The $$L^{r}$$-Variational Integral

It is proved that any function of a Lusin-type class, the class of ACGr-functions, is differentiable almost everywhere in the sense of a derivative defined in the space L r , 1 ≤ r < ∞. This leads to obtaining a full descriptive characterization of a Henstock-Kurzweil-type integral, the HKrintegral, which serves to recover functions from their L rderivatives. The class ACGr is compared with the classical Lusin class ACG and it is shown that a continuous ACGfunction can fail to be L r -differentiable almost everywhere.

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“…holds for 0 < h ≤ h x . So we can assume that h k < h x ≤ γ for all k. Having in mind ( 10), ( 11), (12), and ( 13), we obtain:…”

Section: R -Differentiability Of Acg R -Functionsmentioning

confidence: 99%

“…It was also shown in [6] that the indefinite HK r -integral is L r -differentiable almost everywhere and belongs to a Lusin-type class of ACG r -functions. Some other properties of these integrals are investigated in [9,10,11,12].…”

Section: Introductionmentioning

confidence: 99%

On Descriptive Characterizations of an Integral Recovering a Function from Its $$L^r$$-Derivative

2022

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“…Originally, his ideas were developed as an extension of the Riemann integral, known as the Riemann-Stieltjes integral [9]. In addition, Jong Sul Lim, Ju Han Yoon and Gwang Sik Eun defined the Kurzweil-Henstock-Stieltjes integral on R in which the integrator is an increasing function [10], [11]. This integral is more general compared to the Kurzweil-Henstock integral; in fact, the Kurzweil-Henstock integral is a special case of Kurzweil-Henstock-Stieltjes integral, whenever the integrator is an identity function [12], [13], [14], [15].…”

Section: Introductionmentioning

confidence: 99%

Some Elementary Properties of Kurzweil-Henstock-Stieltjes Integral on $\mathbb{R}^n$

Macaso

Flores

2022

ARJOM

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Kurzweil-Henstock integral is a generalization of the Reimann integral. In this paper, we established the definition of Kurzweil-Henstock-Stieltjes integral on $\mathbb{R}$n via gauge type approach where integrand and integrator are all real-valued functions defined on a compact interval in $\mathbb{R}$n. Moreover, the Cauchy Criterion is established. To this end, some underlying simple properties of this integral are studied, specifically, uniqueness, linearity, monotonocity, integrability over a subset, and additivity. Results gathered in this paper may serve as a foundation to some related studies such as the notion of convergence with respect to this integral, and the formulation of the Saks-Henstock Lemma.

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Comparison of the P-integral with Burkill's integrals and some applications to trigonometric series

Musial

Скворцов

Tulone

2023

Journal of Mathematical Analysis and Applications

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The $$L^{r}$$-Variational Integral

Abstract: We define the $$L^r$$ L r -variational integral and we prove that it is equivalent to the $$HK_r$$ H K r -integral defined in 2004 by P. Musial and Y. Sagher in the Studia Mathematica paper The$$L^{r}$$ L … Show more

Cited by 5 publications

References 12 publications

On Descriptive Characterizations of an Integral Recovering a Function from Its $$L^r$$-Derivative

On Descriptive Characterizations of an Integral Recovering a Function from Its $$L^r$$-Derivative

Some Elementary Properties of Kurzweil-Henstock-Stieltjes Integral on \(\mathbb{R}^n\)

Comparison of the P-integral with Burkill's integrals and some applications to trigonometric series

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