Solve for f(t) using Fourier transform: $$ \int_{-\infty}^\infty f(s)f(t-s)\,ds - 2\sqrt{2} \int_{-\infty}^\infty e^{-s^2/\pi}f(t-s)\,ds = -\sqrt{2}\pi e^{-\frac{t^{2}}{2\pi}} $$
Now, I get the point that we have a convolution in place of both the integrals, giving me (using the fourier transform, denoted by $F$ here):
$$ F(f(t))^{2}-2\sqrt{2}F\left(e^{-t^2/\pi}\right)F(f(t)) = -\sqrt{2}\pi F\left(e^{-\frac{t^{2}}{2\pi}}\right) $$
...but I can't wrap my mind around how to further simplify this, since
$$F(e^{-\frac{t^{2}}{\pi}})\neq F(e^{-\frac{t^{2}}{2\pi}}).$$
What are my next steps - what am I not seeing?