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

 

Existence theorems for sum and product integrals


Author: Jon C. Helton
Journal: Proc. Amer. Math. Soc. 36 (1972), 407-413
MSC: Primary 26A39
MathSciNet review: 0313459
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Abstract: Necessary and sufficient conditions on a function G are determined for the integrals

$\displaystyle \int_a^b {HG,\quad \prod\limits_a^b {(1 + HG),\quad \int_a^b {\vert HG - \int {HG\vert = 0} } } } $

and

$\displaystyle \int_a^b {\vert 1 + HG - \prod {(1 + HG)\vert = 0} } $

to exist, where H and G are functions from $ R \times R$ to R and H is restricted by one or more of the limits $ H({p^ - },p),H({p^ - },{p^ - }),H(p,{p^ + })$ and $ H({p^ + },{p^ + })$. Furthermore, the conditions on G are sufficient for the existence of these integrals when H and G have their range in a normed complete ring N.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1972-0313459-8
PII: S 0002-9939(1972)0313459-8
Keywords: Sum integral, product integral, existence, bounded variation, subdivision-refinement integrals
Article copyright: © Copyright 1972 American Mathematical Society