2014/03/01 by D. Alpay, Daniel Alpay, Fabrizio Colombo +9
Mathematics · #Algebraic and Geometric Analysis #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Mathematics and Applications #math.FA
paper · pdf · doi:10.48550/arxiv.1403.0079
to appear in Journal of Mathematical Analysis and Applications 2014
openalex publication_date 2014/03/01 · arxiv created 2014/07/23 · arxiv updated 2014/07/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A classical theorem of Herglotz states that a function n↦ r(n) from \mathbb Z into \mathbb Cs× s is positive definite if and only there exists a \mathbb Cs× s-valued positive measure dμ on [0,2π] such that r(n)=∫02πeintdμ(t)for n∈ \mathbb 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.