Generalizations of the Riemann-Lebesgue and Cantor-Lebesgue lemmas

Abstract

The purpose of this paper is to consider the problem of evaluating limits of the type (0.1) lim {f{t){h) dt under various assumptions regarding the functions / and and the interval /. Perhaps the most famihar example of such a limit occurs in the Riemann-Lebesgue lemma v^hich asserts that (0.2) lim f{t) sin () dt = 0 provided that / is an integrable function over the interval I; and so our results may be viev^ed as a generalization of that well-known lemma. These results, for / infinit and finite, will be stated and proved in Sections I and 2, respectively. We will then apply them in Section 3 to establish a generalization of the Cantor-Lebesgue lemma.