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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?
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.
