When a real-valued function of one variable is approximated by its $n^{th}$ degree Taylor polynomial, the remainder is estimated using the Alexiewicz and Lebesgue $p$-norms in cases where $f^{(n)}$ or $f^{(n+1)}$ are Henstock--Kurzweil integrable. When the only assumption is that $f^{(n)}$ is Henstock--Kurzweil integrable then a modified form of the $n^{th}$ degree Taylor polynomial is used. When the only assumption is that $f^{(n)}\in C^0$ then the remainder is estimated by applying the Alexiewicz norm to Schwartz distributions of order 1.
The Fourier transform is considered as a Henstock--Kurzweil integral. Sufficient conditions are given for the existence of the Fourier transform and necessary and sufficient conditions are given for it to be continuous. The Riemann--Lebesgue lemma fails: Henstock--Kurzweil Fourier transforms can have arbitrarily large point-wise growth. Convolution and inversion theorems are established. An appendix gives sufficient conditions for interchanging repeated Henstock--Kurzweil integrals and gives an estimate on the integral of a product.
By using the method in [5], the aim of the present note is to generalize the Riemann integral in probability introduced in [7], to Kurzweil-Henstock integral in probability. Properties of the new integral are proved.
In this paper, a generalised integral called the Laplace integral is defined on unbounded intervals, and some of its properties, including necessary and sufficient condition for differentiating under the integral sign, are discussed. It is also shown that this integral is more general than the Henstock-Kurzweil integral. Finally, the Fourier transform is defined using the Laplace integral, and its well-known properties are established.
Necessary and sufficient for $\int_a^bfg_n\to \int_a^bfg$ for all Henstock--Kurzweil integrable functions $f$ is that $g$ be of bounded variation, $g_n$ be uniformly bounded and of uniform bounded variation and, on each compact interval in $(a,b)$, $g_n\to g$ in measure or in the $L^1$ norm. The same conditions are necessary and sufficient for $\|f(g_n-g)\|\to 0$ for all Henstock--Kurzweil integrable functions $f$. If $g_n\to g$ a.e. then convergence $\|fg_n\|\to\|fg\|$ for all Henstock--Kurzweil integrable functions $f$ is equivalent to $\|f(g_n-g)\|\to 0$. This extends a theorem due to Lee P…
We introduce and investigate the Henstock-Kurzweil (HK) integral for Riesz-space-valued functions on time scales. Some basic properties of the HK delta integral for Riesz-space-valued functions are proved. Further, we prove uniform and monotone convergence theorems.
We investigate properties of the fuzzy Henstock-Kurzweil delta integral (shortly, FHK $Δ$-integral) on time scales, and obtain two necessary and sufficient conditions for FHK $Δ$-integrability. The concept of uniformly FHK $Δ$-integrability is introduced. Under this concept, we obtain a uniformly integrability convergence theorem. Finally, we prove monotone and dominated convergence theorems for the FHK $Δ$-integral.
We define a non-absolutely convergent integration on integral currents of dimension 1 in Euclidean space. This integral is closely related to the Henstock-Kurzweil and Pfeffer Integrals. Using it, we prove a generalized Fundamental Theorem of Calculus on these currents. A detailed presentation of Henstock-Kurzweil Integration is given in order to make the paper accessible to non-specialists.
The theory of integration over R is rich with techniques as well as necessary and sufficient conditions under which integration can be performed. Of the many different types of integrals that have been developed since the days of Newton and Leibniz, one relative newcomer is that of the Henstock integral, aka the Henstock-Kurzweil integral, Generalized Riemann integral, or gauge integral, which was discovered independently by Henstock and Kurzweil in the mid-1950s. In this paper, we develop an alternative, sequential definition of the Henstock integral over closed intervals in R that we denote …
If $f$ is a real-valued function on $[-π,π]$ that is Henstock--Kurzweil integrable, let $u_r(θ)$ be its Poisson integral. It is shown that $\|u_r\|_p=o(1/(1-r))$ as $r\to 1$ and this estimate is sharp for $1\leq p\leq\infty$. If $μ$ is a finite Borel measure and $u_r(θ)$ is its Poisson integral then for each $1\leq p\leq \infty$ the estimate $\|u_r\|_p=O((1-r)^{1/p-1})$ as $r\to 1$ is sharp. The Alexiewicz norm estimates $\|u_r\|\leq\|f\|$ ($0\leq r<1$) and $\|u_r-f\|\to 0$ ($r\to 1$) hold. These estimates lead to two uniqueness theorems for the Dirichlet problem in the unit disc with Henstock…
These are the class notes of lectures given by Ralph Henstock at the New University of Ulster in 1970-71. The notes deal with the Riemann-complete integral (also known as the generalized Riemann integral, the gauge integral, and the Henstock-Kurzweil integral). They also introduce Henstock's abstract theory of integration.
