A general integral

R. Estrada; J. Vindas

Displaying similar documents to “A general integral”

Continuity in the Alexiewicz norm

Erik Talvila (2006)

Mathematica Bohemica

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If $f$ is a Henstock-Kurzweil integrable function on the real line, the Alexiewicz norm of $f$ is $∥ f ∥ = {sup}_{I} | \int_{I} f |$ where the supremum is taken over all intervals $I \subset ℝ$ . Define the translation $τ_{x}$ by $τ_{x} f (y) = f (y - x)$ . Then $∥ τ_{x} f - f ∥$ tends to $0$ as $x$ tends to $0$ , i.e., $f$ is continuous in the Alexiewicz norm. For particular functions, $∥ τ_{x} f - f ∥$ can tend to 0 arbitrarily slowly. In general, $∥ τ_{x} f - f ∥ \geq osc f | x |$ as $x \to 0$ , where $osc f$ is the oscillation of $f$ . It is shown that if $F$ is a primitive of $f$ then $∥ τ_{x} F - F ∥ \leq ∥ f ∥ | x |$ . An example shows that the function $y \mapsto τ_{x} F (y) - F (y)$ need not be in $L^{1}$ . However, if...

The $L^{r}$ 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 $L^{r}$ -derivatives are integrable in this method.

Role of the Harnack extension principle in the Kurzweil-Stieltjes integral

Umi Mahnuna Hanung (2024)

Mathematica Bohemica

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In the theories of integration and of ordinary differential and integral equations, convergence theorems provide one of the most widely used tools. Since the values of the Kurzweil-Stieltjes integrals over various kinds of bounded intervals having the same infimum and supremum need not coincide, the Harnack extension principle in the Kurzweil-Henstock integral, which is a key step to supply convergence theorems, cannot be easily extended to the Kurzweil-type Stieltjes integrals with...

Cauchy's residue theorem for a class of real valued functions

Branko Sarić (2010)

Czechoslovak Mathematical Journal

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Let $[a, b]$ be an interval in $ℝ$ and let $F$ be a real valued function defined at the endpoints of $[a, b]$ and with a certain number of discontinuities within $[a, b]$ . Assuming $F$ to be differentiable on a set $[a, b] ∖ E$ to the derivative $f$ , where $E$ is a subset of $[a, b]$ at whose points $F$ can take values $\pm \infty$ or not be defined at all, we adopt the convention that $F$ and $f$ are equal to $0$ at all points of $E$ and show that $𝒦ℋ -vt \int_{a}^{b} f = F (b) - F (a)$ , where $𝒦ℋ -vt$ denotes the total value of the integral. The paper ends with a few examples that illustrate the...

Henstock-Kurzweil integral on $BV$ sets

Jan Malý, Washek Frank Pfeffer (2016)

Mathematica Bohemica

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The generalized Riemann integral of Pfeffer (1991) is defined on all bounded $BV$ subsets of $ℝ^{n}$ , but it is additive only with respect to pairs of disjoint sets whose closures intersect in a set of $σ$ -finite Hausdorff measure of codimension one. Imposing a stronger regularity condition on partitions of $BV$ 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...

A necessary condition for HK-integrability of the Fourier sine transform function

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 $f \in {BV}_{0} (ℝ)$ , where ${BV}_{0} (ℝ)$ is the space of bounded variation functions vanishing at infinity. It is shown that for the Fourier sine transform function of $f$ to be integrable in the Henstock-Kurzweil sense, it is necessary that $f / x \in L^{1} (ℝ)$ . We prove that this condition is optimal through the theoretical scope of the Henstock-Kurzweil integration theory.

Boundary value problems for the Schrödinger equation involving the Henstock-Kurzweil integral

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 $- y^{''} + q y = f$ , 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.

Copies of $ℓ_{\infty}$ in the space of Pettis integrable functions with integrals of finite variation

Juan Carlos Ferrando (2012)

Studia Mathematica

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Let (Ω,Σ,μ) be a complete finite measure space and X a Banach space. We show that the space of all weakly μ-measurable (classes of scalarly equivalent) X-valued Pettis integrable functions with integrals of finite variation, equipped with the variation norm, contains a copy of $ℓ_{\infty}$ if and only if X does.

A version of the law of large numbers

Katusi Fukuyama (2001)

Colloquium Mathematicae

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By the method of Rio [10], for a locally square integrable periodic function f, we prove $(f (μ ₁^{t} x) + . . . + f (μ ₙ^{t} x)) / n \to \int_{0}^{1} f$ for almost every x and t > 0.

Variational Henstock integrability of Banach space valued functions

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 $[0, 1]$ . In the case of functions $f$ given by $\sum_{n = 1}^{\infty} x_{n} χ_{E_{n}}$ , where $x_{n}$ are points of a Banach space and the sets $E_{n}$ are Lebesgue measurable and pairwise disjoint subsets of $[0, 1]$ , there are well known characterizations for Bochner and Pettis integrability of $f$ . The function $f$ is Bochner integrable if and only if the series $\sum_{n = 1}^{\infty} x_{n} | E_{n} |$ is absolutely convergent. Unconditional convergence of the series is equivalent to Pettis integrability...