Addition theorems for spin spherical harmonics: I. Preliminaries

Fast Spin ±2 Spherical Harmonics Transforms

An exact fast algorithm is developed for the direct spin-weighted spherical harmonics transforms of bandlimited spin ±2 functions on the sphere. First, we define spin functions on the sphere and their decomposition in an orthonormal basis of spin-weighted spherical harmonics. Second, we discuss the a priori O(L4) asymptotic complexity of the spin ±2 spherical harmonics transforms, where 2L stan...

Spherical Harmonics

5 Spherical Harmonics 7 5.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 5.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5.2.1 Harmonic expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5.2.2 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . ...

Fast spin ±2 spherical harmonics transforms and application in cosmology

An exact fast algorithm is developed for the spin-weighted spherical harmonics transforms of band-limited spin ±2 functions on the sphere. First, we recall the notion of spin functions on the sphere and their decomposition in an orthonormal basis of spin-weighted spherical harmonics. Second, we discuss the a priori O(L4) asymptotic complexity of the spin ±2 spherical harmonics transforms, where...

3 Generalized Spherical Harmonics

Optimal Spherical Harmonics Projection

We introduce the reader to the mathematics behind projection of n-dimensional vectors into a basis on n-dimensional space, where n can be anything upto and including infinity. We show how ideally one wants to project into the duals of a basis if this basis is not orthonormal, and provide the mathematics to formulate this operation in matrix form. The second part of the article discusses spheric...