Double Fourier sphere method

In mathematics, the double Fourier sphere (DFS) method is a 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 ${\displaystyle f(x,y,z)}${\displaystyle f(x,y,z)} on the sphere is written as ${\displaystyle f(\lambda ,\theta )}${\displaystyle f(\lambda ,\theta )} using spherical coordinates, i.e.,

${\displaystyle f(\lambda ,\theta )=f(\cos \lambda \sin \theta ,\sin \lambda \sin \theta ,\cos \theta ),(\lambda ,\theta )\in [-\pi ,\pi ]\times [0,\pi ].}${\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 ${\displaystyle f(\lambda ,\theta )}${\displaystyle f(\lambda ,\theta )} is ${\displaystyle 2\pi }${\displaystyle 2\pi }-periodic in ${\displaystyle \lambda }${\displaystyle \lambda }, but not periodic in ${\displaystyle \theta }${\displaystyle \theta }. The periodicity in the latitude direction has been lost. To recover it, the function is "doubled up" and a related function on ${\displaystyle [-\pi ,\pi ]\times [-\pi ,\pi ]}${\displaystyle [-\pi ,\pi ]\times [-\pi ,\pi ]} is defined as

${\displaystyle {\tilde {f}}(\lambda ,\theta )={\begin{cases}g(\lambda +\pi ,\theta ),&(\lambda ,\theta )\in [-\pi ,0]\times [0,\pi ],\\h(\lambda ,\theta ),&(\lambda ,\theta )\in [0,\pi ]\times [0,\pi ],\\g(\lambda ,-\theta ),&(\lambda ,\theta )\in [0,\pi ]\times [-\pi ,0],\\h(\lambda +\pi ,-\theta ),&(\lambda ,\theta )\in [-\pi ,0]\times [-\pi ,0],\\\end{cases}}}${\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}}}

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

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

${\displaystyle {\tilde {f}}\approx \sum _{j=-n}^{n}\sum _{k=-n}^{n}a_{jk}e^{ij\theta }e^{ik\lambda }}${\displaystyle {\tilde {f}}\approx \sum _{j=-n}^{n}\sum _{k=-n}^{n}a_{jk}e^{ij\theta }e^{ik\lambda }}

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]

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