An extension of Herglotz's theorem to the quaternions

A classical theorem of Herglotz states that a function n. r(. n) from Z into Cs×s is positive definite if and only if there exists a Cs×s-valued positive measure μ on [0, 2π] such that r(n)=∫02πeintdμ(t) for n∈Z. We prove a quaternionic analogue of this result when the function is allowed to have a number of negative squares. A key tool in the argument is the theory of slice hyperholomorphic functions, and the representation of such functions which have a positive real part in the unit ball of the quaternions. We study in great detail the case of positive definite functions.