Positive convolution squares form a dense inversion core

($g\ast \tilde{g}$ is a compactly supported continuous function.) For $g\in C_c(G;\mathbb{C})$ the conjugate reflection $\tilde{g}$ is continuous with compact support $-\mathrm{supp}g$. For $u,v\in C_c(G;\mathbb{C})$, the defining integral converges everywhere and $\mathrm{\mid }(u\ast v)(x+z)-(u\ast v)(x)\mathrm{\mid }\le \mathrm{\parallel }u{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }{T}_{z}v-v{\mathrm{\parallel }}_1\to 0$ by translation continuity [F3]. Thus the convolution $g\ast \tilde{g}$ is continuous, and its support lies in the compact set $\mathrm{supp}g-\mathrm{supp}g$, so $g\ast \tilde{g}\in C_c(G;\mathbb{C})$.

(Positive definiteness and the value at $0$.) Since $\tilde{g}(z-y)=\overline{g(y-z)}$, the convolution can be written $(g\ast \tilde{g})(z)={\int }_{G}g(y)\overline{g(y-z)}d{m}_G(y)$; at $z=0$ this is ${\int }_G\mathrm{\mid }g(y){\mathrm{\mid }}^{2}d{m}_G(y)\ge 0$. For a finite family $x_1,\dots ,x_n$ and coefficients $c_1,\dots ,c_n$, translation invariance of $m_G$ ([F2]) and the finite sum rule give ${\sum }_{j,k}{c}_j\overline{c_k}(g\ast \tilde{g})(x_j-x_k)={\int }_G{\sum }_{j,k}{c}_j\overline{c_k}g(y)\overline{g(y-x_j+x_k)}d{m}_G(y)={\int }_G\mid {\sum }_{j=1}^n{c}_{j}g(y+x_j){\mid }^{2}d{m}_G(y)\ge 0,$ the second equality by substituting $y\mapsto y+x_j$ term by term (a translation) and expanding the square. Hence $g\ast \tilde{g}$ is positive definite by [F2].

(The span contains every $f\ast \tilde{h}$.) Let $f,h\in C_c(G;\mathbb{C})$. Writing $Q(v)=v\ast \tilde{v}$, direct expansion gives $f\ast \tilde{h}=\frac{1}{4}(Q(f+h)-Q(f-h)+iQ(f+ih)-iQ(f-ih))\mathrm{.}$ Each square lies in $E$ and $E$ is a complex vector space, so $f\ast \tilde{h}\in E$. Hence $E$ contains the complex span $E^{\mathrm{\prime }}$ of $\{f\ast \tilde{h}:f,h\in C_c(G;\mathbb{C})\}$.

(Density.) Let $p\in \{1,2\}$, let $f\in L^p(G,m_G)$ and let $\epsilon >0$. By [F3] choose $h\in C_c(G;\mathbb{C})$ with $\mathrm{\parallel }f-h{\mathrm{\parallel }}_p<\epsilon \mathrm{/}3$, and then, applying the approximate identity of [F3] to $h$, choose $u\in C_c(G;\mathbb{C})$ with $\mathrm{\parallel }u\ast h-h{\mathrm{\parallel }}_p<\epsilon \mathrm{/}3$. By step 1.3 the function $u\ast h=u\ast (\widetilde{\stackrel{~}{h}})$ lies in $E$, and $\mathrm{\parallel }u\ast h-f{\mathrm{\parallel }}_p\le \mathrm{\parallel }u\ast h-h{\mathrm{\parallel }}_p+\mathrm{\parallel }h-f{\mathrm{\parallel }}_p<2\epsilon \mathrm{/}3<\epsilon \mathrm{.}$ Therefore $E$ is dense in $L^p(G,m_G)$ for $p=1$ and $p=2$.

Steps 1.1, 1.2 and 2.1 establish that every $g\ast \tilde{g}$ with $g\in C_c(G;\mathbb{C})$ is a compactly supported continuous positive definite function with $(g\ast \tilde{g})(0)={\int }_G\mathrm{\mid }g{\mathrm{\mid }}^{2}d{m}_G\ge 0$, and that the complex span of these squares is dense in $L^1(G,m_G)$ and in $L^2(G,m_G)$.