The (C, α) integrability of functions by weighted mean methods

Abstract

Let p(x) be a nondecreasing continuous function on [0, ?) such that p(0) = 0 and p(t) ? ? as t ? ?. For a continuous function f (x) on [0, ?), we define s(t)= ?0t f(u)du and ?? (t) =?0t? (1- p(u)/p(t))? f(u)du. We say that a continuous function f (x) on [0, ?) is (C, ?) integrable to a by the weighted mean method determined by the function p(x) for some ? > ?1 if the limit limt?? ?? (t) = a exists. We prove that if the limit limt?? ?? (t) = a exists for some ? > ?1, then the limit limt?? ??+h (t) = a exists for all h > 0. Next, we prove that if the limit limt?? ?? (t) = a exists for some ? > 0 and p(t)/p?(t) f(t)= O(1), t ? ?, then the limit limt?? ???1 (t) = a exists.