Various kinds of Stieltjes integrals using gauge integration have become highly popular in the field of differential equations and other applications. In the theories of integration and of ordinary differential equations, convergence theorems provide one of the most widely used tools. The Harnack extension principle, which discusses a sufficient condition for Kurzweil-Henstock integrable functions on particular subsets of $(a,b)$ to be integrable on $[a,b]$, is a key step to supply convergence theorems. The Kurzweil-Stieltjes integral reduces to the Kurzweil-Henstock integral whenever the inte…
Ralph Henstock (1923 - 2007), with (independently) Jaroslav Kurzweil, was the originator of the Riemann-complete or generalized Riemann integral. This material consists of four chapters of a book proposal to Cambridge University Press, which remained unpublished in the Henstock Archive at the University of Ulster. Section 5 presents a novel approach to taking limits under the integral sign, and goes on to place the traditional monotone and dominated convergence theorems of Lebesgue in this new context.
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 $τ_xf(y)=f(y-x)$. Then $\|τ_xf-f\|$ tends to 0 as $x$ tends to 0, i.e., $f$ is continuous in the Alexiewicz norm. For particular functions, $\|τ_xf-f\|$ can tend to 0 arbitrarily slowly. In general, $\|τ_xf-f\|\geq {\rm osc}f |x|$ as $x\to 0$, where ${\rm osc}f$ is the oscillation of $f$. It is shown that if $F$ is a primitive of $f$ then $\|τ_xF-F\|\leq \|f\||x|$. An example show…
In generalized Riemann integration (or Henstock-Kurzweil integration), the formation of the partitions used in Riemann sum construction is regulated by rules known as gauges. This article examines gauges for multi-dimensional Cartesian product domains.
Ralph Henstock (1923 - 2007) worked in non-absolute integration, including the Riemann-complete or gauge integral which, independently, Jaroslav Kurzweil also discovered in the 1950's. As a Cambridge undergraduate Henstock took a course of lectures, by J.C. Burkill, on the integration of interval functions. Later, under the supervision of Paul Dienes in Birkbeck College, London, he undertook research into the ideas of Burkill (interval function integrands) and Dienes (Stieltjes integrands); and he presented this thesis in December 1948. The thesis contains the germ of Henstock's later work, in…
Let $f$ be a distribution (generalised function) on the real line. If there is a continuous function $F$ with real limits at infinity such that $F'=f$ (distributional derivative) then the distributional integral of $f$ is defined as $\int_{-\infty}^\infty f = F(\infty) - F(-\infty)$. It is shown that this simple definition gives an integral that includes the Lebesgue and Henstock--Kurzweil integrals. The Alexiewicz norm leads to a Banach space of integrable distributions that is isometrically isomorphic to the space of continuous functions on the extended real line with uniform norm. The dual …
Fourier series are considered on the one-dimensional torus for the space of periodic distributions that are the distributional derivative of a continuous function. This space of distributions is denoted $\alext$ and is a Banach space under the Alexiewicz norm, $\|f\|_\T =\sup_{|I|\leq 2π}|\int_I f|$, the supremum being taken over intervals of length not exceeding $2π$. It contains the periodic functions integrable in the sense of Lebesgue and Henstock-Kurzweil. Many of the properties of $L^1$ Fourier series continue to hold for this larger space, with the $L^1$ norm replaced by the Alexiewicz …
A distribution on the real line has a continuous primitive integral if it is the distributional derivative of a function that is continuous on the extended real line. The space of distributions integrable in this sense is a Banach space that includes all functions integrable in the Lebesgue and Henstock--Kurzweil senses. The one-dimensional heat equation is considered with initial data that is integrable in the sense of the continuous primitive integral. Let $Θ_t(x)=\exp(-x^2/(4t))/\sqrt{4πt}$ be the heat kernel. With initial data $f$ that is the distributional derivative of a continuous funct…
