Compactly supported nonnegative transform bumps on the dual

Given: A locally compact Hausdorff abelian group $G$ with Haar measure $m_G$, its dual $\widehat{G}$ with the compatible dual Haar measure $m_{\widehat{G}}$, a character ${\gamma }_0$ and a compact neighbourhood $K$ of ${\gamma }_0$.

$\widehat{G}$ is a locally compact Hausdorff abelian group; the dual Haar measure is positive on nonempty open sets and finite on compact sets, and every neighbourhood of the identity contains an open symmetric relatively compact neighbourhood whose closure product is as small as desired: for $K$ a compact neighbourhood of ${\gamma }_0$ there is a symmetric open relatively compact $V\ni 1$ with ${\gamma }_0\bar{V}{\bar{V}}^{-1}\subseteq K$. To obtain it, take an open identity neighbourhood $O\subseteq {\gamma }_0^{-1}K$ and use continuity of $(a,b)\mapsto a{b}^{-1}$ to choose symmetric open $W\ni 1$ with $W{W}^{-1}\subseteq O$. Compact shrinking gives open $V_0\ni 1$ with compact closure in $W$; set $V=V_0\cap V_0^{-1}$. (The dual of a locally compact abelian group is locally compact abelian, In a locally compact Hausdorff space every open set containing a point contains an open set containing it whose closure is compact and still inside; such a space is regular, Haar measure is positive on nonempty open sets and finite on compact sets, Locally compact topological space: every point has a compact neighbourhood; and what this says in a metric space, Neighbourhood of a point and neighbourhood base, with this library's convention that a neighbourhood need not be open)

The Plancherel transform $\mathcal{F}:L^2(G)\to L^2(\widehat{G})$ is a unitary operator and its inverse is again unitary; on $L^1(G)\cap L^2(G)$ it agrees with the Fourier transform $\widehat{f}(\chi )={\int }_{G}f(x)\overline{\chi (x)}d{m}_G(x)$. (The Plancherel theorem for locally compact abelian groups, The Fourier transform on an LCA group, The space $L^p(\mu )$ as the quotient by null functions)

Modulation in $L^2$. For $\omega \in \widehat{G}$ and $v\in L^2(G)$ the function $\omega v$ (pointwise product) lies in $L^2(G)$, and $\mathcal{F}(\omega v)(\chi )=\mathcal{F}v({\omega }^{-1}\chi )$ in $L^2(\widehat{G})$. Indeed for $v\in C_c(G)$ this is the $L^1$ modulation identity, both sides are continuous functions of $v\in L^2(G)$ into $L^2(\widehat{G})$ (multiplication by the character and translation of the argument are isometries), and $C_c(G)$ is dense in $L^2(G)$, so the identity extends by continuity. (Fourier transform intertwines translation, modulation and convolution, Translation continuity and normalised local approximate identities on an LCA group, C_c(X) is dense in L^p(mu) for a Radon measure, Compact support, $C_c(X)$, and $C_0(X)$)

If $u,v\in L^2(G)$ then $u\bar{v}\in L^1(G)$ with $\mathrm{\parallel }u\bar{v}{\mathrm{\parallel }}_1\le \mathrm{\parallel }u{\mathrm{\parallel }}_2\mathrm{\parallel }v{\mathrm{\parallel }}_2$. (Cauchy-Schwarz inequality for $L^2$)