  We give explicit transforms for Hilbert spaces associated with positive
definite functions on $\mathbb{R}$, and positive definite tempered
distributions, incl., generalizations to non-abelian locally compact groups.
Applications to the theory of extensions of p.d. functions/distributions are
included. We obtain explicit representation formulas for positive definite
tempered distributions in the sense of L. Schwartz, and we give applications to
Dirac combs and to diffraction. As further applications, we give parallels
between Bochner's theorem (for continuous p.d. functions) on the one hand, and
the generalization to Bochner/Schwartz representations for positive definite
tempered distributions on the other; in the latter case, via tempered positive
measures. Via our transforms, we make precise the respective reproducing kernel
Hilbert spaces (RKHSs), that of N. Aronszajn and that of L. Schwartz. Further
applications are given to stationary-increment Gaussian processes.
