yet space-localized basis. Moreover, particular weak limits related to the structure of the spherical harmonics provide information on the proportion of basis functions needed to approximate localized functions. Finally, a scheme for the fast computation of the coefficients in the new localized basis is provided.