Double Fourier sphere method

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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.

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

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