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I already showed b item using the fact that it is $h\left(0\right)=\int \:f\left(t\right)g\left(0-t\right)dt$

I struggle a lot of hours trying to find the trick in item C.

Can anyone help please ?

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  • $\begingroup$ If you know that $\mathcal{G}$ is the Fourier transform of $g$, do you know the inverse Fourier transform of $g$? $\endgroup$ – TimWescott Jan 12 at 19:48
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Hint:

You only need to use the definition of the Fourier transform and its inverse transform to show that if

$$G(f)=\mathscr{F}\{g(t)\}$$

then

$$g(f)=\mathscr{F}\{G(-t)\}$$

holds. Then just use the result from $b$.

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