Convolution of complicated exponential functions

Question

Im trying to convolve the functions :

$f(x)=((-2\pi x)^{3}+3(2\pi)^2x)e^{-\pi x^{2}}$

$g(x)=e^{-\pi x^{2}} $

Thinking maybe I could find their fourier transforms, multiply them then find the inverse fourier transform? is there something simpler?

Answer 1

You can start by observing two things: First, you can rewrite the multiplications as derivatives as follows: $$ ((-2\pi x)^3 + (6\pi)(2\pi x) e^{-\pi x^2} = \left(\left(\frac{d}{dx}\right)^3 - 6\pi\left(\frac{d}{dx}\right)\right)e^{-\pi x^2}. $$ Second, it is well known, or can easily be checked that a convolution interacts very nicely with differential operators, i.e. $$ h_1 * D (h_2) = D(h_1* h_2), $$ for any linear differential operator and suitable functions $h_1,h_2$. As a consequence, $$ f*g = \left(\left(\frac{d}{dx}\right)^3 - 6\pi\left(\frac{d}{dx}\right)\right) (g * g), $$ where I used your notation for $f$ and $g$. Now $g*g$ can be computed with the convolution theorem and what you know about the Fourier transform of a Gaussian. You have already pointed this out in your question so I won't go into detail here. The rest is simply a matter of taking derivatives.