A new characterization of characteristic functions of absolutely continuous distributions

It is well known that if g belongs to L 2 , then g(x)g(x +y)dx / (x)\ 2 dx is the characteristic function of an absolutely continuous distribution function. Conversely, every such characteristic function has the representation given above. Here we shown that if R(s,t) is a covariance function such that R(s,s) belongs to L,, then r R(s,s +t)ds R(s,s)ds is the characteristic function of an absolutely continuous distribution. Conversely, every such characteristic function has the latter representation (put R(s,t) = g(s)g(t)).