In mathematics, 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)
f(λ,\theta)
f(λ,\theta)=f(\cosλ\sin\theta,\sinλ\sin\theta,\cos\theta),(λ,\theta)\in[-\pi,\pi] x [0,\pi].
The function
f(λ,\theta)
2\pi
λ
\theta
[-\pi,\pi] x [-\pi,\pi]
\tilde{f}(λ,\theta)=\begin{cases}g(λ+\pi,\theta),&(λ,\theta)\in[-\pi,0] x [0,\pi],\\ h(λ,\theta),&(λ,\theta)\in[0,\pi] x [0,\pi],\\ g(λ,-\theta),&(λ,\theta)\in[0,\pi] x [-\pi,0],\\ h(λ+\pi,-\theta),&(λ,\theta)\in[-\pi,0] x [-\pi,0],\\ \end{cases}
where
g(λ,\theta)=f(λ-\pi,\theta)
h(λ,\theta)=f(λ,\theta)
(λ,\theta)\in[0,\pi] x [0,\pi]
\tilde{f}
2\pi
λ
\theta
\theta=0
\theta=\pm\pi
The function
\tilde{f}
\tilde{f} ≈
n | |
\sum | |
j=-n |
n | |
\sum | |
k=-n |
ajkeij\thetaeikλ
The DFS method was proposed by Merilees[1] and developed further by Steven Orszag.[2] The DFS method has been the subject of relatively few investigations since (a notable exception is Fornberg's work),[3] 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[4] and to novel space-time spectral analysis.[5]