Computing with functions in spherical and polar geometries I. The sphere

Understand

A collection of algorithms is described for numerically computing with smooth functions defined on the unit sphere.

• Functions are approximated to essentially machine precision by using a structure-preserving iterative variant of Gaussian elimination together with the double Fourier sphere method.
• We show that this procedure allows for stable differentiation, reduces the oversampling of functions near the poles, and converges for certain analytic functions.
• Operations such as function evaluation, differentiation, and integration are particularly efficient and can be computed by essentially one-dimensional algorithms.