# Compact Groups

• M. W. Wong
Chapter
Part of the Operator Theory: Advances and Applications book series (OT, volume 136)

## Abstract

We look at left regular representations L: G → U (L 2 (G)) of compact and Haus-dorff groups G in this chapter. Let ϕ ∈ L 2(G). Then, using Minkowski’s inequality in integral form, the unimodularity of the group G and Schwarz’ inequality, we get
$$\begin{array}{*{20}{c}} \hfill {{{{\left\{ {\int_{G} {|(\varphi ,L(g)\varphi ){{|}^{2}}d\mu (g)} } \right\}}}^{{\tfrac{1}{2}}}} = {{{\left\{ {\int_{G} {\left| {\int_{G} {\varphi (h)\overline {\varphi ({{g}^{{ - 1}}}h)} d\mu (h)\left| {^{2}d} \right.\mu (g)} } \right.} } \right\}}}^{{\tfrac{1}{2}}}}} \\ \hfill { \leqslant \int_{G} {{{{\left\{ {\int_{G} {|\varphi (h){{|}^{2}}|\varphi ({{g}^{{ - 1}}}h){{|}^{2}}d\mu (g)} } \right\}}}^{{\tfrac{1}{2}}}}d\mu (h)} } \\ \hfill { = \int_{G} {|\varphi (h)|{{{\left\{ {\int_{G} {|\varphi ({{g}^{{ - 1}}}h){{|}^{2}}d\mu (g)} } \right\}}}^{{\tfrac{1}{2}}}}} d\mu (h)} \\ \hfill { = \parallel \varphi {{\parallel }_{{{{L}^{1}}(G)}}}\parallel \varphi {{\parallel }_{{{{L}^{2}}(G)}}} \leqslant \mu {{{(G)}}^{{\tfrac{1}{2}}}}\parallel \varphi \parallel _{{{{L}^{2}}(G)}}^{2} < \infty .} \\ \end{array}$$
Thus, every function go in L 2 (G) with $$\parallel \varphi {{\parallel }_{{{{L}^{2}}(G)}}} = 1$$ is an admissible wavelet for the left regular representation L: G → U(L 2 (G)) of G.