• pamparana
In summary, Spherical harmonics 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. This is because it is not possible to represent unbounded functions using bounded ones due to the triangle inequality. Additionally, the angular frequency of spherical harmonics increases with harmonic order n, meaning that to capture more complex and fast changing functions, higher harmonic orders will be needed.
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

Last edited:
pamparana said:
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

## What is a bounded function?

A bounded function is a mathematical function where the output values are limited or "bounded" within a certain range. The values of the function cannot exceed this range, making it useful for analyzing and predicting behaviors in various systems.

## What are the properties of a bounded function?

There are a few important properties of a bounded function. First, the range of the function must be finite, meaning that the values cannot go to infinity. Second, the function must have a maximum and minimum value within its range. Finally, the function must be continuous, meaning that there are no breaks or gaps in its graph.

## Why are bounded functions important in science?

Bounded functions are important in science because they help us understand and predict the behavior of natural systems. Many physical and biological processes can be modeled using bounded functions, allowing scientists to make accurate predictions about how these systems will change and evolve over time.

## What are some common examples of bounded functions?

Some common examples of bounded functions include the logistic function, which is used to model population growth, and the sine function, which is used to model oscillating systems such as waves and sound. Other examples include polynomial functions, exponential functions, and trigonometric functions.

## How can we determine if a function is bounded?

To determine if a function is bounded, we can look at its graph and see if the values are limited within a certain range. We can also use mathematical techniques, such as finding the maximum and minimum values of the function, to determine if it is bounded. Additionally, we can check if the function meets the properties of a bounded function.

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