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.
Introduction
First, a function
on the sphere is written as
using spherical coordinates, i.e.,
![{\displaystyle f(\lambda ,\theta )=f(\cos \lambda \sin \theta ,\sin \lambda \sin \theta ,\cos \theta ),(\lambda ,\theta )\in [-\pi ,\pi ]\times [0,\pi ].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/943c62468a9c362f553f76faf5960f542a27dfc2)
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
![{\displaystyle {\tilde {f))(\lambda ,\theta )={\begin{cases}g(\lambda +\pi ,\theta ),&(\lambda ,\theta )\in [-\pi ,0]\times [0,\pi ],\\h(\lambda ,\theta ),&(\lambda ,\theta )\in [0,\pi ]\times [0,\pi ],\\g(\lambda ,-\theta ),&(\lambda ,\theta )\in [0,\pi ]\times [-\pi ,0],\\h(\lambda +\pi ,-\theta ),&(\lambda ,\theta )\in [-\pi ,0]\times [-\pi ,0],\\\end{cases))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac7c8ce23921906a494e2dc05bdea5ced98d4396)
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
![{\displaystyle {\tilde {f))\approx \sum _{j=-n}^{n}\sum _{k=-n}^{n}a_{jk}e^{ij\theta }e^{ik\lambda ))](https://wikimedia.org/api/rest_v1/media/math/render/svg/dcfcf1c5bcea5231c8cd3da7410c4847267b1364)
History
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]