# solving integral equation by fourier transform

this is fredholm type of integral equation that im currently triying to solve

$$\int_{-\infty}^{\infty}e^{-(t-x)^2}\phi(t)dt=e^{-x^2/8}$$

the question is that how to derive $\phi(t)$

i tried convolution theorem of fourier transform to solve this equation

and what i got is $\phi(t)=\sqrt{8\over7\pi}e^{-t^2\over7}$

but it seems suspicious.. is it correct answer?

• Your answer is correct. – Juan Ospina Dec 16 '15 at 14:10

There is an easy solution through trial and error. By completing the square and exploing $\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$ we have that for any $A\in (-\infty,1)$ $$\int_{\mathbb{R}}\exp\left(-(t-x)^2+At^2\right)\,dt = \sqrt{\frac{\pi}{1-A}}\exp\left(\frac{Ax}{1-A}\right)$$ holds, hence by solving $\frac{A}{1-A}=\frac{1}{8}$ we get $A=-\frac{1}{7}$, so $$\phi(t) = \sqrt{\frac{8}{7\pi}}\,e^{-t^2/7}$$ is a solution for sure.