I desperately need some help in understanding spherical harmonics and I would be really grateful if someone could help me understand them intuitively.

So, as I understand SH are another way to represent a function as a linear combination of some basis functions but the function lies on a sphere.

Here lies my first point of confusion. What does it mean when someone says a function is on a sphere. Does that mean that if we parameterize it using polar coordinates, then all the set of points have a constant radius. Is that a correct way to look at it?

Now, in Fourier analysis, we can break a function down as a combination of sin and cosine function. What is the equivalant in spherical harmonics? Are they some elementary functions like the sin and cosine? Also, what does it mean by the order of the spherical harmonics? What is the equivalant in the rectilinear fourier side?

Another thing that I have trouble with is the process of determining the coefficients for both fourier and SH. Normally, if there is a simple image processing software, it can do a fourier transform of the image and the inverse. I appreciate that the signal we supply has to be bandlimited and we can only recover upto a certain frequency. However, there are as many number of coefficients as there are the basis functions. How can one determine the whole set of fourier coefficients given one image? Wouldn't that be a under-determined system? We have one observation(one image) and more than one coefficient to find? I am not even sure if this question makes any sense. I hope it does and someone can help me understand it!

Thanks,

Luca