On the Generalizations of a Theorem of Beppo Levi

In this paper we give two generalizations of a theorem of Beppo Levi ([1], p. 347, Formula (12)). This theorem affirms that, under certain conditions, the following assertion is true: where ϕ(x) is a function that verifies ϕ(0) > 0; f(x) is defined and bounded in the interval (a, b) and continuou...

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Autor principal: Trion, S.E.
Formato: JOUR
Acceso en línea:http://hdl.handle.net/20.500.12110/paper_00222526_v87_n3_p195_Trion
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Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
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spelling todo:paper_00222526_v87_n3_p195_Trion2023-10-03T14:29:56Z On the Generalizations of a Theorem of Beppo Levi Trion, S.E. In this paper we give two generalizations of a theorem of Beppo Levi ([1], p. 347, Formula (12)). This theorem affirms that, under certain conditions, the following assertion is true: where ϕ(x) is a function that verifies ϕ(0) > 0; f(x) is defined and bounded in the interval (a, b) and continuous in the point 0 with f(0) ≠ 0; f(x) and ϕ(x) are integrable functions in the interval [a, b]; c >, 0 and υ > 1. This problem was studied by Laplace [2], Darboux [3], Stieltjes [4], Lebesgue [5], Romanovsky [6], and Fowler [7]. The first generalization (Section 1, Theorem 1.2, Formula (1.35)) says that, under certain conditions, the following formula is valid: where ϕ<inf>n</inf>(x) is a sequence of functions and Bn(a) designates the n-dimentional ball of radius a and center in the origin. The extension follows by Romanovsky's method. The absolute maximum of ϕ(x) in the extremes of the interval of definition is treated in the second generalization of the Theorem of Beppo Levi (Section 2, Theorem 2.2, Formulas (2.1), (2.2)). We note that Beppo Levi proves this assertion in the interior of the interval. © 2015 Wiley Periodicals, Inc., A Wiley Company. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_00222526_v87_n3_p195_Trion
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
description In this paper we give two generalizations of a theorem of Beppo Levi ([1], p. 347, Formula (12)). This theorem affirms that, under certain conditions, the following assertion is true: where ϕ(x) is a function that verifies ϕ(0) > 0; f(x) is defined and bounded in the interval (a, b) and continuous in the point 0 with f(0) ≠ 0; f(x) and ϕ(x) are integrable functions in the interval [a, b]; c >, 0 and υ > 1. This problem was studied by Laplace [2], Darboux [3], Stieltjes [4], Lebesgue [5], Romanovsky [6], and Fowler [7]. The first generalization (Section 1, Theorem 1.2, Formula (1.35)) says that, under certain conditions, the following formula is valid: where ϕ<inf>n</inf>(x) is a sequence of functions and Bn(a) designates the n-dimentional ball of radius a and center in the origin. The extension follows by Romanovsky's method. The absolute maximum of ϕ(x) in the extremes of the interval of definition is treated in the second generalization of the Theorem of Beppo Levi (Section 2, Theorem 2.2, Formulas (2.1), (2.2)). We note that Beppo Levi proves this assertion in the interior of the interval. © 2015 Wiley Periodicals, Inc., A Wiley Company.
format JOUR
author Trion, S.E.
spellingShingle Trion, S.E.
On the Generalizations of a Theorem of Beppo Levi
author_facet Trion, S.E.
author_sort Trion, S.E.
title On the Generalizations of a Theorem of Beppo Levi
title_short On the Generalizations of a Theorem of Beppo Levi
title_full On the Generalizations of a Theorem of Beppo Levi
title_fullStr On the Generalizations of a Theorem of Beppo Levi
title_full_unstemmed On the Generalizations of a Theorem of Beppo Levi
title_sort on the generalizations of a theorem of beppo levi
url http://hdl.handle.net/20.500.12110/paper_00222526_v87_n3_p195_Trion
work_keys_str_mv AT trionse onthegeneralizationsofatheoremofbeppolevi
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