I am trying to find all solutions to
$$f\star f=f$$ with $f\in L^2(\mathbb{R})$.
I can solve this equation for $f\in L^1$ (and hence for $f\in L^1\cap L^2$). If I want a solution in$f\in L^2\setminus L^1$, I know that we cannot directly use the Fourier transform with $f\in L^2(\mathbb{R})$, since $f\star f$ only belongs to $L^2\cap L^\infty$.
I have tried to use an approximating sequence $f_n\in L^1\cap L^2$ with no success yet.
So it is given that $f*f=f$ with $f \in L^2(\Bbb R)$ . So in particular $f*f \in L^2(\Bbb R)$
Thus applying the Fourier transform on both sides, we have $(\hat{f}(\xi))^2=\hat{f}(\xi)$ for a.e. $\xi$ i.e. $$\hat{f}(\xi)(\hat{f}(\xi)-1)=0 \text{ for a.e. } \xi$$ Thus either $\hat{f}(\xi) \equiv 0 \text{ for a.e. } \xi \implies f \equiv 0 $ a.e.
Or, $\hat{f}(\xi) \equiv 1 \text{ for a.e. } \xi$ but then $\hat f \notin L^2(\Bbb R)$
So the only possibility left with $\hat{f}(\xi)(\hat{f}(\xi)-1)=0 \text{ for a.e. } \xi$ and $\hat f \in L^2(\Bbb R)$ is $\hat f = \chi_E$ where $E$ is some measurable set with finite, positive measure.But then $\hat f \in L^1(\Bbb R) \cap L^2(\Bbb R)$ thus you can apply inverse Fourier transform on that to obtain $$f(x)=\int_{\Bbb R} \chi_E(\xi)e^{i\xi x} d\xi=\int_{E} e^{i\xi x} d\xi \text{ a.e. } x \in \Bbb R$$