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

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

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

where $g(\lambda ,\theta )=f(\lambda -\pi ,\theta )$ and $h(\lambda ,\theta )=f(\lambda ,\theta )$ for $(\lambda ,\theta )\in [0,\pi ]\times [0,\pi ]$. The new function ${\tilde {f}}$ is $2\pi$-periodic in $\lambda$ and $\theta$, and is constant along the lines $\theta =0$ and $\theta =\pm \pi$, corresponding to the poles.

The function ${\tilde {f}}$ can be expanded into a double Fourier series

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-timespectral analysis.^{[5]}

References

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^P. E. Merilees, The pseudospectral approximation applied to the shallow water equations on a sphere, Atmosphere, 11 (1973), pp. 13–20

^S. A. Orszag, Fourier series on spheres, Mon. Wea. Rev., 102 (1974), pp. 56–75.

^B. Fornberg, A pseudospectral approach for polar and spherical geometries, SIAM J. Sci. Comp, 16 (1995), pp. 1071–1081

^R. Bartnik and A. Norton, Numerical methods for the Einstein equations in null quasispherical coordinates, SIAM J. Sci. Comp, 22 (2000), pp. 917–950

^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)