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Herglotz's theorem and quaternion series of positive term
Author(s) -
Kou Kit Ian,
Liu MingSheng,
Tao ShuZhen
Publication year - 2016
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.3945
Subject(s) - quaternion , mathematics , series (stratigraphy) , sequence (biology) , measure (data warehouse) , term (time) , function (biology) , pure mathematics , geometry , computer science , paleontology , physics , genetics , quantum mechanics , database , evolutionary biology , biology
The present paper first introduces the notion of quaternion infinite series of positive term and establishes its several tests. Next, we give the definitions of the positive‐definite quaternion sequence and the positive semi‐definite quaternion function, and we extend the classical Herglotz's theorem to the quaternion linear canonical transform setting. Then we investigate the properties of the two‐sided quaternion linear canonical transform, such as time shift characteristics and differential characteristics. Finally, we derive its several basic properties of the quaternion linear canonical transform of a probability measure, in particular, and establish the Bochner–Minlos theorem. Copyright © 2016 John Wiley & Sons, Ltd.