The Henstock-Kurzweil integral
Paul M. Musial, Yoram Sagher (2004)
Studia Mathematica
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We present a method of integration along the lines of the Henstock-Kurzweil integral. All -derivatives are integrable in this method.
Paul M. Musial, Yoram Sagher (2004)
Studia Mathematica
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We present a method of integration along the lines of the Henstock-Kurzweil integral. All -derivatives are integrable in this method.
Juan H. Arredondo, Manuel Bernal, Maria G. Morales (2025)
Czechoslovak Mathematical Journal
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The paper is concerned with integrability of the Fourier sine transform function when , where is the space of bounded variation functions vanishing at infinity. It is shown that for the Fourier sine transform function of to be integrable in the Henstock-Kurzweil sense, it is necessary that . We prove that this condition is optimal through the theoretical scope of the Henstock-Kurzweil integration theory.
Jan Malý, Washek Frank Pfeffer (2016)
Mathematica Bohemica
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The generalized Riemann integral of Pfeffer (1991) is defined on all bounded subsets of , but it is additive only with respect to pairs of disjoint sets whose closures intersect in a set of -finite Hausdorff measure of codimension one. Imposing a stronger regularity condition on partitions of sets, we define a Riemann-type integral which satisfies the usual additivity condition and extends the integral of Pfeffer. The new integral is lipeomorphism-invariant and closed with respect...
Erik Talvila (2006)
Mathematica Bohemica
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If is a Henstock-Kurzweil integrable function on the real line, the Alexiewicz norm of is where the supremum is taken over all intervals . Define the translation by . Then tends to as tends to , i.e., is continuous in the Alexiewicz norm. For particular functions, can tend to 0 arbitrarily slowly. In general, as , where is the oscillation of . It is shown that if is a primitive of then . An example shows that the function need not be in . However, if...
Branko Sarić (2010)
Czechoslovak Mathematical Journal
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Let be an interval in and let be a real valued function defined at the endpoints of and with a certain number of discontinuities within . Assuming to be differentiable on a set to the derivative , where is a subset of at whose points can take values or not be defined at all, we adopt the convention that and are equal to at all points of and show that , where denotes the total value of the integral. The paper ends with a few examples that illustrate the...
Salvador Sánchez-Perales, Francisco J. Mendoza-Torres (2020)
Czechoslovak Mathematical Journal
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In the present paper, we investigate the existence of solutions to boundary value problems for the one-dimensional Schrödinger equation , where and are Henstock-Kurzweil integrable functions on . Results presented in this article are generalizations of the classical results for the Lebesgue integral.
Aneta Sikorska-Nowak (2004)
Annales Polonici Mathematici
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We prove some existence theorems for nonlinear integral equations of the Urysohn type and Volterra type , , where f and φ are functions with values in Banach spaces. Our fundamental tools are: measures of noncompactness and properties of the Henstock-Kurzweil integral.
Luisa Di Piazza, Valeria Marraffa, Kazimierz Musiał (2016)
Mathematica Bohemica
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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...
Martin Rmoutil (2025)
Czechoslovak Mathematical Journal
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For any with we provide a simple construction of an -Hölde function and a -Hölder function such that the integral fails to exist even in the Kurzweil-Stieltjes sense.