### A general integral

Ricardo Estrada and Jasson Vindas Diaz UGent (2012) 483.
abstract
We define an integral, the distributional integral of functions of one real variable, that is more general than the Lebesgue and the Denjoy-Perron-Henstock Kurzweil integrals, and which allows the integration of functions with distributional values everywhere or nearly everywhere. Our integral has the property that if f is locally distributionally integrable over the real line and psi is an element of D(R) is a test function, then f psi is distributionally integrable, and the formula < f, psi > = (dist)integral(infinity)(-infinity) f(x)psi(x)dx, defines a distribution f is an element of D'(R) that has distributional point values almost everywhere and actually f(x) = f(x) almost everywhere. The indefinite distributional integral F(x) = (dist) integral(x)(a) f(t)dt corresponds to a distribution with point values everywhere and whose distributional derivative has point values almost everywhere equal to f (x). The distributional integral is more general than the standard integrals, but it still has many of the useful properties of those standard ones, including integration by parts formulas, substitution formulas, even for infinite intervals (in the Cesaro sense), mean value theorems, and convergence theorems. The distributional integral satisfies a version of Hake's theorem. Unlike general distributions, locally distributionally integrable functions can be restricted to closed sets and can be multiplied by power functions with real positive exponents.
author
organization
year
type
journalArticle (original)
publication status
published
subject
keyword
POINTWISE CONVERGENCE, FOURIER-SERIES, Lojasiewicz point values, distributions, non-absolute integrals, general integral, distributional integration, DISTRIBUTIONS, EXPANSIONS, BEHAVIOR
journal title
DISSERTATIONES MATHEMATICAE
Diss. Math.
volume
483
pages
49 pages
Web of Science type
Article
Web of Science id
000301828600001
JCR category
MATHEMATICS
JCR impact factor
0.167 (2012)
JCR rank
291/296 (2012)
JCR quartile
4 (2012)
ISSN
0012-3862
DOI
10.4064/dm483-0-1
language
English
UGent publication?
no
classification
A1
I have transferred the copyright for this publication to the publisher
id
2082321
handle
http://hdl.handle.net/1854/LU-2082321
date created
2012-04-06 16:34:59
date last changed
2016-12-19 15:46:22
```@article{2082321,
abstract     = {We define an integral, the distributional integral of functions of one real variable, that is more general than the Lebesgue and the Denjoy-Perron-Henstock Kurzweil integrals, and which allows the integration of functions with distributional values everywhere or nearly everywhere.
Our integral has the property that if f is locally distributionally integrable over the real line and psi is an element of D(R) is a test function, then f psi is distributionally integrable, and the formula
{\textlangle} f, psi {\textrangle} = (dist)integral(infinity)(-infinity) f(x)psi(x)dx,
defines a distribution f is an element of D'(R) that has distributional point values almost everywhere and actually f(x) = f(x) almost everywhere.
The indefinite distributional integral F(x) = (dist) integral(x)(a) f(t)dt corresponds to a distribution with point values everywhere and whose distributional derivative has point values almost everywhere equal to f (x).
The distributional integral is more general than the standard integrals, but it still has many of the useful properties of those standard ones, including integration by parts formulas, substitution formulas, even for infinite intervals (in the Cesaro sense), mean value theorems, and convergence theorems. The distributional integral satisfies a version of Hake's theorem. Unlike general distributions, locally distributionally integrable functions can be restricted to closed sets and can be multiplied by power functions with real positive exponents.},
author       = {Estrada, Ricardo and Vindas Diaz, Jasson},
issn         = {0012-3862},
journal      = {DISSERTATIONES MATHEMATICAE},
keyword      = {POINTWISE CONVERGENCE,FOURIER-SERIES,Lojasiewicz point values,distributions,non-absolute integrals,general integral,distributional integration,DISTRIBUTIONS,EXPANSIONS,BEHAVIOR},
language     = {eng},
pages        = {49},
title        = {A general integral},
url          = {http://dx.doi.org/10.4064/dm483-0-1},
volume       = {483},
year         = {2012},
}

```
Chicago
Estrada, Ricardo, and Jasson Vindas Diaz. 2012. “A General Integral.” Dissertationes Mathematicae 483.
APA
Estrada, R., & Vindas Diaz, J. (2012). A general integral. DISSERTATIONES MATHEMATICAE, 483.
Vancouver
1.
Estrada R, Vindas Diaz J. A general integral. DISSERTATIONES MATHEMATICAE. 2012;483.
MLA
Estrada, Ricardo, and Jasson Vindas Diaz. “A General Integral.” DISSERTATIONES MATHEMATICAE 483 (2012): n. pag. Print.