I am using spherical harmonics to approximate the radii of $N$ points on a sphere: $$\sum_{l=0}^{L_{max}}\sum_{m=-l}^{l}\hat{a}_l^mY_l^m(\theta_i,\phi_i) \approx r(\theta_i,\phi_i)\quad \text{where } i =(1,\dots,N)$$
However the points are unevenly distributed (randomly, to be specific). Does this influence the quality of my solution?
I already thought about approximating $r(\theta_i, \phi_i)\cdot s_i$ instead, where $s_i$ is an estimate of the surface area of the points. But I don't know if this is right and sufficient, or if there is some better way.
Edit: Sorry, I probably should have provided more context to make clear why I'm concerned about spherical harmonics in particular:
I'm approximating spheres with slightly noisy points (thus slightly different radii). Then I calculate the amplitudes by taking the L2-Norm $||\hat{a}_l||$ with respect to $m$. By looking at the amplitudes I want to be able to tell a very smooth surface from a very noisy one.
But right now, the amplitude curves in a plot (over $l=0\dots L_{max})$ look very similar for different surfaces. That's why I wondered how the distribution of the points affects my result. A large distortion in a very dense surrounding should have a stronger effect on the amplitude than one in a very sparsely sampled area. The other reason, I can think of why it doesn't quite work is that my amplitude calculation is possibly problematic.
Independent of the fact that they're spherical harmonics: you have a system of $N$ equations in a set of unknowns $\hat{a}^m_l$. There are $(L_\max + 1)^2$ such unknowns, so unless $N \leq (L_\max+1)^2$, your system will likely be underdetermined (multiple solutions will exist); and if $N > (L_\max+1)^2$, you run the risk of having your system be overdetermined (no solutions will exist.)
Viewed this way, using randomly selected points on the sphere is probably a better choice, since you'll be less likely to run into "accidental" degeneracies in your system of equations. As (somewhat trivial) example of this: suppose that instead of picking your point all over the surface of the sphere, you happened to only pick points along the equator. Then none of the coefficients $\hat{a}_l^0$ (where $l$ is odd) would appear in the system of equations, since since $Y_l^0(\pi/2, \phi) = 0$ for all odd values of $l$. Thus, these coefficients would be completely undetermined, no matter how many points you had on (the equator of) the sphere.