Lemma 2

Let K be a compact subset of G. For every $\eta >0$, there exists a function $j\in L^1(G)$ such that:

a) $\|j\|_1\leqslant$ 2 ;

b) the function $\mathscr{F}_G(j)$ is equal to 1 in a neighbourhood of the identity element of $\widehat{G}$;

c) for every $x\in K$, we have $\|j-j*\varepsilon_x\|_1\leqslant \eta$.

The set $U_1$ of the elements $\widehat{x}\in \widehat{G}$ such that

for every $x\in$ K is a neighbourhood of $e$ in $\widehat{G}$. Let U $\subset U_1$ be an open, symmetric, and integrable neighbourhood for the Haar measure $m=d\widehat{x}$ of $\widehat{G}$ dual to the measure $dx$. Let $V\subset U$ be a compact symmetric neighbourhood of $e$ such that $m(V)\geqslant \frac{1}{2}m(U)$. Let us denote by $\varphi_U$ (resp. $\varphi_V$) the characteristic function of U (resp. of V). Since $\varphi_U$ belongs to $L^2(G)$, there exists $u\in L^2(G)$ such that $\varphi_U=\mathscr{F}_G(u)$ (th. 1 of II, p. 215). Analogously, there exists a function $v\in L^2(G)$ such that $\varphi_V=\mathscr{F}_G(v)$. We now prove that the function $j=\frac{1}{m(V)}uv$ verifies the required properties. We have $j\in L^1(G)$.

a) By the Plancherel theorem and the condition $m(V)\geqslant \frac{1}{2}m(U)$, we have

b) There exists a neighbourhood W of $e$ in $\widehat{G}$ such that WV $\subset U$ (TG, II, p. 31, prop. 4). For every $\widehat{x}\in W$, we have $\widehat{x}V\subset U$, and prop. 14 of II, p. 223 implies

since V is symmetric.

c) If $x\in K$, we have

since $U\subset U_1$, and analogously $\|v-v*\varepsilon_x\|^2_2\leqslant m(V)(\frac{\eta}{4})^2$. Hence