Document Type
Article
Publication Date
2015
Abstract
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.
Recommended Citation
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.
Peer Reviewed
1
Copyright
Elsevier
Creative Commons License
This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License.
Included in
Algebra Commons, Discrete Mathematics and Combinatorics Commons, Other Mathematics Commons
Comments
NOTICE: this is the author’s version of a work that was accepted for publication in Journal of Mathematical Analysis and Applications. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Mathematical Analysis and Applications, volume 421, in 2015. DOI: 10.1016/j.jmaa.2014.07.025
The Creative Commons license below applies only to this version of the article.