Proposition 1

Every square integrable complex function $f$ on G is the sum in the Hilbert space $L^2(G)$ of the family $(f_u)_{u\in\widehat{G}}$ where, for all $h\in G$ and all $u\in \widehat{G}$,

$=d(u)\int_Gf(g$)Tr($u(gh^{-1})$)$dg=d(u)\int_Gf(gh$)Tr($u(g)$)$dg$. (10)

For all $u\in \widehat{G}$ choose an orthonormal basis $Bu$ of $E_u$, and denote by $(u_{ij}(g))$ the matrix of $u(g)$ in this basis. Prop. 1 also means that the family of functions $\surd\overline{d(u)}u_{ij}$, for $u$ in $\widehat{G}$ and $i, j$ in $B_u$, is an orthonormal basis of the space $L^2(G)$.

If $f$ is an integrable function on G such that the family $(f_u)$ is uniformly summable, then the sum of this family is a continuous function which coincides almost everywhere with $f$; in other words, if we assume in addition that $f$ is continuous, then for all $h\in G$, $f(h) =\sum_{u\in\widehat{G}}d(u)\int_Gf(gh$)Tr($u(g)$)$dg$. (11)

Conversely, let $A\in F( \widehat{G})$; if the family $(\mathscr{F}_uA)_{u\in\widehat{G}}$ is uniformly summable, the function

$g \rightarrow \sum_{u\in\widehat{G}}(\mathscr{F}_uA)(g) =\sum_{u\in\widehat{G}}d(u$)Tr(A$_uu(g)^{-1}$)

is a continuous function on G whose Fourier cotransform is A.

Let $f$ be an integrable function on G, and let $s\in G$. Denote by $\gamma (s)f$ and $\delta (s)f$ the functions on G defined by $\gamma (s)f=\varepsilon_s*f,\delta (s)f=f*\varepsilon_{s^{-1}}$, that is,

$(\gamma (s)f)(g) =f(s^{-1}g),(\delta (s)f)(g) =f(gs)$ for $g\in G$,

(Chap. III, §3, no. 4 and Integration, Chap. VII, §1, no. 1). We have

so

and similarly

When G is commutative, $\widehat{G}$ is the underlying set of the dual group of G (Spectral Theories, Chap. II, §1, no. $1$)$,d(u) = 1$ for all $u\in \widehat{G}$, and we recover the definitions of the Fourier transform given in Spectral Theories, Chap. II.