Question about spherical harmonics

[pamparana]

Hello,

I had posted this in the 'General math' section and did not get any response. Maybe it belongs in this group as it is more related to function decomposition. I hope I am not breaking any forum rules and it is not my intention to cross-post.

Just reading an essay about spherical harmonics and it says that spherical harmonic form a complete orthonormal basis set of functions over the sphere and can be used to represent any bounded single-valued function over a sphere.

I am not sure I understand why we can only represent bounded functions by spherical harmonics. Is it because otherwise we would need an infinite number of the spherical basis functions?

It also says about Spherical harmonics that the angular frequency increases with harmonic order n. Does this mean that to capture fast changing functions, one would need higher harmonic orders?

Thanks,

Luca

 

[Zaphos]

I am not sure I understand why we can only represent bounded functions by spherical harmonics. Is it because otherwise we would need an infinite number of the spherical basis functions?

No, I think you'd potentially need an infinite number of them for a bounded function as well, so that doesn't sound like a good explanation.

It also says about Spherical harmonics that the angular frequency increases with harmonic order n. Does this mean that to capture fast changing functions, one would need higher harmonic orders?

Yes.