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Define $\tilde{\phi}=\phi(-x)$ and $\langle\tilde{f},\phi\rangle=\langle f, \tilde{\phi}\rangle$. Show this definition is consistent for distributions defined by functions.

This is a question from Strichartz - A guide to distribution theory and Fourier Transforms.

I couldn't find how to start to solve it. Any suggestion will be very helpful.

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Recall: If $f \in L^1_{\def\loc{{\operatorname{loc}}}\loc}(\Omega)$, we define the corresponding distribution $T_f \colon \mathcal D(\Omega) \to \mathbf C$ by $$ \def\<#1>{\left<#1\right>}\<T_f, \phi> = \int_\Omega f\phi\, dx $$ If now $f \in L^1_\loc(\Omega)$ is given, the question wants you to show that $\tilde T_f = T_{\tilde f}$, using the definitions, that is for every $\phi \in \mathcal D(\Omega)$ $$ \<\tilde T_f, \phi> = \<T_f, \tilde \phi> = \int_\Omega f\tilde \phi \, dx \stackrel ?= \int_\Omega \tilde f \phi\, dx = \<T_{\tilde f}, \phi > $$

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  • $\begingroup$ But there is no assumption about $f \in L^1_{\def\loc{{\operatorname{loc}}}\loc}(\Omega)$ ? $\endgroup$ Nov 11, 2015 at 19:15
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    $\begingroup$ Otherwise you cannot associate a distribution to $f$. $\endgroup$
    – martini
    Nov 12, 2015 at 11:18

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