A note on positive definite distributions

We provide necessary and sufficient conditions for a tempered distribution F∈S′(R)F∈S′(R) to be positive definite. A generalized Cauchy transform F˜ of FF is used as a numerical continuation of FF to the open upper and lower half-planes in CC. In fact, our necessary and sufficient conditions for FF are determined completely by the properties of the restriction of F˜ to the imaginary axis in CC. The main result is given in terms of completely monotonic and absolutely monotonic functions.