Compute the S2-Fourier/harmonic coefficients of an given S2Fun or given evaluations on a specific quadrature grid.
Therefore we obtain the Fourier coefficients with numerical integration (quadrature), i.e. we choose a quadrature scheme of meaningful quadrature nodes \(v_m\) and quadrature weights \(\omega_m\) and compute
\[ \hat f_n^{k} = \int_{S^2} f(v)\, \overline{Y_n^{k}(v)} \mathrm{d}\my(v) \approx \sum_{m=1}^M \omega_m \, f(v_m) \, \overline{Y_n^{k}(v_m)}, \]
for all \(n=0,\dots,N\) and \(k=-n,\dots,n\).
Therefore this method evaluates the given S2Fun on the quadrature grid. Afterwards it uses the adjoint NFSFT (nonequispaced fast spherical Fourier transform) to quickly compute the above sums.
Syntax
sF = S2FunHarmonic.quadrature(nodes,values,'weights',w)
sF = S2FunHarmonic.quadrature(f)
sF = S2FunHarmonic.quadrature(f, 'bandwidth', bandwidth)Input
| values | double (first dimension has to be the evaluations) |
| nodes | vector3d |
| f | function handle in vector3d (first dimension has to be the evaluations) |
Output
| sF | S2FunHarmonic |
Options
| bandwidth | minimal degree of the spherical harmonic (default: 128) |