Fourier analysis for compact groups

## Fourier analysis for compact groups

A function $f:G\to ℂ$ is

1. representative if there is a finite dimensional representation $V$ of $G$ and vectors $v,w\in V$ such that $f\left(g\right)=⟨v,gw⟩$ for all $g\in G.$
2. square integrable if $∥f∥ 2 2 = ∫ G f g$ f g dμ g <.
3. smooth if all derivatives exist.
4. real analytic if $f$ has a power series expansion at each point.

We have a map

The set $\stackrel{^}{G}$ has a norm $\parallel .\parallel :\stackrel{^}{G}\to {ℝ}_{\ge 0}.$ For $\left({\stackrel{^}{f}}^{\lambda }\right)\in \prod _{\lambda \in \stackrel{^}{G}}{M}_{{d}_{\lambda }}\left(ℂ\right)$ define

1. $\left({\stackrel{^}{f}}^{\lambda }\right)$ is finite if all but a finite number of the blocks ${\stackrel{^}{f}}^{\lambda }$ in $\left({\stackrel{^}{f}}^{\lambda }\right)$ are 0,
2. $\left({\stackrel{^}{f}}^{\lambda }\right)$ is square summable if $∑ λ∈ G ^ 1 d λ ∥ f λ ∥ 2 <∞.$
3. $\left({\stackrel{^}{f}}^{\lambda }\right)$ is rapidly decreasing if, for all $k\in {ℤ}_{>0},\left\{{\parallel \lambda \parallel }^{k}\parallel {\stackrel{^}{f}}^{\lambda }\parallel \phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}\lambda \in \stackrel{^}{G}\right\}$ is bounded,
4. $\left({\stackrel{^}{f}}^{\lambda }\right)$ is exponentially decreasing if, for some $K\in {ℝ}_{>1},\left\{{K}^{\parallel \lambda \parallel }\parallel {\stackrel{^}{f}}^{\lambda }\parallel \phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}\lambda \in \stackrel{^}{G}\right\}$ is bounded.

Under the map

The space $C{\left(G\right)}^{\mathrm{rep}}$ is dense in $C\left(G\right)$ and $C\left(G\right)\subseteq {L}^{2}\left(G\right).$ In fact the sup norm on $C\left(G\right)$ is related to the ${L}^{2}$ norm on ${L}^{2}\left(G\right)$ and $C\left(G\right)$ is dense in ${L}^{2}\left(G\right).$