86,847 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
Variations on the Author
No full text“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
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
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