Consider the case that $f \in L^1(\mathbb{R})$ and $g \in L^1_{loc}(\mathbb{R})$. Then look at the equation $$ f*f=g\cdot f. $$ I know that if $g$ is constant, then $f=0$. But what about other $g$'s? One can also consider other function spaces, for example $L^2$. But in any case, I don't really have an idea how one could compute $f$. Are there any good references or articles on such convolution equation problems? I know that it's quite a general question, but maybe some of you have seen such things before.
I simply have some ideas on existence of solutions to such equations. For example if $g \not= 0$ and $ || g^{-1}||_{L^{\infty}} < 1 $ one could use a the contraction map theorem.
Thanks a lot for your input!
You can of course answer the question if and only if you can do it for some function $f \in L^1(R)$ with $\|f\|_1 = 1$. For non-negative functions with this normalization (i.e. for densities of probability distributions) one can easily show that $$\| f \ast f - f\|_1 \geq 1/4$$ because by the normalization one has $\hat{f}(0)= \int_R f(t) dt = 1$, but due to the Riemann-Lebesgue Lemma $\lim_{|s| \to \infty} \hat{f} (s) = 0$, so the intermediate value theorem ensures that there exists $s_0$ such that $|\hat{f}(s_0)| = 1/2$. But this implies that the Fourier transform of $h := f \ast f - f$ takes the value $$|\hat{h}(s_0)| \geq |\hat{f}(s_0)| - |\hat{f}(s_0)|^2 = 1/4 , $$ but it is true in general that $\| \hat{h}\|_\infty \leq \|h\|_1$. so we end up with $$ \|h\|_1 \geq \| \hat{h}\|_\infty \geq 1/4. $$