The (C, α) integrability of functions by weighted mean methods
Copyright policy )The (C, α) integrability of functions by weighted mean methods
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.
- Adnan Menderes University Turkey
- Ege University Turkey
Tauberian conditions and theorems, Cesàro integrability, Divergent integrals, (C,?) integrability, Cesaro integrability, (C, alpha) integrability
Tauberian conditions and theorems, Cesàro integrability, Divergent integrals, (C,?) integrability, Cesaro integrability, (C, alpha) integrability
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