  This paper is concerned with the spectral characteristics of quaternionic
positive definite functions on the real line. We generalize the Stone's theorem
to the case of a right quaternionic linear one-parameter unitary group via two
different types of functional calculus. From the generalized Stone's theorems
we obtain a correspondence between continuous quaternionic positive definite
functions and spectral systems, i.e., unions of a spectral measure and a
unitary anti-self-adjoint operator that commute with each other; and then
deduce that the Fourier transform of a continuous quaternionic positive
definite function is an unusual type of quaternion-valued measure which can be
described equivalently in two different ways. One is related to spectral
systems (induced by the first generalized Stone's theorem), the other is
related to non-negative finite Borel measures (induced by the second
generalized Stone's theorem). An application to weakly stationary quaternionic
random processes is also presented.
