The limit set of the Henstock-Kurzweil integral sums of a vector-valued function
DOI:
https://doi.org/10.26565/2221-5646-2013-1081-02Keywords:
Henstock-Kurzweil integral, Banach space, limit set of integral sumsAbstract
We introduce the notion of the limit set $I_{H-K}(f)$ of the Henstock-Kurzweil integral sums of a function $f: [0, 1]\to X$, where $X$ is a Banach space, and study its properties. In particular, we construct an example of function $f$, which is not integrable, but its limit set consists exactly of one point. We find sufficient conditions that guarantee the convexity of the limit set.
References
H.W. Ellis, On the limits of Riemann sums, J. London Math. Soc. 34 (1959), - no. 1, - P. 93-100.
S. Heikkila, Di erential and integral equations with Henstock-Kurzweil integrable functions, J. Math. Anal. Appl. 379. - 2011. - P. 171-179.
M. Nakamura, I. Amemiya, On the limit of Riemann sums of functions in Banach spaces, J. Fac. Sci. Hokkaido Univ. Ser. I 19 (1966), - no. 3.4, - P. 135-145.
M.I. Kadets, V.M. Kadets, Series in Banach spaces. Conditional and unconditional convergence, Birkhauser, Basel - Boston - Berlin, 1997. - 156 p.
V.M. Kadets, The starness of the domain of limits of Riemann integral sums of a vector-valued function, in: Operator Theory, Subharmonic Functions, - Naukova Dumka, - Kiev, - 1991. - P. 60-67 (Russian).
V.M. Kadets, M.I. Kadets, Conditions for the convexity of the set of limits of Riemann sums of a vector-valued function, Mat. Zametki 35 (1984), - no. 2, - P. 161-167 (Russian).
J.L. Kelley, General Topology, - Springer, - 1975. - 298 p.
D.S. Kurtz, Ch.W. Swartz. Series in real analysis - volume 9: theories of integration. The integrals of Riemann, Lebesgue, Henstock-Kurzweil and McShane, - World scienti c, - 2004. - 268 p.
Downloads
Published
Issue
Section
License
The copyright holder is the author.
Authors who publish with this journal agree to the following terms:
1. Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal. Creative Commons Attribution License International CC-BY (CC BY 4.0).
2. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
3. Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (see The Effect of Open Access).
