Consider a function $f$ on $S^2,$ and its spherical transform $\hat{f}.$ Let $r$ be a rotation by some $\rho \in SO(3).$ Is there some nice formula for $\widehat{f \circ r}?$ I have found some allusions to "Wigner D-matrices"... I am sure there are good references...



Spherical transforms and rotations
Suppose your function is $f(s)$ and its spherical transform, as you call it, is $\hat{f}\_{lm}=\int\_{S^2} Y\_{lm}(s)^\* f(s)\,\mathrm{d}s$, where the $Y\_{lm}(S)$ are spherical harmonics. Then for fixed $l$, the components of $\hat{f}$ will transform under a rotation $r\in SO(3)$ in an irreducible representation of $SO(3)$ of spin $l$:
$$
 \hat{f\circ r}\_{lm} = \sum\_{n} D(r)^l\_{m,n} \hat{f}\_{ln}.
$$
The representation maps $D({\cdot})^l\_{m,n}$ are presumably what you are looking for and are precisely the Wigner D-matrices (they are matrices in the $m,n$ indices).


An explicit expression for these matrices depends on how you choose to parametrize your rotations $r\in SO(3)$. If you use Euler angles, then explicit expressions are given on the Wikipedia page for [Wigner D-matrices](http://en.wikipedia.org/wiki/Wigner_D-matrix). If you prefer a different parametrization of $SO(3)$, then another reference might be more convenient. In that case, please refine your question.


