Norvidas, Saulius

We propose necessary and sufficient conditions for a distribution (generalized function) f of several variables to be positive definite. For this purpose, certain analytic extensions of f to tubular domains in complex space Cn are studied. The main result is given in terms of the Cauchy transform and completely monotonic functions.

We propose necessary and sufficient conditions for a distribution (generalized function) f of several variables to be positive definite. For this purpose, certain analytic extensions of f to tubular domains in complex space C n are studied. The main result is given in terms of completely monotonic functions on convex cones in R n

We propose necessary and sufficient conditions for a complex-valued function f on Rn to be a characteristic function of a probability measure. Certain analytic extensions of f to tubular domains in Cn are studied. In order to extend the class of functions under study, we also consider the case where f is a generalized function (distribution). The main result is given in terms of completely monotonic functions on convex cones in Rn.

We provide necessary and sufficient conditions for a tempered distribution F∈S′(R)F∈S′(R) to be positive definite. A generalized Cauchy transform View the MathML sourceF˜ 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 View the MathML sourceF˜ to the imaginary axis in CC. The main result is given in terms of completely monotonic and absolutely monotonic functions.