Using the concept of the H1-integral, we consider a similarly defined Stieltjes integral. We prove a Riemann-Lebesgue type theorem for this integral and give examples of adjoint classes of functions.

Adjoint classes of functions in the H1 sense

Based on the total H$_{1}$-integrability concept, which is established in this paper, we shall try to show that at any point of a compact interval $% (a,b]$ in $\mathbb{R}$, at which a point function $F$ has no a discontinuity, $F$ is the total H$_{1}$-indefinite integral of a function $% dF_{ex}$ being the limit of $\Delta F_{ex}\left( I\right) $, where $% I\subseteq \left[ a,b\right] $, on $\left[ a,b\right] $, without additional hypotheses on $F$. A residue function of $F$ is introduced. The paper ends with a few of examples that illustrate the theory.

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On Totalization of the $H_1$-Integral

In this note we show that the uniform convergence theorem for the H1-integral is false.

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Uniform convergence theorem for the h1-integral revisited

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Integration with One-Dimensional Space of Gauges

Using the total H1-integrability concept we shall show that functions, which take on infinite values in the interval (−π, π) at only finitely many places, can be expanded into a Fourier series over this interval.

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Fourier series of functions with infinite discontinuities

We present two convergence theorems for the H1-integral. The Henstock integral is now relatively well-known. An attempt has been made by Garces, Lee, and Zhao (2) to define the Henstock integral as the Moore-Smith limit of Riemann sums. The resulting integral is the so-called H1-integral. It has the property that a function f is Henstock integrable on (a;b) if and only if there is an H1-integrable function g such that f(x) = g(x) almost everywhere in (a;b). Every integral has a corresponding convergence theorem. For example, the Denjoy integral has the controlled convergence the- orem, whereas the Perron integral has the generalized dominated convergence theorem. Corresponding to the Henstock integral, which is equivalent to both the integrals of Denjoy and Perron, is the equi-integrability theorem with the strong Lusin condition. It is the purpose of the current paper to present two (well-known) convergence theorems that hold for the H1-integral. We assume that the reader is familiar with the definition of the Henstock integral (5). A division D of (a;b) is a finite set of interval-point pairs ((u;v);») such that the intervals (u;v) are non-overlapping, (a;b) = ((u;v), and » 2 (u;v). If -(x) > 0 for x 2 (a;b), then a division D = f((u;v);»)g is said to be --fine if » 2 (u;v) ‰ (» i -(»);» + -(»)) for each ((u;v);») 2 D. A function f is said to be Henstock integrable to a real number A on (a;b) if for every † > 0 there exists a positive function - on (a;b) such that for every --fine division D, we have

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P. Y. Lee

Papers

Convergence theorems for the h1-integral