S2FunHarmonic.sphericalHarmonicTrafo

The spherical harmonic transform transfers a given harmonic series \[ \sum_{n=0}^N\sum_{k=-n}^n \hat{f}_n^{k} Y_n^{k}(\theta,\phi)\] into a bivariate Fourier series \[ \sum_{k,j=-N}^N \hat{g}_{k,j} e^{-i \, (k\alpha+j\beta+l\gamma)}.\] Therefore we just transform the harmonic coefficients \(\hat{f}_n^{k}\) into Fourier coefficients \(\hat{g}_{k,j}\) by the linear operator \[\hat{g}_{k,j} = i^{k} \, \sum_{n = \max \{|k|,|j|\} }^N \sqrt{2n+1}\, \hat{f}_n^{k} \, d_n^{j,k}(0) \, d_n^{j,0}(0).\]

Normally the indices of the output Fourier array ghat(k,j) runs over k,j=-N,...,N.

If SO3F is real valued the Fourier array ghat(k,j) is of size k = 0,...,N j = -N,...,N.

If we want to use the NFFT on this Fourier array, we have to make the size even, as the index set of the NFFT is -(N+1),...,N. Hence the flag 2^1 (make output even) yields ghat(k,j) of size k = 0,...,N+mod(N+1,2) j = -(N+1),...,N

flags: 2^0 -> use L_2-normalized Wigner-D functions 2^1 -> make size of output Fourier array (ghat) even in every dimension 2^2 -> fhat are the Fourier coefficients of a real valued function 2^3 -> fhat are the Fourier coefficients of a antipodal function 2^4 -> use symmetry property (not implemented yet)

Syntax

ghat = sphericalHarmonicTrafo(sF)
ghat = sphericalHarmonicTrafo(sF,flags,'bandwidth',N)

Input

Output