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I am working through some class notes on Fourier transforms and convolutions. Currently, we are trying to motivate the definition $\langle S * T, \phi \rangle := \langle S_x, \langle T_y, \phi(x+y) \rangle \rangle$, where $S$ and $T$ are distributions (not necessarily tempered). The lecturer did it in this way: take $u,v \in L^1(\mathbb{R}^n)$ such that either $u$ or $v$ has compact support (this allows us to use Fubini-Tonelli later on). Then \begin{align*} \langle u * v , \phi \rangle &= \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} u(x-y) v(y) \phi(x) dy dx \\ &= \int_{\mathbb{R}^n}\int_{\mathbb{R}^n} u(y) v(x) \phi(x+y) dx dy. \end{align*}

Everything makes sense except the last line. The use of Fubini-Tonelli is clear. However, the inner portion confuses me. It seems like we are using some sort of u-substitution, but this isn't enough. Any insight would be appreciated. Thanks!

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From the top line, make the substitutions $w:=y$ and $z:=x-y$.

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  • $\begingroup$ That's the same as just making the substitution $z:= x-y$, though, which still leaves us with $\int_{\mathbb{R}^n} \int_{\mathbb{R}^n} u(z) v(x-z) \phi(z+y) dz dy$. $\endgroup$ – mathishard.butweloveit May 27 '18 at 21:48
  • $\begingroup$ Never mind, I see the trick! Thank you! $\endgroup$ – mathishard.butweloveit May 27 '18 at 21:56

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