# Find a function $f$ such that $f \star f = g$ for a given $g$.

Let $$g(x)=\frac{1}{x^2 |\ln x|}$$ for $$x \in ]0,a]$$ with $$a$$ fixed in $$]0,1[$$. We know, thanks to Bertrand's integrale convergence, that $$g \in L^1([0,a])$$.

I'm looking for a positive measurable function defined on $$\mathbb{R}$$ such as $$f \in L^1([0,a])$$ and such that :

$$g(x) = \int_{0}^a f(t) f(x-t) \ \mathrm{d}t = f \star f(x), \quad 0 < x \leq a$$ The values that $$f$$ takes on $$\mathbb{R} \setminus]0,a]$$ are not very important, as long as $$g$$ and $$f \star f$$ coincides on $$]0,a].$$

Have you ever heard of similar problems ? I have serious doubts about this problem and it is not necessarily well-posed. I tried using Fourier series so as to determine $$f$$ but this wasn't very successfull... What are your thoughts about this problem ?