An extension of Herglotz's theorem to the quaternions

A classical theorem of Herglotz states that a function n↦r(n)n↦r(n) from ZZ into Cs×sCs×s is positive definite if and only if there exists a Cs×sCs×s-valued positive measure μ   on [0,2π][0,2π] such that r(n)=∫02πeintdμ(t) for n∈Zn∈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.