As Example 6.8 clearly illustrates, the pdf of a random variable that is symmetric about the origin is not in general a valid reproducing kernel. Take two such iid random variables \(X\) and \(X^{\prime}\) with common pdf \(f\), and define \(Z=X+X^{\prime}\). Denote by \(\psi_{z}\) and \(f_{Z}\) the characteristic function and pdf of \(Z\), respectively.

Show that if \(\psi_{z}\) is in \(L^{1}(\mathbb{R}), f_{Z}\) is a positive semidefinite function. Use this to show that \(\kappa(x\), \(\left.x^{\prime}\right)=f_{Z}\left(x-x^{\prime}\right)=1\left\{\left|x-x^{\prime}\right| \leqslant 2\right\}\left(1-\left|x-x^{\prime}\right| / 2\right)\) is a valid reproducing kernel.