Decomposition and harmonic analysis of $L^2(S^n)$ Deitmar and Echterhoff write in their book Principles of Harmonic Analysis that `It follows from the Peter-Weyl theorem that the $SU(2)$ representation on $L^2(S^3)$ is isomorphic to the orthogonal sum $\oplus_{m\ge 0}(m+1)P_m$ where $P_m$ is the space of homogeneous polynomials of degree $m$.'
They provide no further explanation of this statement, so here is my attempt at understanding it: we can use the homeomorphism $SU(2)\simeq S^3$ and let $SU(2)$ act on $L^2(S^3)\simeq L^2(SU(2))$ by 
$$(\pi(g)f)(x)=f(xg).$$
Since the group acting, namely $SU(2)$ is compact the Peter-Weyl theorem says that $L^2(S^3)$ decomposes as a hilbert space direct sum of irreducible representations, namely $(\pi_m, P_m)$ each appearing with multiplicity $\dim \text{Hom} (\pi_m, \pi)$. So it remains to prove that $\dim \text{Hom} (\pi_m, \pi)=m+1.$
Now my first question is whether computing the dimension of the space of intertwiners is elementary.
Second, are there any concrete applications for harmonic analysis on $S^3$ that this result might be helpful for? 
Lastly, how can one come to grips with the decomposition of $L^2(S^n)$ for $n>3$?
Any (partial) answers to any of these questions is highly appreciated.
 A: 1) Your formula for the decomposition of the regular representation is not the whole statement of the Peter-Weyl theorem.  It is part of the Peter-Weyl theorem that the multiplicity $\dim(Hom(\pi_m,\pi))$ in the regular representation equals the dimension of the representation $\dim(\pi_m)$ which is of course $(m+1)$.
2) Computation and properties of spherical harmonics is the most classical applications, and most others stem from that.  See below.
3)  In that case $S^n$ is not itself isomorphic to a Lie group; however, it is still a symmetric space $\textrm{SO}(n)/\textrm{SO}(n-1)$ and we can still use Peter-Weyl theory, only in a more delicate way.  Spherical harmonics arise here as well; see http://www.springer.com/cda/content/document/cda_downloaddocument/9781461479710-c1.pdf?SGWID=0-0-45-1445132-p175259457 for an introduction to these things.
A: Peter-Weyl says a bit more than even what guest states: it tells you how $L^2(G)$ decomposes under the natural $G \times G$ action given by multiplication by $g \in G$ on the left and multiplication by $g^{-1} \in G$ on the right. This decomposition is as a Hilbert space direct sum
$$\bigoplus_V V \boxtimes V^{\ast}$$
where $\boxtimes$ denotes the external tensor product and the direct sum runs over all irreducible representations of $G$. This is exactly what one expects from the finite case, since in the finite case $\mathbb{C}[G]$ is naturally isomorphic to the direct sum $\bigoplus_V \text{End}(V)$ of the endomorphism algebras of the irreducible representations of $G$. Restricting to one copy of $G$ gives the desired multiplicity result, that $V$ appears with multiplicity $\dim V^{\ast} = \dim V$. 
