 Solution

Assume the Laplace integrals converge absolutely in a common right half-plane. The Fubini theorem and the substitution $t=u-v$ on $u\ge v\ge 0$ give the convolution theorem for Laplace transforms:For the specified inverse-square-root function, the supplied identity says $f\ast f=\pi$ on $t>0$. Consequently $\hat{f}(p{)}^2=\pi /p$. For real $p>0$ the defining integral is positive, fixing the positive square root; analytic continuation into $\mathrm{Re}p>0$ giveswith the branch positive on the positive real axis. The square identity alone would not select the sign.