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 $[-\pi ,\pi ]\times [-\pi ,\pi ]$ is defined as

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