This article may be too technical for most readers to understand.(August 2021) |
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.
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
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]
This article may be too technical for most readers to understand.(August 2021) |
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.
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
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]