On analytic continuation of characteristic functions of probability measures

We derive necessary and sufficient conditions for a continuous bounded function f:R→C to be a characteristic function of a probability measure. The Cauchy transform Kf of f is used as analytic continuation of f to the upper and lower half-planes in C. The conditions depend on the behavior of Kf(z) and its derivatives on the imaginary axis in C. The main results are given in terms of completely monotonic and absolutely monotonic functions.