Henstock-Kurzweil Integrable Functions Research Articles

Some characterization of $L^r-$ Henstock-Kurzweil integrable functions

In this article, we discuss few properties of $L^r$-Henstock-Kurzweil (in short $L^r$-HK) integrable functions, introduced by Paul Musial in \cite{MS}. We re-defined $L^r$-bounded variations. We have proved that $L^r$-Henstock-Kurzweil integrable functions are Denjoy integrable.

A new characterization of the dual space of the HK-integrable functions

<abstract><p>We construct the Henstock-Kurzweil (HK) integral as an extension of a linear form initially defined on $ L^{1} $, but which is not continuous in this space. This gives us an alternative way to prove existing results. In particular, we give a new characterization of the dual space of Henstock-Kurzweil integrable functions in terms of a quotient space.</p></abstract>

Sufficient Conditions for Convergence of Sequences of Henstock-Kurzweil Integrable Functions

The main aim of this paper is to present our approach of obtaining sufficient conditions for convergence of sequences of Henstock-Kurzweil integrable functions. Our approach involves the use of the concept of multiplier functions, where we define a class Φ of multipliers for the Henstock-Kurzweil integral. We consider a sequence (fn) of Henstock-Kurzweil integrable functions on a non-degenerate interval [a, b] and we assume that (fn) converges point wise to a function f. Then we show that f is Henstock-Kurzweil integrable and its integral is equal to the limit of the sequence (∫abfn) if there exists φ∈Φ such that the defined functionals of the type F (φ, fn) satisfy the imposed conditions. Beside the fact that the results regarding the convergence under the integral sign are always of great importance, the method introduced here can be imitated and used to obtain other results on the related areas.

Henstock-Kurzweil Integration on Metric Spaces Revisited

We fill up some gaps in the existing literature on the Henstock-Kurzweil integration on metric measure spaces. The most important one is the choice of suitable candidates for `intervals' in metric spaces for which the conclusion of Cousin's lemma holds. We also provide some characterizations of compactness and completeness in terms of Cousin's lemma, along with some alternative proofs of a few related results. Then we improve the result regarding differentiability of the primitives of Henstock-Kurzweil integrable functions on metric spaces, and a few consequences. We propose shorter proofs of measure theoretic characterizations of the HK-integral in terms of the variational measure $V_F$ as well as the $ACG^\Delta$ functions. Finally, we present alternative proofs of some generalizations of two results on Lebesgue integration. The first one bypasses Vitali covering lemma for a result on absolute continuity, while the second one is a version of the fundamental theorem of calculus in our setting.

Inclusion Properties of Henstock-Orlicz Spaces

Henstock-Orlicz spaces were generally introduced by Hazarika and Kalita in 2021. In general, a function is Lebesgue integral if only if that function and its modulus are Henstock-Kurzweil integrable functions. Moreover, suppose a function is a finite measurable function with compact supports. In that case, the function is a Henstock-Kurzweil integrable function if only if the function is a Lebesgue integrable function. Due to these properties, Henstock-Orlicz spaces were constructed by utilizing Young functions. This definition is almost similar to the definition of Orlicz spaces, but by embedding the Henstock-Kurzweil integral, and the norm used is the Luxembourg norm. Therefore, an analysis of properties in these spaces is needed carried out more deeply. This research was using a literature study on inclusion properties from scientific journals, especially those related to the Orlicz Spaces. And based on the definition of Henstock-Orlicz spaces and its norm, we formulate a hypothesis regarding the inlcusion properties. By deductive proof, we proof the hypothesis and state it as theorem. In this study, we obtain sufficient and necessary conditions for the inclusion properties in Henstock-Orlicz spaces.

Canonical Orlicz class 𝒦𝒮−𝜃[ℝJn

The objective of this paper is to construct canonical Orlicz class and study their fundamental properties. Also, we prove that this space contains Henstock–Kurzweil integrable functions.

Banach-Steinhaus Theorem for the Space P of All Primitives of Henstock-Kurzweil Integrable Functions

In this paper, it is shown how the Banach-Steinhaus theorem for the space P of all primitives of Henstock-Kurzweil integrable functions on a closed bounded interval, equipped with the uniform norm, can follow from the Banach-Steinhaus theorem for the Denjoy space by applying the classical Hahn-Banach theorem and Riesz representation theorem.

