Double Fourier sphere method

The Double Fourier Sphere (DFS) method is a simple 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 $f(x,y,z)$ on the sphere is written as $f(\lambda ,\theta )$ using spherical coordinates, i.e.,

$f(\lambda ,\theta )=f(\cos \lambda \sin \theta ,\sin \lambda \sin \theta ,\cos \theta ),(\lambda ,\theta )\in [-\pi ,\pi ]\times [0,\pi ].$

The function $f(\lambda ,\theta )$ is $2\pi$-periodic in $\lambda$, but not periodic in $\theta$. The periodicity in the latitude direction has been lost. To recover it, the function is "doubled up” and a related function on Failed to parse (syntax error): {\displaystyle [−\pi, \pi]\times[−\pi, \pi]} is defined as

Failed to parse (unknown function "\begin{cases}"): {\displaystyle f(\lambda,\theta) = \begin{cases} g(\lambda + \pi, \theta), & (\lambda, \theta) \in [−\pi, 0] × [0, \pi],\\ h(\lambda, \theta), &(\lambda, \theta) \in [0, \pi] × [0, \pi],\\ g(\lambda, −\theta), &(\lambda, \theta) \in [0, \pi] × [−\pi, 0],\\ h(\lambda + \pi, −\theta), &(\lambda, \theta) \in [−\pi, 0] × [−\pi, 0],\\ \end{cases}}