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## Homework Statement

Find a function ##u## such that

##\int_{-\infty}^\infty u(x-y)e^{-|y|}dy=e^{-x^4}##.

## Homework Equations

Not really sure how to approach this but here's a few of the formulas I tried to use.

Fourier transform of convolution

##\mathscr{F} (f*g)(x) \to \hat f(\xi ) \hat g(\xi )##.

##\mathscr{F} e^{-a|x|} \to 2a(\xi² +a^2)^{-1}##

Plancherel's Theorem

If ##f,g \in L^2## then

##\langle \hat f,\hat g \rangle = 2\pi \langle f, g \rangle##.

## The Attempt at a Solution

The left side is the convolution ##u*(e^{-|x|}## Taking the Fourier transform I have

##\hat u2(\xi^2+1)^{-1}= \mathscr{F}(e^{-x^4})##.

Which I don't know the fourier transform of nor how to calculate the inverse.

I also tried to somehow look at the integral as a scalar product and use Plancherel's formula but that didn't work at all. Any ideas on how I should approach the problem? I don't even know if I should use the Fourier transform for this, the problem was just grouped with other exercises on Fourier transforms.