Quaternion-valued positive definite functions on locally compact Abelian groups and nuclear spaces

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

doi:10.1016/j.amc.2016.03.034

Quaternion-valued positive definite functions on locally compact Abelian groups and nuclear spaces

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

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

5 Scopus citations

Abstract

In this paper we study quaternion-valued positive definite functions on locally compact Abelian groups, real countably Hilbertian nuclear spaces and on the space ^RN={(_x1,_x2,...):_xdϵR} endowed with the Tychonoff topology. In particular, we prove a quaternionic version of the Bochner-Minlos theorem. A tool for proving this result is a classical matricial analogue of the Bochner-Minlos theorem, which we believe is new. We will see that in all these various settings the integral representation is with respect to a quaternion-valued measure which has certain symmetry properties.

Original language	English
Pages (from-to)	115-125
Number of pages	11
Journal	Applied Mathematics and Computation
Volume	286
DOIs	https://doi.org/10.1016/j.amc.2016.03.034
State	Published - 5 Aug 2016

Keywords

Bochner's theorem
Bochner-Minlos theorem
nuclear spaces
quaternionic analysis

ASJC Scopus subject areas

Computational Mathematics
Applied Mathematics

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10.1016/j.amc.2016.03.034

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title = "Quaternion-valued positive definite functions on locally compact Abelian groups and nuclear spaces",

abstract = "In this paper we study quaternion-valued positive definite functions on locally compact Abelian groups, real countably Hilbertian nuclear spaces and on the space RN={(x1,x2,...):xdϵR} endowed with the Tychonoff topology. In particular, we prove a quaternionic version of the Bochner-Minlos theorem. A tool for proving this result is a classical matricial analogue of the Bochner-Minlos theorem, which we believe is new. We will see that in all these various settings the integral representation is with respect to a quaternion-valued measure which has certain symmetry properties.",

keywords = "Bochner's theorem, Bochner-Minlos theorem, nuclear spaces, quaternionic analysis",

author = "Daniel Alpay and Fabrizio Colombo and Kimsey, {David P.} and Irene Sabadini",

note = "Funding Information: D. Alpay thanks the Earl Katz family for endowing the chair which supported his research. D. Kimsey gratefully acknowledges support by the ISF within the ISF-UGC joint research framework (grant No. 1775/14). F. Colombo and I. Sabadini acknowledge the Center for Advanced Studies of the Mathematical Department of the Ben-Gurion University of the Negev for the support and the kind hospitality during the period in which part of this paper has been written. Publisher Copyright: {\textcopyright} 2016 Elsevier Inc. All rights reserved.",

year = "2016",

month = aug,

day = "5",

doi = "10.1016/j.amc.2016.03.034",

language = "English",

volume = "286",

pages = "115--125",

journal = "Applied Mathematics and Computation",

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T1 - Quaternion-valued positive definite functions on locally compact Abelian groups and nuclear spaces

AU - Alpay, Daniel

AU - Colombo, Fabrizio

AU - Kimsey, David P.

AU - Sabadini, Irene

N1 - Funding Information: D. Alpay thanks the Earl Katz family for endowing the chair which supported his research. D. Kimsey gratefully acknowledges support by the ISF within the ISF-UGC joint research framework (grant No. 1775/14). F. Colombo and I. Sabadini acknowledge the Center for Advanced Studies of the Mathematical Department of the Ben-Gurion University of the Negev for the support and the kind hospitality during the period in which part of this paper has been written. Publisher Copyright: © 2016 Elsevier Inc. All rights reserved.

PY - 2016/8/5

Y1 - 2016/8/5

N2 - In this paper we study quaternion-valued positive definite functions on locally compact Abelian groups, real countably Hilbertian nuclear spaces and on the space RN={(x1,x2,...):xdϵR} endowed with the Tychonoff topology. In particular, we prove a quaternionic version of the Bochner-Minlos theorem. A tool for proving this result is a classical matricial analogue of the Bochner-Minlos theorem, which we believe is new. We will see that in all these various settings the integral representation is with respect to a quaternion-valued measure which has certain symmetry properties.

AB - In this paper we study quaternion-valued positive definite functions on locally compact Abelian groups, real countably Hilbertian nuclear spaces and on the space RN={(x1,x2,...):xdϵR} endowed with the Tychonoff topology. In particular, we prove a quaternionic version of the Bochner-Minlos theorem. A tool for proving this result is a classical matricial analogue of the Bochner-Minlos theorem, which we believe is new. We will see that in all these various settings the integral representation is with respect to a quaternion-valued measure which has certain symmetry properties.

KW - Bochner's theorem

KW - Bochner-Minlos theorem

KW - nuclear spaces

KW - quaternionic analysis

UR - http://www.scopus.com/inward/record.url?scp=84964267063&partnerID=8YFLogxK

U2 - 10.1016/j.amc.2016.03.034

DO - 10.1016/j.amc.2016.03.034

M3 - Article

AN - SCOPUS:84964267063

SN - 0096-3003

VL - 286

SP - 115

EP - 125

JO - Applied Mathematics and Computation

JF - Applied Mathematics and Computation

ER -

Quaternion-valued positive definite functions on locally compact Abelian groups and nuclear spaces

Abstract

Keywords

ASJC Scopus subject areas

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