371 research outputs found
Integration by parts for the Lr Henstock-Kurzweil integral
No full textMusial and Sagher [4] described a Henstock-Kurzweil type integral that integrates Lr-derivatives. In this article, we develop a product rule for the Lr-derivative and then an integration by parts formula
A new descriptive characterization of the HKr-integral and its inclusion in Burkill's integrals
No full textWe introduce a new class of functions, ACGr⁎, and compare it to the class of ACGr-functions which had been previously introduced by Musial and Sagher to characterize their Henstock–Kurzweil-type integral, the HKr-integral. We show that these two classes coincide and thereby we obtain a new descriptive characterization of the class of HKr-integrable functions. We then compare the HKr-integral with Burkill's CP-integral and obtain a de la Vallée Poussin-type theorem for the HKr-integral
Dual of the Class of HKr Integrable Functions
No full textWe define for 1 <= r < infinity a norm for the class of functions which are Henstock-Kurzweil integrable in the L-r sense. We then establish that the dual in this norm is isometrically isomorphic to L-r' and is therefore a Banach space, and in the case r = 2, a Hilbert space. Finally, we give results pertaining to convergence and weak convergence in this space
On Descriptive Characterizations of an Integral Recovering a Function from Its -Derivative
No full textThe notion of Lr-variational measure generated by a function F ∈ Lr[a, b] is introduced and, in terms of absolute continuity of this measure, a descriptive characterization of the HKr -integral recovering a function from its Lr-derivative is given. It is shown that the class of functions generating absolutely continuous Lr-variational measure coincides with the class of ACGr -functions which was introduced earlier, and that both classes coincide with the class of the indefinite HKr-integrals under the assumption of Lr-differentiability almost everywhere of the functions consisting these classe
A decomposition theorem for compact-valued Henstock integral
No full textWe prove that if X is a separable Banach space, then a measurable multifunction Γ : [0, 1] → ck(X) is Henstock integrable if and only if Γ can be represented as Γ = G + f, where G : [0, 1] → ck(X) is McShane integrable and f is a Henstock integrable selection of
THE HKr-INTEGRAL IS NOT CONTAINED IN THE Pr-INTEGRAL
No full textWe compare a Perron-type integral with a Henstock-Kurzweiltype integral, both having been introduced to recover functions from their generalized derivatives defined in the metric Lr. We give an example of an HKr-integrable function which is not Pr-integrable, thereby showing that the first integral is strictly wider than the second one
Comparison of the Pr-integral with Burkill's integrals and some applications to trigonometric series
No full textIt is proved that the Pr-integral [9] which recovers a function from its derivative defined in the space Lr, 1 <= r < infinity, is properly included in Burkill's trigonometric CP-and SCP-integrals. As an application to harmonic analysis, a de La Vallee-Poussin-type theorem for the Pr-integral is obtained: convergence nearly everywhere of a trigonometric series to a Pr-integrable function f implies that this series is the Pr-Fourier series of f.(c) 2023 Elsevier Inc. All rights reserved
Variational Henstock integrability of Banach space valued functions
No full textsummary:We study the integrability of Banach space valued strongly measurable functions defined on . In the case of functions given by , where are points of a Banach space and the sets are Lebesgue measurable and pairwise disjoint subsets of , there are well known characterizations for Bochner and Pettis integrability of . The function is Bochner integrable if and only if the series is absolutely convergent. Unconditional convergence of the series is equivalent to Pettis integrability of . In this paper we give some conditions for variational Henstock integrability of a certain class of such functions
Radon-Nikodym derivatives of finitely additive interval measures taking values in a Banach space with basis
No full textLet X be a Banach space with a Schauder basis {en}, and let Φ(I)= ∑n en ∫I fn(t)dt be
a finitely additive interval measure on the unit interval [0, 1], where the integrals are taken in the sense
of Henstock–Kurzweil. Necessary and sufficient conditions are given for Φ to be the indefinite integral of
a Henstock–Kurzweil–Pettis (or Henstock, or variational Henstock) integrable function f:[0, 1] → X
A full characterization of multipliers for the strong -integral in the euclidean space
No full textsummary:We study a generalization of the classical Henstock-Kurzweil integral, known as the strong -integral, introduced by Jarník and Kurzweil. Let be the space of all strongly -integrable functions on a multidimensional compact interval , equipped with the Alexiewicz norm . We show that each element in the dual space of can be represented as a strong -integral. Consequently, we prove that is strongly -integrable on for each strongly -integrable function if and only if is almost everywhere equal to a function of bounded variation (in the sense of Hardy-Krause) on
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