A Tauberian Theorem For The Riemann-Liouville Integral Of Integer Order

Abstract

1. Notation. Let s(x) be a function integrable in every finite interval of x ≥ 0. Then the Riemann-Liouville integral of s(x), of order a > 0, is defined for x ≥ 0 by(1).The object of this note is to prove a Tauberian theorem for sα(x) in the case in which α is a positive integer p, employing certain difference formulae due to Karamata (4, Lemma 2) and Bosanquet (1, Theorem 1) used already for a broadly similar purpose in an earlier paper (12) where a is any positive number.