- Introduction
- History
- References
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.,
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
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
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]
References
2. ^S. A. Orszag, Fourier series on spheres, Mon. Wea. Rev., 102 (1974), pp. 56–75.
3. ^B. Fornberg, A pseudospectral approach for polar and spherical geometries, SIAM J. Sci. Comp, 16 (1995), pp. 1071–1081
4. ^R. Bartnik and A. Norton, Numerical methods for the Einstein equations in null quasispherical coordinates, SIAM J. Sci. Comp, 22 (2000), pp. 917–950
5. ^C. Sun, J. Li, F.-F. Jin, and F. Xie, Contrasting meridional structures of stratospheric and tropospheric planetary wave variability in the northern hemisphere, Tellus A, 66 (2014)