A full characterization of multipliers for the strong ρ -integral in the euclidean space

Lee Tuo-Yeong

Czechoslovak Mathematical Journal (2004)

  • Volume: 54, Issue: 3, page 657-674
  • ISSN: 0011-4642

Abstract

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We study a generalization of the classical Henstock-Kurzweil integral, known as the strong ρ -integral, introduced by Jarník and Kurzweil. Let ( 𝒮 ρ ( E ) , ∥ · ∥ ) be the space of all strongly ρ -integrable functions on a multidimensional compact interval E , equipped with the Alexiewicz norm ∥ · ∥ . We show that each element in the dual space of ( 𝒮 ρ ( E ) , ∥ · ∥ ) can be represented as a strong ρ -integral. Consequently, we prove that f g is strongly ρ -integrable on E for each strongly ρ -integrable function f if and only if g is almost everywhere equal to a function of bounded variation (in the sense of Hardy-Krause) on E .

How to cite

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Tuo-Yeong, Lee. "A full characterization of multipliers for the strong $\rho $-integral in the euclidean space." Czechoslovak Mathematical Journal 54.3 (2004): 657-674. <http://eudml.org/doc/30889>.

@article{Tuo2004,
abstract = {We study a generalization of the classical Henstock-Kurzweil integral, known as the strong $\rho $-integral, introduced by Jarník and Kurzweil. Let $(\mathcal \{S\}_\{\rho \} (E), \Vert \cdot \Vert )$ be the space of all strongly $\rho $-integrable functions on a multidimensional compact interval $E$, equipped with the Alexiewicz norm $\Vert \cdot \Vert $. We show that each element in the dual space of $(\mathcal \{S\}_\{\rho \} (E), \Vert \cdot \Vert )$ can be represented as a strong $\rho $-integral. Consequently, we prove that $fg$ is strongly $\rho $-integrable on $E$ for each strongly $\rho $-integrable function $f$ if and only if $g$ is almost everywhere equal to a function of bounded variation (in the sense of Hardy-Krause) on $E$.},
author = {Tuo-Yeong, Lee},
journal = {Czechoslovak Mathematical Journal},
keywords = {strong $\rho $-integral; multipliers; dual space; strong -integral; multipliers; dual space; Kurzweil-Henstock integral},
language = {eng},
number = {3},
pages = {657-674},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A full characterization of multipliers for the strong $\rho $-integral in the euclidean space},
url = {http://eudml.org/doc/30889},
volume = {54},
year = {2004},
}

TY - JOUR
AU - Tuo-Yeong, Lee
TI - A full characterization of multipliers for the strong $\rho $-integral in the euclidean space
JO - Czechoslovak Mathematical Journal
PY - 2004
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 54
IS - 3
SP - 657
EP - 674
AB - We study a generalization of the classical Henstock-Kurzweil integral, known as the strong $\rho $-integral, introduced by Jarník and Kurzweil. Let $(\mathcal {S}_{\rho } (E), \Vert \cdot \Vert )$ be the space of all strongly $\rho $-integrable functions on a multidimensional compact interval $E$, equipped with the Alexiewicz norm $\Vert \cdot \Vert $. We show that each element in the dual space of $(\mathcal {S}_{\rho } (E), \Vert \cdot \Vert )$ can be represented as a strong $\rho $-integral. Consequently, we prove that $fg$ is strongly $\rho $-integrable on $E$ for each strongly $\rho $-integrable function $f$ if and only if $g$ is almost everywhere equal to a function of bounded variation (in the sense of Hardy-Krause) on $E$.
LA - eng
KW - strong $\rho $-integral; multipliers; dual space; strong -integral; multipliers; dual space; Kurzweil-Henstock integral
UR - http://eudml.org/doc/30889
ER -

References

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