### A general integral

(2012) DISSERTATIONES MATHEMATICAE. 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.

Please use this url to cite or link to this publication:
http://hdl.handle.net/1854/LU-2082321

- author
- Estrada, Ricardo and Vindas Diaz, Jasson UGent
- organization
- year
- 2012
- 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
- copyright statement
*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.