• pamparana
In summary, the conversation discusses the use of spherical harmonics as a basis set of functions over a sphere and their ability to represent any bounded single-valued function. The question is raised about why only bounded functions can be represented and it is suggested that an infinite number of basis functions may be needed for both bounded and unbounded functions. The conversation also touches on the relationship between harmonic order and angular frequency, with higher orders being needed to capture fast changing functions.
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

pamparana said:
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.

pamparana said:
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.

## Question 1: What are spherical harmonics?

Spherical harmonics are mathematical functions that describe the behavior of waves on a sphere. They are used in many fields of science, including physics, chemistry, and geophysics, to model the behavior of physical phenomena on a spherical surface.

## Question 2: How are spherical harmonics different from regular harmonics?

Spherical harmonics are different from regular harmonics in that they are defined on a spherical surface instead of a one-dimensional line. This allows them to represent more complex patterns and behaviors, such as those found in three-dimensional systems.

## Question 3: What are some applications of spherical harmonics?

Spherical harmonics have many applications in science and engineering. They are commonly used in quantum mechanics to describe the behavior of electrons in an atom, in geophysics to model the Earth's magnetic field, and in computer graphics to generate realistic lighting effects on spherical objects.

## Question 4: How are spherical harmonics calculated?

Spherical harmonics are calculated using a series of mathematical equations that involve trigonometric functions and Legendre polynomials. These equations allow us to determine the amplitude and phase of the spherical harmonics at different points on the sphere.

## Question 5: Can spherical harmonics be used in other shapes besides spheres?

Yes, spherical harmonics can be used to model waves on other shapes besides spheres. They can be extended to other surfaces, such as ellipsoids or cylinders, and can also be used in higher dimensions to describe complex three-dimensional shapes.

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