Is there a way of expanding a scalar field defined on a sphere of radius R in the base of spherical harmonic functions? Everywhere I read about expansions on the unit sphere. What changes if one would try this for a function on a sphere of arbitrary radius?
Edit: (I want to precise my question) The particular problem I am dealing with is numerical and as follows:
I have two functions $f(\theta, \phi)$ and $g(\theta, \phi)$, they are both defined on a sphere of radius R. I would like to compute the convolution $f *g$ of these two functions. If I they are expanded in the a series of spherical harmonic functions I can use the following convolution theorem:
$\hat{(f*g)}_{l,m} = \sqrt{\frac{4 \pi}{2l+1}} \hat{f}_{l,m} . \hat{g}_{l,0}$
As far as I understood the expansion and the convolution theorem only work for functions on the unit sphere. Is there any way I use this method for a sphere of radius R?
Edit2: Numerically the functions are defined on a $\theta$-$\phi$-grid. So my idea was to compute the coefficients $a_{l,m}$, then convolve in the spherical harmonic space and then expand to be in normal space again.
If $f(x)$ is defined over a sphere of radius R, then $f(Rx)$ is defined over the unit sphere. Try expanding this scalar field first, and then scale appropriately.