Existence theorems for sum and product integrals
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- by Jon C. Helton PDF
- Proc. Amer. Math. Soc. 36 (1972), 407-413 Request permission
Abstract:
Necessary and sufficient conditions on a function G are determined for the integrals \[ \int _a^b {HG,\quad \prod \limits _a^b {(1 + HG),\quad \int _a^b {|HG - \int {HG| = 0} } } } \] and \[ \int _a^b {|1 + HG - \prod {(1 + HG)| = 0} } \] 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
- William D. L. Appling, Interval functions and real Hilbert spaces, Rend. Circ. Mat. Palermo (2) 11 (1962), 154–156. MR 154081, DOI 10.1007/BF02843951
- Burrell W. Helton, Integral equations and product integrals, Pacific J. Math. 16 (1966), 297–322. MR 188731
- Burrell W. Helton, A product integral representation for a Gronwall inequality, Proc. Amer. Math. Soc. 23 (1969), 493–500. MR 248310, DOI 10.1090/S0002-9939-1969-0248310-8
Additional Information
- © Copyright 1972 American Mathematical Society
- 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