Parseval pairing on the integrable core

Given: The Axiom of Choice and Dependent Choice, a locally compact Hausdorff abelian group $G$ with Haar measure $m_G$ and compatible dual Haar measure $m_{\widehat{G}}$ on $\widehat{G}$, and $f,g\in L^1(G,m_G)\cap L^2(G,m_G)$ with $\widehat{f},\widehat{g}\in L^1(\widehat{G},m_{\widehat{G}})$.

$A=L^1(G,m_G)$ is a commutative Banach $\ast$-algebra under convolution and $u^{\ast }(x)=\overline{u(-x)}$; for $u,v\in A$ the class $u\ast v$ is given $m_G$-a.e. by an absolutely convergent integral and $\mathrm{\parallel }u\ast v{\mathrm{\parallel }}_1\le \mathrm{\parallel }u{\mathrm{\parallel }}_1\mathrm{\parallel }v{\mathrm{\parallel }}_1$, and $\mathrm{\parallel }{u}^{\ast }{\mathrm{\parallel }}_1=\mathrm{\parallel }u{\mathrm{\parallel }}_1$, $u^{\ast \ast }=u$ (L^1 of an LCA group is a commutative Banach star algebra under convolution); for $g\in L^2$ also $g^{\ast }\in L^2$ with $\mathrm{\parallel }{g}^{\ast }{\mathrm{\parallel }}_2=\mathrm{\parallel }g{\mathrm{\parallel }}_2$ because inversion preserves Haar measure and conjugation preserves moduli (Haar measure on an abelian group is invariant under inversion, The space $L^p(\mu )$ as the quotient by null functions).

The transform satisfies $\widehat{u\ast v}=\widehat{u}\widehat{v}$ and $\widehat{u^{\ast }}=\bar{\widehat{u}}$ (Fourier transform intertwines translation, modulation and convolution); in particular $\widehat{f\ast g^{\ast }}=\widehat{f}\bar{\widehat{g}}$, and since $\widehat{f}$ is bounded and $\bar{\widehat{g}}\in L^1(\widehat{G},m_{\widehat{G}})$, this product lies in $L^1(\widehat{G},m_{\widehat{G}})$ with $\mathrm{\parallel }\widehat{f}\bar{\widehat{g}}{\mathrm{\parallel }}_1\le \mathrm{\parallel }\widehat{f}{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }\widehat{g}{\mathrm{\parallel }}_1$ (The Fourier transform on an LCA group).

For $f,g\in L^2(G,m_G)$ the integral $H(x):={\int }_{G}f(y)\overline{g(y-x)}d{m}_G(y)$ converges absolutely for every $x$ with $\mathrm{\mid }H(x)\mathrm{\mid }\le \mathrm{\parallel }f{\mathrm{\parallel }}_2\mathrm{\parallel }g{\mathrm{\parallel }}_2$ by Cauchy-Schwarz (Cauchy-Schwarz inequality for $L^2$), and $H$ is continuous: $\mathrm{\mid }H(x)-H(x_0)\mathrm{\mid }\le \mathrm{\parallel }f{\mathrm{\parallel }}_2\mathrm{\parallel }{T}_{x}g-T_{x_0}g{\mathrm{\parallel }}_2\to 0$ by norm continuity of translations in $L^2$ (Translation continuity and normalised local approximate identities on an LCA group, The space $L^p(\mu )$ as the quotient by null functions).

The convolution $h:=f\ast g^{\ast }$ lies in $L^1(G,m_G)$ with $\mathrm{\parallel }h{\mathrm{\parallel }}_1\le \mathrm{\parallel }f{\mathrm{\parallel }}_1\mathrm{\parallel }g{\mathrm{\parallel }}_1$, its representative $H$ of [F3] satisfies $H=h$ $m_G$-a.e., and its transform $\widehat{h}=\widehat{f}\bar{\widehat{g}}$ lies in $L^1(\widehat{G},m_{\widehat{G}})$ (L^1 of an LCA group is a commutative Banach star algebra under convolution, Fourier transform intertwines translation, modulation and convolution, Integrable real and complex functions, and their integrals).

Fourier inversion for integrable transforms: if $h\in L^1(G,m_G)$ has $\widehat{h}\in L^1(\widehat{G},m_{\widehat{G}})$, then $h^{\vee }(x)={\int }_{\widehat{G}}\widehat{h}(\gamma )\gamma (x)d{m}_{\widehat{G}}(\gamma )$ is a bounded uniformly continuous function with $h^{\vee }=h$ $m_G$-a.e., and $h^{\vee }$ is the unique continuous representative of the class of $h$ (Fourier inversion for integrable transforms on LCA groups, Compatible dual Haar normalisation).