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Let $G$ be a locally compact abelian group (non compact) and $\mu$ a Haar measure on $\hat{G}$ given by the Plancherel Theorem. Suppose $f\in L^2(G)$. I would like to write a formula as $$ f=\int_{\hat{G}} \gamma(\cdot) \hat{f}(\gamma)d\mu(\gamma) $$ in $L^2(G)$. What is allowed?

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I think you need to modify Q1, because on a non-compact group, characters never lie in $L^p(G)$ for $p<\infty$. –  user16299 Jun 20 '12 at 20:06

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