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 ?



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