An extension of Herglotz's theorem to the quaternions

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:421

Article

Alpay, Daniel¹  Colombo, Fabrizio²  Kimsey, David P.¹  Sabadini, Irene² 

[1] Ben Gurion Univ Negev, Dept Math, IL-84105 Beer Sheva, Israel

[2] Politecn Milan, Dipartimento Matemat, I-20133 Milan, Italy

关键词: Bochner theorem;    Herglotz integral representation theorem;    Negative squares;    Pontryagin spaces;    Reproducing kernels;    Slice hyperholomorphic functions;   

DOI  :  10.1016/j.jmaa.2014.07.025

来源: Elsevier

【 摘 要 】

A classical theorem of Herglotz states that a function n -> r(n) from Z into C-sxs is positive definite if and only if there exists a C-sxs-valued positive measure mu on [0, 2 pi] such that r(n) = integral(2 pi)(0)e(int) d mu(t) for n is an element of 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. (c) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

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