Abstract:

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 ϕn(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.