Editing Peter–Weyl theorem (section)

===An example: SU(2)===
We use the standard representation of the group [[Special unitary group#The group SU(2)|SU(2)]] as
:<math> \operatorname{SU}(2) = \left \{ \begin{pmatrix} \alpha&-\overline{\beta}\\ \beta & \overline{\alpha} \end{pmatrix}: \ \ \alpha,\beta\in\mathbb{C},\, |\alpha|^2 + |\beta|^2 = 1\right \}  ~,</math>
Thus, SU(2) is represented as the [[3-sphere]] <math>S^3</math> sitting inside <math>\mathbb{C}^2=\mathbb{R}^4</math>.
The irreducible representations of SU(2), meanwhile, are labeled by a non-negative integer <math>m</math> and can be realized as the natural action of SU(2) on the space of [[homogeneous polynomials]] of degree <math>m</math> in two complex variables.<ref>{{harvnb|Hall|2015}} Example 4.10</ref> The matrix coefficients of the <math>m</math>th representation are [[Spherical harmonics#Higher dimensions|hyperspherical harmonics]] of degree <math>m</math>, that is, the restrictions to <math>S^3</math> of homogeneous harmonic polynomials of degree <math>m</math> in <math>\alpha</math> and <math>\beta</math>. The key to verifying this claim is to compute that for any two complex numbers <math>z_1</math> and <math>z_2</math>, the function 
:<math>(\alpha,\beta)\mapsto (z_1\alpha+z_2\beta)^m</math> 
is harmonic as a function of <math>(\alpha,\beta)\in\mathbb{C}^2=\mathbb{R}^4</math>.

In this case, finding an orthonormal basis for <math>L^2(\operatorname{SU}(2))=L^2(S^3)</math> consisting of matrix coefficients amounts to finding an orthonormal basis consisting of hyperspherical harmonics, which is a standard construction in analysis on spheres.