A New Extension of the Riemann-Lebesgue Lemma

The asymptotic behavior for a sequence of integrals ∫ a b f( x )g( nx )dx , a,b∈ℝ, as n→∞, is investigated, where f is continuous differentiable function and g is a continuous and periodic function. The first two terms asymptotic estimates have been given by [D. Andrica, M. Piticari. Intern. Conf. on Theory and Appl. of Math. and Inf. Thessaloniki, Acta Universitatis Apulensis, 2004(8), 26-39.] with some different assumptions. This paper obtains the complete asymptotic expansions. Some examples are provided.