TY - JOUR UR - http://lib.ugent.be/catalog/pug01:2082321 ID - pug01:2082321 LA - eng TI - A general integral PY - 2012 JO - (2012) DISSERTATIONES MATHEMATICAE SN - 0012-3862 PB - 2012 AU - Estrada, Ricardo AU - Vindas Diaz, Jasson WE01 802000645185 0000-0002-3789-8577 AB - 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. ER -Download RIS file
00000nam^a2200301^i^4500 | |||
001 | 2082321 | ||
005 | 20180813141417.0 | ||
008 | 120406s2012------------------------eng-- | ||
022 | a 0012-3862 | ||
024 | a 000301828600001 2 wos | ||
024 | a 1854/LU-2082321 2 handle | ||
024 | a 10.4064/dm483-0-1 2 doi | ||
040 | a UGent | ||
245 | a A general integral | ||
260 | c 2012 | ||
520 | a 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. | ||
598 | a A1 | ||
100 | a Estrada, Ricardo | ||
700 | a Vindas Diaz, Jasson u WE01 0 802000645185 0 0000-0002-3789-8577 9 2150C3A8-F0EE-11E1-A9DE-61C894A0A6B4 | ||
650 | a Mathematics and Statistics | ||
653 | a POINTWISE CONVERGENCE | ||
653 | a FOURIER-SERIES | ||
653 | a Lojasiewicz point values | ||
653 | a distributions | ||
653 | a non-absolute integrals | ||
653 | a general integral | ||
653 | a distributional integration | ||
653 | a DISTRIBUTIONS | ||
653 | a EXPANSIONS | ||
653 | a BEHAVIOR | ||
773 | t DISSERTATIONES MATHEMATICAE g Diss. Math. 2012. 483 q 483:< | ||
856 | 3 Full Text u https://biblio.ugent.be/publication/2082321/file/2082324 z [open] y A_general_integral.pdf | ||
920 | a article | ||
Z30 | x WE 1 WE01 | ||
922 | a UGENT-WE |
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