Displaying similar documents to “Cauchy's residue theorem for a class of real valued functions”

The L r Henstock-Kurzweil integral

Paul M. Musial, Yoram Sagher (2004)

Studia Mathematica

Similarity:

We present a method of integration along the lines of the Henstock-Kurzweil integral. All L r -derivatives are integrable in this method.

Henstock-Kurzweil integral on BV sets

Jan Malý, Washek Frank Pfeffer (2016)

Mathematica Bohemica

Similarity:

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...

Continuity in the Alexiewicz norm

Erik Talvila (2006)

Mathematica Bohemica

Similarity:

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...

The M α and C -integrals

Jae Myung Park, Hyung Won Ryu, Hoe Kyoung Lee, Deuk Ho Lee (2012)

Czechoslovak Mathematical Journal

Similarity:

In this paper, we define the M α -integral of real-valued functions defined on an interval [ a , b ] and investigate important properties of the M α -integral. In particular, we show that a function f : [ a , b ] → R is M α -integrable on [ a , b ] if and only if there exists an A C G α function F such that F ' = f almost everywhere on [ a , b ] . It can be seen easily that every McShane integrable function on [ a , b ] is M α -integrable and every M α -integrable function on [ a , b ] is Henstock integrable. In addition, we show that the M α -integral is equivalent to...

Variational Henstock integrability of Banach space valued functions

Luisa Di Piazza, Valeria Marraffa, Kazimierz Musiał (2016)

Mathematica Bohemica

Similarity:

We study the integrability of Banach space valued strongly measurable functions defined on [ 0 , 1 ] . In the case of functions f given by ∑ n = 1 ∞ 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 ∑ n = 1 ∞ x n | E n | is absolutely convergent. Unconditional convergence of the series is equivalent to Pettis integrability...

On coincidence of Pettis and McShane integrability

Marián J. Fabián (2015)

Czechoslovak Mathematical Journal

Similarity:

R. Deville and J. Rodríguez proved that, for every Hilbert generated space X , every Pettis integrable function f : [ 0 , 1 ] → X is McShane integrable. R. Avilés, G. Plebanek, and J. Rodríguez constructed a weakly compactly generated Banach space X and a scalarly null (hence Pettis integrable) function from [ 0 , 1 ] into X , which was not McShane integrable. We study here the mechanism behind the McShane integrability of scalarly negligible functions from [ 0 , 1 ] (mostly) into C ( K ) spaces. We focus in more detail on...

Compact operators and integral equations in the ℋ𝒦 space

Varayu Boonpogkrong (2022)

Czechoslovak Mathematical Journal

Similarity:

The space ℋ𝒦 of Henstock-Kurzweil integrable functions on [ a , b ] is the uncountable union of Fréchet spaces ℋ𝒦 ( X ) . In this paper, on each Fréchet space ℋ𝒦 ( 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 ℋ𝒦 ( X ) space. It is known that every control-convergent sequence in the ℋ𝒦 space always belongs to a ℋ𝒦 ( X ) space for some X . We illustrate how...