I have been trying to find a solution to the following convolution equation: \begin{align*} e^{-ax^2/2}*\ln \left(f(x)*e^{-ax^2/2}\right)+e^{-bx^2/2}*\ln \left(f(x)*e^{-bx^2/2}\right)=cx^2 \end{align*} I am trying to solve for $f(x)$ where $a,b,c$ are constants. Note,that '*' denotes convolution. We can assume that $f(x)>0$ and $f(x) \in L^1$.
I have been attempting to use Fourier analysis method but so far unsuccessful. For example I have found that \begin{align*} \mathcal{F}\left (\ln(f(t)\right)=\frac{1}{i\omega} \mathcal{F} \left( \frac{f'(t)}{f(t)}\right) \end{align*} Applying this we get \begin{align*} \sqrt{\frac{2\pi}{a} }e^{-\omega^2/(2a)} \mathcal{F} \left(\ln \left(f(x)*e^{-ax^2/2}\right) \right)+ \sqrt{\frac{2\pi}{b} }e^{-\omega^2/(2b)} \mathcal{F} \left(\ln \left(f(x)*e^{-bx^2/2}\right) \right)=-2\pi c \delta^{(2)}(\omega) \end{align*} by using $\mathcal{F}\left (\ln(f(t)\right)=\frac{1}{i\omega} \mathcal{F} \left( \frac{f'(t)}{f(t)}\right)$ we get
\begin{align*} \sqrt{\frac{2\pi}{a} }e^{-\omega^2/(2a)} \frac{1}{i\omega} \mathcal{F} \left( \frac{f(x)*(-ax)e^{-ax^2/2}}{f(x)*e^{-ax^2/2}}\right)+ \sqrt{\frac{2\pi}{b} }e^{-\omega^2/(2b)} \frac{1}{i\omega} \mathcal{F} \left( \frac{f(x)*(-bx)e^{-bx^2/2}}{f(x)*e^{-bx^2/2}}\right)=-2\pi c \delta^{(2)}(\omega) \end{align*} But how do I proceed now?
I am also looking for any related reference. Is also reminds of functional equations, but I am don't know much in that area. Thank you for any help in advance.