2020/09/06 by Саулюс Норвидас, Norvidas, Saulius
Mathematics · #Advanced Harmonic Analysis Research #Advanced Banach Space Theory #Mathematical Analysis and Transform Methods
paper · pdf · doi:10.48550/arxiv.2009.02802
We provide necessary and sufficient conditions for a tempered distribution F∈ S'(R) to be positive definite. A generalized Cauchy transform \widetildeF of F is used as a numerical continuation of F to the open upper and lower complex half-planes in C. In fact, our necessary and sufficient conditions for F are determined completely by the properties of the restriction of \widetildeF to the imaginary axis in C. The main result is given in terms of completely monotonic and absolutely monotonic functions.