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
DOI:
https://doi.org/10.1090/S0002-9939-1970-0267068-8
MathSciNet review:
0267068
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Abstract | References | Similar Articles | Additional Information
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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- J. S. MacNerney, Integral equations and semigroups, Illinois J. Math. 7 (1963), 148–173. MR 144179
- H. S. Wall, Concerning harmonic matrices, Arch. Math. (Basel) 5 (1954), 160–167. MR 61268, DOI https://doi.org/10.1007/BF01899333
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Keywords:
Exponentials,
product integrals,
subdivision-refinement type integrals,
bounded variation
Article copyright:
© Copyright 1970
American Mathematical Society