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Journal of Geodesy

, Volume 92, Issue 2, pp 123–130 | Cite as

Transformation between surface spherical harmonic expansion of arbitrary high degree and order and double Fourier series on sphere

Original Article

Abstract

In order to accelerate the spherical harmonic synthesis and/or analysis of arbitrary function on the unit sphere, we developed a pair of procedures to transform between a truncated spherical harmonic expansion and the corresponding two-dimensional Fourier series. First, we obtained an analytic expression of the sine/cosine series coefficient of the \(4 \pi \) fully normalized associated Legendre function in terms of the rectangle values of the Wigner d function. Then, we elaborated the existing method to transform the coefficients of the surface spherical harmonic expansion to those of the double Fourier series so as to be capable with arbitrary high degree and order. Next, we created a new method to transform inversely a given double Fourier series to the corresponding surface spherical harmonic expansion. The key of the new method is a couple of new recurrence formulas to compute the inverse transformation coefficients: a decreasing-order, fixed-degree, and fixed-wavenumber three-term formula for general terms, and an increasing-degree-and-order and fixed-wavenumber two-term formula for diagonal terms. Meanwhile, the two seed values are analytically prepared. Both of the forward and inverse transformation procedures are confirmed to be sufficiently accurate and applicable to an extremely high degree/order/wavenumber as \(2^{30}\,{\approx }\,10^9\). The developed procedures will be useful not only in the synthesis and analysis of the spherical harmonic expansion of arbitrary high degree and order, but also in the evaluation of the derivatives and integrals of the spherical harmonic expansion.

Keywords

Associated Legendre function Fourier series expansion Recurrence formula Spherical harmonic expansion Wigner d function 

Notes

Acknowledgements

The author appreciates valuable suggestions and fruitful comments by anonymous referees to improve the readability of the article.

Supplementary material

190_2017_1049_MOESM1_ESM.pdf (320 kb)
Supplementary material 1 (pdf 320 KB)

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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.National Astronomical Observatory/SOKENDAIMitakaJapan

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