Mutual existence of product integrals
Author:
Jon C. Helton
Journal:
Proc. Amer. Math. Soc. 42 (1974), 96103
MSC:
Primary 26A39
MathSciNet review:
0349925
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Abstract: Definitions and integrals are of the subdivisionrefinement type, and functions are from to R, where R represents the real numbers. Let be the class of functions G such that exists for and . Let be the class of functions G such that is bounded for refinements of a suitable subdivision of [a, b]. If F and G are functions from to R such that on [a, b], and exist and are zero for , each of and exist for , and G has bounded variation on [a, b], then any two of the following statements imply the other: (1) on [a, b], (2) on [a, b], and (3) on [a, b].
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 , A product integral representation for a Gronwall inequality, Proc. Amer. Math. Soc. 23 (1969), 493500. MR 40 #1562. MR 0248310 (40:1562)
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 , Bounds for products of interval functions, Pacific J. Math. (to appear). MR 0360969 (50:13416)
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 A. Kolmogoroff, Untersuchungen über den Integralbegriff, Math. Ann. 103 (1930), 654696. MR 1512641
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 J. S. MacNerney, Integral equations and semigroups, Illinois J. Math. 7 (1963), 148173. MR 26 #1726. MR 0144179 (26:1726)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939197403499250
PII:
S 00029939(1974)03499250
Keywords:
Product integral,
sum integral,
subdivisionrefinement integral,
interval function
Article copyright:
© Copyright 1974 American Mathematical Society