English

The space $${\cal H}{\cal K}$$ of Henstock-Kurzweil integrable functions on [a, b] is the uncountable union of Frechet spaces $${\cal H}{\cal K}$$ (X). In this paper, on each Frechet space $${\cal H}{\cal K}$$ (X), an F-norm is defined for a continuous linear operator. Hence, many important results in functional analysis, like the Banach-Steinhaus theorem, the open mapping theorem and the closed graph theorem, hold for the $${\cal H}{\cal K}$$ (X) space. It is known that every control-convergent sequence in the $${\cal H}{\cal K}$$ space always belongs to a $${\cal H}{\cal K}$$ (X) space for some X. We illustrate how to apply results for Frechet spaces $${\cal H}{\cal K}$$ (X) to control-convergent sequences in the $${\cal H}{\cal K}$$ space. Examples of compact linear operators are given. Existence of solutions to linear and Hammerstein integral equations is proved.

The Continuous Primitive Integral in the Plane

An integral is defined on the plane that includes the Henstock-Kurzweil and Lebesgue integrals (with respect to Lebesgue measure). A space of primitives is taken as the set of continuous real-valued functions \(F(x,y)\) defined on the extended real plane \([-\infty,\infty]^2\) that vanish when \(x\) or \(y\) is \(-\infty\). With usual pointwise operations this is a Banach space under the uniform norm. The integrable functions and distributions (generalised functions) are those that are the distributional derivative \(\partial^2/(\partial x\partial y)\) of this space of primitives. If \(f=\partial^2/(\partial x\partial y) F\) then the integral over interval \([a,b]\times [c,d] \subseteq[-\infty,\infty]^2\) is \(\int_a^b\int_c^d f=F(a,c)+F(b,d)-F(a,d)-F(b,c)\) and \(\int_{-\infty}^\infty \int_{-\infty}^\infty f=F(\infty,\infty)\). The definition then builds in the fundamental theorem of calculus. The Alexiewicz norm is \({\lVert f\rVert}={\lVert F\rVert}_\infty\) where \(F\) is the unique primitive of \(f\). The space of integrable distributions is then a separable Banach space isometrically isomorphic to the space of primitives. The space of integrable distributions is the completion of both \(L^1\) and the space of Henstock-Kurzweil integrable functions. The Banach lattice and Banach algebra structures of the continuous functions in \({\lVert \cdot\rVert}_\infty\) are also inherited by the integrable distributions. It is shown that the dual space is the functions of bounded Hardy-Krause variation. Various tools that make these integrals useful in applications are proved: integration by parts, Hölder’s inequality, second mean value theorem, Fubini’s theorem, a convergence theorem, change of variables, convolution. The changes necessary to define the integral in \({\mathbb R}^n\) are sketched out.

Fourier Analysis with Generalized Integration

We generalize the classic Fourier transform operator F p by using the Henstock–Kurzweil integral theory. It is shown that the operator equals the H K -Fourier transform on a dense subspace of L p , 1 < p ≤ 2 . In particular, a theoretical scope of this representation is raised to approximate the Fourier transform of functions on the mentioned subspace numerically. Besides, we show the differentiability of the Fourier transform function F p ( f ) under more general conditions than in Lebesgue’s theory. Additionally, continuity of the Fourier Sine transform operator into the space of Henstock-Kurzweil integrable functions is proved, which is similar in spirit to the already known result for the Fourier Cosine transform operator. Because our results establish a representation of the Fourier transform with more properties than in Lebesgue’s theory, these results might contribute to development of better algorithms of numerical integration, which are very important in applications.

The N-Integral

In this paper, we introduced a Henstock-type integral named N-integral of a real valued function f on a closed and bounded interval [a,b]. The set N-integrable functions lie entirely between Riemann integrable functions and Henstock-Kurzweil integrable functions. Furthermore, this new integral integrates all improper Riemann integrable functions even if they are not Lebesgue integrable. It was shown that for a Henstock-Kurzweil integrable function f on [a,b], the following are equivalent:The function f is N-integrable;There exists a null set S for which given epsilon >0 there exists a gauge delta such that for any delta-fine partial division D={(xi,[u,v])} of [a,b] we have [(phi_S(D) Gamma_epsilon) sum |f(v)-f(u)||v-u|<epsilon] where phi_S(D)={(xi,[u,v])in D:xi not in S} and [Gamma_epsilon={(xi,[u,v]): |f(v)-f(u)|<= epsilon}] andThe function f is continuous almost everywhere. A characterization of continuous almost everywhere functions was also given.

