Imagine a pdf $f$ that is defined on the surface of the sphere, and that I have the theoretical mathematical description and would also like to experimentally measure. The results obtained from experiments are not values at individual points though, but the spherical harmonic expansion coefficients of the $f$ function.
So on one hand, I have the smooth mathematical description of the function and can calculate the SH coefficients as reference values; on the other hand, I have the SH coefficients measured via experiments and can reconstruct the smooth function f accordingly too. If I just want a quantitative value to evaluate how similar my experimental results are with the theoretical solutions, what tool should I use please? I've been suggested Hellinger distance for the indication of similarity between PDFs, is that the best way to go please? Many thanks in advance!