Question about bounded functions

[pamparana]

Hello,

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?

EDIT: One more question. It 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

 

[fresh_42]

Hello,

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?

You cannot get unbounded functions from bounded ones because of the triangle inequality.

EDIT: One more question. It 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?

No, whatever capture means. In the end it is a matter of scaling. It is: the more complicated the pattern the more high order functions will be needed.

Thanks,

Luca