Tauberian theorems concerning weighted mean summable integrals
Makaleler > Tam Makale · 2023
Yazarlar: Sefa Anıl Sezer, İbrahim ÇANAK
Periodica Mathematica Hungarica, SCI Expanded İndekslerine Giren Dergi, Uluslararası, Wos, 2023, cilt 87, ss 315-323
DOI: 10.1007/s10998-023-00519-4
Özet
Let p be a positive real-valued continuous function on R+ such that the function P(x)=∫0xp(t)dt,x>0,is regularly varying with a positive index in the Karamata sense. For a real- or complex-valued continuous function f on R+, we define s(x)=∫0xf(y)dyandσp(x)=1P(x)∫0xs(y)p(y)dy.It is known that if the finite limit lim x→∞s(x) = L exists, then so does lim x→∞σp(x) = L. In this paper, we introduce some Tauberian conditions in terms of the weighted classical control modulo and the weighted general control modulo of order one under which the converse implication and its extensions hold. Our results generalize some classical type Tauberian theorems existing in the literature.