# spherical harmonics

## 1. basis functions

The spherical harmonics are a basis for all square integrable functions on \(S^2\) So for \(f: S^2 \rightarrow \mathbb{C}\) \(f(\theta, \phi) = \sum_{l=0}^{\infty}\sum_{m=-l}^{m=l}f_l^m Y_l^m(\theta, \phi)\)

## 2. what does it mean for spherical harmonics to be a representation of SO(3)

My rough understanding of things: for a given \(l\), consider all the \(Y_l^m\) where \(m=-l,-l+1,...,l\). Consider the space of functions in \(span(Y_l^m)\). Then, this span is a vector space. And there is some representation \(D^l\) such that SO(3) acts on this vector space through \(D^l\). Importantly, what happens when you apply a rotation to a \(Y_l^m\)? You get a new function, which can be written as a linear combination of all \(Y_l^m\). Bottom line: a "rotated" \(Y_l^m\) can be written as a linear function of the "unrotated" \(Y_l^m\) 's

- see stack overflow answer – this gives a brief description of the vector space spanned by spherical harmonics and the operators that act on this vector space
- stack overflow answer – summarizes some definitions, but the question I didn't read completely
- stack overflow answer – the spherical harmonics get rotated and are still a linear combination of spherical harmonics
- see group representation
- see lie group representation

## 3. coding it up

- given a vector \(x\) – a point in 3D space
- \(Y_l: \mathbb{R}^3 \rightarrow \mathbb{C}^{2l+1}\)
- at each index \(m\), \(Y_l[m] =\) the value of spherical harmonic \(Y_l^m\) at \(x\)

- can we see how \(Y_l(x)\) is equivariant to rotations? (see group invariant and group equivariant functions)
- imagine applying rotation \(R\) to \(x\). To have an equivariant function, we want \(Y_l(Rx) = D^lY_l(x)\)
- What is \(D_l\)? It is a transformation. More precisely, it is the
*matrix representation*of the transformation that takes "unrotated" spherical harmonics and gives the "rotated" spherical harmonics in terms of linear combinations of the "unrotated" spherical harmonics. - This is a point that that confused me on the homework: specifically, it takes the values of the spherical harmonics at \(x\) and gives us the values of the "rotated" spherical harmonics, in terms of the values of the "unrotated" spherical harmonics at \(x\)

## 4. spherical harmonics equivariant under SO(3)

- see stack overflow