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