An extension of Herglotz's theorem to the quaternions

Daniel Alpay, Fabrizio Colombo, David P. Kimsey, Irene Sabadini

Research output: Contribution to journalArticlepeer-review

14 Scopus citations


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.

Original languageEnglish
Pages (from-to)754-778
Number of pages25
JournalJournal of Mathematical Analysis and Applications
Issue number1
StatePublished - 1 Jan 2015


  • Bochner theorem
  • Herglotz integral representation theorem
  • Negative squares
  • Pontryagin spaces
  • Reproducing kernels
  • Slice hyperholomorphic functions

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics


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