Spin-weighted spherical harmonics and their application for the construction of tensor slepian functions on the spherical cap

The spin-weighted spherical harmonics of Newman and Penrose (1966) form an orthonormal basis of L²(Ω) on the unit sphere Ω and have a huge field of applications. We present a unified mathematical theory, which implies the collection of already known properties of the spin-weighted spherical harmonics, recapitulated in a mathematical way, and connected to the notation of the spherical harmonics. In addition, we use spin-weighted spherical harmonics to construct tensor Slepian functions on the sphere. Slepian functions are spatially concentrated and spectrally limited. For scalar and vectorial data on the sphere, they are utilized in a variety of disciplines, including geodesy, cosmology, and biomedical imaging. Their concentration within a chosen region of the sphere allows for local inversions when only regional data are available, or enable the extraction of regional information. We focus on the analysis of tensorial fields, as collected e.g.~in the GOCE mission, by means of Slepian functions. For tensorial data, Slepian functions have already been constructed by Eshagh (2009) in the basis of the tensor spherical harmonics of Martinec (2003). By using spin-weighted spherical harmonics, our theory offers several numerical advantages. Furthermore, we present a method for an efficient construction of tensor Slepian functions for spherical caps. In this context, we are able to construct a localized basis on the spherical cap for the cosmic microwave background (CMB) polarization.