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Nonequispaced fast Fourier transform
NFFT on the sphere
NFSFT

NFSFT - NFFT on the sphere

This library of C functions computes evaluates a function $f \in \mathrm{L}^2(\mathbb{S}^2)$ with finite orthogonal expansion \begin{equation*}
f(\vartheta,\varphi) =
\sum_{k=0}^M \sum_{n=-k}^k a_k^n
Y_{k}^n(\vartheta,\varphi) %=
\quad (M \in \mathbb{N}_0)
\end{equation*} in terms of spherical harmonics $Y_k^n$ on a set of arbitary nodes $\left(\vartheta_d,\varphi_d\right)$, $d=1,\ldots,D$, $D \in \mathbb{N}$, in spherical coordinates. Furthermore, the fast evaluation of sums \begin{equation*}
\tilde{a}_{k}^n := \sum_{d=1}^{D} f\left(\vartheta_d,\varphi_d\right)
\overline{Y_{k}^n\left(\vartheta_d,\varphi_d\right)}
\end{equation*} for given function values $f\left(\vartheta_d,\varphi_d\right) \in \mathbb{C}$ and all indices $k=0,\ldots,M$, $n = -k,\ldots,k$ is possible. The algorithm is also kwnon as fast spherical harmonic transform for nonequispaced data.



The algorithms are implemented by Jens Keiner in ./kernel/nfsft. Related paper are


oGräf, M., Kunis, S., Potts, D.
On the computation of nonnegative quadrature weights on the sphere.
Appl. Comput. Harm. Anal., accepted,   (full paper ps, pdf),   2009

oKeiner, J., Potts, D.
Fast evaluation of quadrature formulae on the sphere
Math. Comput. 77, 397 - 419,   (full paper ps, pdf),   2008

oKunis, S. and Potts, D.
Fast spherical Fourier algorithms.
J. Comput. Appl. Math. 161, 75-98. (full paper ps, pdf),   2003

oPotts, D., Steidl G., and Tasche M.
Fast and stable algorithms for discrete spherical Fourier transforms.
Linear Algebra Appl. 275, 433-450. (full paper ps.Z, pdf),   1998