# Spherical harmonics filtering

Consider a signal $$U$$ defined on the 2-sphere that is expressed as the product of two functions $$A,B$$, or \begin{aligned} U(\theta,\phi) &= \sum_{n=0}^{\infty}\sum_{m=-n}^{n}u_{nm}Y_n^m(\theta,\phi)\\ &= \left[\sum_{n'=0}^{\infty}\sum_{m'=-n'}^{n'}a_{n'm'}Y_{n'}^{m'}(\theta,\phi)\right]\cdot\left[\sum_{n''=0}^{\infty}\sum_{m''=-n''}^{n''}b_{n''m''}Y_{n''}^{m''}(\theta,\phi)\right], \end{aligned} where the coefficients $$a_{nm},b_{nm}$$ are known. Now, imagine my signal $$A$$ is filtered, e.g. $$a_{nm}\rightarrow a_{nm} h_{nm}$$. Is there a modification of any kind that can be performed on the signal $$B$$, for instance some kind of compensating filter $$b_{nm} \rightarrow b_{nm} k_{nm}$$ to retain the same value of $$U$$? If so, can we derive an expression for the filter coefficients $$k_{nm}$$? In a strict Fourier basis, this is trivial, but the spherical harmonics add quite a bit of complexity to this problem unfortunately.

The product of spherical functions can be expanded to series of spherical functions again. It is usually done with Wigner 3-j symbols or Clebsch-Gordan coefficients (formulas there are for complex spherical functions, but one can derive from them real-valued ones). This will give you an infinite linear system on coefficients $$b_{mn}$$. However, you can trim your expansion at some azimuthal number and solve a finite system