Proposition 15

Let $f\in L^1(G)$ be such that the restriction to $H^{\bot}$ of the continuous function $\mathscr{F}_G(f)$ is integrable with respect to $\widehat{\gamma}$. Then, for almost every $x\in G$, the function $y\mapsto f(xy)$ on H is $\alpha$-integrable, and one has:

From the above, the function $f^{\flat}$ defined almost everywhere on $G/H$ by

belongs to $L^1(G/H)$. The Fourier transform of $f^{\flat}$ is identified with the function on $H^{\bot}=\widehat{G}/H$ given for $\chi \in H^{\bot}$ by

by formula (33), applied to the integrable function $x\mapsto$ $\overline{\langle\chi , x\rangle}f(x)$. By hypothesis, the function $\mathscr{F}(f)|H^{\bot}=\mathscr{F}_{G/H}(f^{\flat})$ belong to $L^1(H^{\bot})$, and therefore the function $f^{\flat}$ belong to the space $B(G/H)$. It follows (Theorem 3 of II, p. 222) that $f^{\flat}$ coincides almost everywhere with $\overline{\mathscr{F}}_{\widehat{G}/H}(\mathscr{F}_{G/H}(f^{\flat}))$. For almost all $\dot{x}\in G/H$, one therefore has

This completes the proof.