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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Concerning product integrals and exponentials


Authors: W. P. Davis and J. A. Chatfield
Journal: Proc. Amer. Math. Soc. 25 (1970), 743-747
MSC: Primary 28.40
MathSciNet review: 0267068
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Abstract: Suppose $ S$ is a linearly ordered set, $ N$ is the set of real numbers, $ G$ is a function from $ S \times S$ to $ N$, and all integrals are of the subdivision-refinement type. We show that if $ \int_a^b {{G^2} = 0} $ and either integral exists, then the other exists and $ a\prod {^b(1 + G) = \exp \int_a^b G } $. We also show that the bounded variation of $ G$ is neither necessary nor sufficient for $ \int_a^b {{G^2}} $ to be zero.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1970-0267068-8
PII: S 0002-9939(1970)0267068-8
Keywords: Exponentials, product integrals, subdivision-refinement type integrals, bounded variation
Article copyright: © Copyright 1970 American Mathematical Society