Mutual existence of product integrals
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- by Jon C. Helton PDF
- Proc. Amer. Math. Soc. 42 (1974), 96-103 Request permission
Abstract:
Definitions and integrals are of the subdivision-refinement type, and functions are from $R \times R$ to R, where R represents the real numbers. Let $O{M^ \circ }$ be the class of functions G such that $_x\prod {^y} (1 + G)$ exists for $a \leqq x < y \leqq b$ and $\smallint _a^b|1 + G - \prod {(1 + G)| = 0}$. Let $O{P^ \circ }$ be the class of functions G such that $|\prod \nolimits _{q = i}^j {(1 + {G_q})|}$ is bounded for refinements $\{ {x_q}\} _{q = 0}^n$ of a suitable subdivision of [a, b]. If F and G are functions from $R \times R$ to R such that $F \in O{P^ \circ }$ on [a, b], ${\lim _{x,y \to {p^ + }}}F(x,y)$ and ${\lim _{x,y \to {p^ - }}}F(x,y)$ exist and are zero for $p \in [a,b]$, each of ${\lim _{x \to {p^ + }}}F(p,x),{\lim _{x \to {p^ - }}}F(x,p),{\lim _{x \to {p^ + }}}G(p,x)$ and ${\lim _{x \to {p^ - }}}G(x,p)$ exist for $p \in [a,b]$, and G has bounded variation on [a, b], then any two of the following statements imply the other: (1) $F + G \in O{M^ \circ }$ on [a, b], (2) $F \in O{M^ \circ }$ on [a, b], and (3) $G \in O{M^\circ }$ on [a, b].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 J. C. Helton, Product integrals, bounds and inverses, Texas J. Sci. (to appear).
- Jon C. Helton, Bounds for products of interval functions, Pacific J. Math. 49 (1973), 377–389. MR 360969
- A. Kolmogoroff, Untersuchungen über denIntegralbegriff, Math. Ann. 103 (1930), no. 1, 654–696 (German). MR 1512641, DOI 10.1007/BF01455714
- J. S. MacNerney, Integral equations and semigroups, Illinois J. Math. 7 (1963), 148–173. MR 144179
Additional Information
- © Copyright 1974 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 42 (1974), 96-103
- MSC: Primary 26A39
- DOI: https://doi.org/10.1090/S0002-9939-1974-0349925-0
- MathSciNet review: 0349925