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

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.