Double Fourier sphere method

In mathematics, the double Fourier sphere 'method' is a technique that transforms a function defined on the surface of the sphere to a function defined on a rectangular domain while preserving periodicity in both the longitude and latitude directions.

Introduction

First, a function on the sphere is written as using spherical coordinates, i.e.,
The function is -periodic in, but not periodic in. The periodicity in the latitude direction has been lost. To recover it, the function is "doubled up” and a related function on is defined as
where and for. The new function is -periodic in and, and is constant along the lines and, corresponding to the poles.
The function can be expanded into a double Fourier series

History

The DFS method was proposed by Merilees and developed further by Steven Orszag. The DFS method has been the subject of relatively few investigations since, perhaps due to the dominance of spherical harmonics expansions. Over the last fifteen years it has begun to be used for the computation of gravitational fields near black holes and to novel space-time spectral analysis.