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.

First, a function $f(x, y, z)$ on the sphere is written as $f(\lambda,\theta)$ using spherical coordinates, i.e., 

Failed to parse (syntax error): {\displaystyle 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

$$ 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}