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I'm trying to solve an integral equation by identify the convolution and then transforming, but I'm getting to a really confusing expression, where I'm not sure how to continue:

$$ \int_{-\infty}^{\infty}f(t-y)e^{-|y|}dy=e^{-t^2/2} $$

Any ideas? Solving for the transformed f is easy, but finding the original function f seems difficult.

Curtain
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$\newcommand{\F}{\mathcal{F}}$I suppose that you refer to the Fourier transform? If so, you are on the right track: The solution is $$f = \F^{-1}(\F(\exp(-|y|^2/2)/\F(\exp(-|y|)).$$ Is suspect, that you know the Fourier transforms that appear on the right hand side. Do you also know the Fourier transforms of the derivatives of $\exp(-y^2/2)$? Would be helpful here…

Dirk
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  • Thanks for taking your time, Dirk. I don't know the Fourier transforms of the derivatives of the function you mention, I only know that $$F(f'(t))(\omega)=i\omega F(\omega)$$ – Curtain Sep 23 '13 at 19:28
  • Well, that's all you need… – Dirk Sep 23 '13 at 19:29
  • Dirk, yeah, I missed that. Probably because of that the $$ \omega $$ was squared and I didn't realize it was the derivative. I will try to solve it now. – Curtain Sep 23 '13 at 19:30