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Existence theorems for sum and product integrals


Author: Jon C. Helton
Journal: Proc. Amer. Math. Soc. 36 (1972), 407-413
MSC: Primary 26A39
DOI: https://doi.org/10.1090/S0002-9939-1972-0313459-8
MathSciNet review: 0313459
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Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] W. D. L. Appling, Interval functions and real Hilbert spaces, Rend. Circ. Mat. Palermo (2) 11 (1962), 154-156. MR 27 #4040. MR 0154081 (27:4040)
  • [2] B. W. Helton, Integral equations and product integrals, Pacific J. Math. 16 (1966), 297-322. MR 32 #6167. MR 0188731 (32:6167)
  • [3] -, A product integral representation for a Gronwall inequality, Proc. Amer. Math. Soc. 23 (1969), 493-500. MR 40 #1562. MR 0248310 (40:1562)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0313459-8
Keywords: Sum integral, product integral, existence, bounded variation, subdivision-refinement integrals
Article copyright: © Copyright 1972 American Mathematical Society

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