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. Here, we not only collect already known properties in a mathematical way, but also show new ones as well.
All of this is 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. 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. 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.