4 ms·

"Spherical harmonics are orthogonal functions on the sphere, just like Fourier modes are orthogonal functions in normal space" I am stupid. What does this mean

"Spherical harmonics are orthogonal functions on the sphere, just like Fourier modes are orthogonal functions in normal space" I am stupid. What does this mean?

xeonmc 3y ago
A vibrating string form sine waves. A vibrating bubble form spherical harmonics.
- russh 3y ago
  Thank you.
mjhay 3y ago
It doesn't mean you're stupid! You're probably familiar with the concept of two lines (or vectors) being orthogonal (perpendicular) to each other. e.g., two 2-d vectors (0, 1) and (1, 0) are perpendicular. This is equivalent to saying that their dot product is 0: 0*1 + 1*0 = 0 Those are finite, 2d vectors. They also happen to be orthogonal unit length (orthonormal) basis vectors, because you can write any 2-d vector with linear combinations of those: (2, 1) = 2*(1, 0) + 1*(0, 1) We could do the same with any pair of non-parallel 2d vectors, but it's much easier to have an orthonormal basis. The same intuition almost completely carries over to infinite-dimensional vector spaces. Here, vectors are well-behaved functions on the sphere. The inner product (or dot product) of two such functions (e.g. f and g) is the integral of their product over the sphere, which pretty much multiplies the value of both f and g at every point, and adds up the pointwise values by area weighting (some other orthogonal function series are orthogonal under other weightings or spaces). Spherical harmonic functions form an orthonormal basis over the sphere (viewed as an infinite-dimensional vector space), just like those two vectors do over a 2d space. So, two different spherical harmonic functions have an inner product of zero. This makes a lot of things much easier! Fourier modes are very similar. Any nice periodic function on the real line can be written as a sum of the trigonmetric basis functions, called its Fourier series.*

"Spherical harmonics are orthogonal functions on the sphere, just like Fourier modes are orthogonal functions in normal space" I am stupid. What does this mean?

xeonmc 3y ago

A vibrating string form sine waves. A vibrating bubble form spherical harmonics.

russh 3y ago
Thank you.

mjhay 3y ago

It doesn't mean you're stupid! You're probably familiar with the concept of two lines (or vectors) being orthogonal (perpendicular) to each other. e.g., two 2-d vectors (0, 1) and (1, 0) are perpendicular. This is equivalent to saying that their dot product is 0: 0*1 + 1*0 = 0 Those are finite, 2d vectors. They also happen to be orthogonal unit length (orthonormal) basis vectors, because you can write any 2-d vector with linear combinations of those: (2, 1) = 2*(1, 0) + 1*(0, 1) We could do the same with any pair of non-parallel 2d vectors, but it's much easier to have an orthonormal basis. The same intuition almost completely carries over to infinite-dimensional vector spaces. Here, vectors are well-behaved functions on the sphere. The inner product (or dot product) of two such functions (e.g. f and g) is the integral of their product over the sphere, which pretty much multiplies the value of both f and g at every point, and adds up the pointwise values by area weighting (some other orthogonal function series are orthogonal under other weightings or spaces). Spherical harmonic functions form an orthonormal basis over the sphere (viewed as an infinite-dimensional vector space), just like those two vectors do over a 2d space. So, two different spherical harmonic functions have an inner product of zero. This makes a lot of things much easier! Fourier modes are very similar. Any nice periodic function on the real line can be written as a sum of the trigonmetric basis functions, called its Fourier series.*