Convolution square root of $\delta $

I want to somehow classify the distributional solutions of the equation $$ f \ast f = \delta $$ where $\delta = \delta _0$ is the Dirac delta distribution. Clearly, by Fourier transformation, we have $$ \widehat{f}^2 = 1, $$ but I'm wondering whether it is possible to obtain a more explicit solution?

Solution 1:

Let $A\subset\mathbb{R}$ be a measurable set. Define $$ f_A=\mathcal{F}^{-1}\bigl(\chi_A-\chi_{\mathbb{R}\setminus A}\bigl), $$ where $\mathcal{F}$ denotes the Fourier transform and $\chi_B$ is the characteristic fnction of the set $B$. Then $$ f_A\ast f_A=\delta. $$ Some explicit examples are:

• $A=\mathbb{R}$, $f_A=\delta$.
• $A=\emptyset$, $f_A=-\delta$.
• $A=[0,\infty)$, $f_A=\dfrac{i}{\pi}\operatorname{Principal Value}\dfrac1x$.