The HK-Sobolev space and applications to one-dimensional boundary value problems

In this paper, we introduce the HK-Sobolev space and establish a fundamental theorem of calculus and an integration by parts formula, then we give sufficient conditions for the existence and uniqueness of a solution to a variational problem associated with a Sturm–Liouville type equation involving Henstock-Kurzweil integrable functions as source terms.

Numerical Solution of Some Differential Equations with Henstock–Kurzweil Functions

In this work, the Finite Element Method is used for finding the numerical solution of an elliptic problem with Henstock–Kurzweil integrable functions. In particular, Henstock–Kurzweil high oscillatory functions were considered. The weak formulation of the problem leads to integrals that are calculated using some special quadratures. Definitions and theorems were used to guarantee the existence of the integrals that appear in the weak formulation. This allowed us to apply the above formulation for the type of slope bounded variation functions. Numerical examples were developed to illustrate the ideas presented in this article.

English

In the present paper, we investigate the existence of solutions to boundary value problems for the one-dimensional Schrodinger equation $-y''+qy=f$, where $q$ and $f$ are Henstock-Kurzweil integrable functions on $[a,b]$. Results presented in this article are generalizations of the classical results for the Lebesgue integral.

Banach Spaces for the Schwartz Distributions

This paper is a survey of a new family of Banach spaces \(\mathcal{B}\) that provide the same structure for the Henstock-Kurzweil (HK) integrable functions as the \(L^p\) spaces provide for the Lebesgue integrable functions. These spaces also contain the wide sense Denjoy integrable functions. They were first use to provide the foundations for the Feynman formulation of quantum mechanics. It has recently been observed that these spaces contain the test functions \(\mathcal{D}\) as a continuous dense embedding. Thus, by the Hahn-Banach theorem, \(\mathcal{D}' \subset \mathcal{B}'\). A new family that extend the space of functions of bounded mean oscillation \(BMO[\mathbb{R}^n]\), to include the HK-integrable functions are also introduced.

On Vector Valued Multipliers for the Class of Strongly ℋ𝒦-Integrable Functions

Abstract We investigate the space of vector valued multipliers of strongly Henstock-Kurzweil integrable functions. We prove that if X is a commutative Banach algebra with identity e such that ‖e‖ = 1 and g : [a, b] → X is of strongly bounded variation, then the multiplication operator defined by Mg (f) := fg maps 𝒮ℋ𝒦 to ℋ𝒦. We also prove a partial converse, when X is a Gel’fand space.

Some results about completion of the space of Henstock-Kurzweil integrable functions

Continuing with our study begun in [6] and making use of some results of that article, in this paper we establish new properties for the completion of the space of Henstock-Kurzweil integrable functions with Alexiewicz norm. Mathematics Subject Classification: Primary 26A39; Secondary 46B03, 46B04

The Use of an Isometric Isomorphism on the Completion of the Space of Henstock-Kurzweil Integrable Functions

Employing an isometrically isomorphic space, we determine new properties for the completion of the space of the Henstock-Kurzweil integrable functions with the Alexiewicz norm.

The Closed Graph Theorem and the Space of Henstock-Kurzweil Integrable Functions with the Alexiewicz Norm

We prove that the cardinality of the spaceℋ𝒦([a,b])is equal to the cardinality of real numbers. Based on this fact we show that there exists a norm onℋ𝒦([a,b])under which it is a Banach space. Therefore if we equipℋ𝒦([a,b])with the Alexiewicz topology thenℋ𝒦([a,b])is not K-Suslin, neither infra-(u) nor a webbed space.

Equations containing locally Henstock-Kurzweil integrable functions

A fixed point theorem in ordered spaces and a recently proved monotone convergence theorem are applied to derive existence and comparison results for solutions of a functional integral equation of Volterra type and a functional impulsive Cauchy problem in an ordered Banach space. A novel feature is that equations contain locally Henstock-Kurzweil integrable functions.