Displaying similar documents to “The L r Henstock-Kurzweil integral”

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 ± ∞ 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 ∫ 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...

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...

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 | ∫ I f | where the supremum is taken over all intervals I ⊂ 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 ∥ ≥ osc f | x | as x → 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 ∥ ≤ ∥ f ∥ | x | . An example shows that the function y ↦ τ x F ( y ) - F ( y ) need not be in L 1 . However, if...

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.

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 ∈ 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 ∈ L 1 ( ℝ ) . We prove that this condition is optimal through the theoretical scope of the Henstock-Kurzweil integration theory.

On the existence of solutions of nonlinear integral equations in Banach spaces and Henstock-Kurzweil integrals

Aneta Sikorska-Nowak (2004)

Annales Polonici Mathematici

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We prove some existence theorems for nonlinear integral equations of the Urysohn type x ( t ) = φ ( t ) + λ ∫ 0 a f ( t , s , x ( s ) ) d s and Volterra type x ( t ) = φ ( t ) + ∫ 0 t f ( t , s , x ( s ) ) d s , t ∈ I a = [ 0 , a ] , 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.