A classical theorem of Herglotz states that a function n↦r(n) from Z into Cs×s is positive definite if and only there exists a Cs×s-valued positive measure dμ on [0,2π] such that r(n)=∫2π0eintdμ(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.
D. Alpay, F. Colombo,D. Kimsey and I. Sabadini. An extension of Herglotz's theorem to the quaternions. Journal of Mathematical Analysis and Applications, vol. 421 (2015), no. 1, 754-778.
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