Tauberian theorems for the weighted mean method of summability of integrals
Bildiriler > Tam Metin Bildiri · 2019
Yazarlar: İbrahim ÇANAK, Firat Ozsarac
3rd International Conference of Mathematical Sciences (ICMS), Türkiye, İstanbul, Uluslararası, Sözlü Sunum, Tam Metin Bildiri, 2019-2019, Wos, cilt 2183, ss 040007-040007
DOI: 10.1063/1.5136127
Özet
Let q be a positive weight function on R+ := [0, ∞) which is integrable in Lebesgue’s sense over every finite interval (0, x) for 0 < x < ∞, in symbol: q∈Lloc1(R+) such that Q(x):=∫0xq(t)dt≠0 for each x > 0, Q(0) = 0 and Q(x) → ∞ as x → ∞. Given a real or complex-valued function f∈Lloc1(R+), we define s(x):=∫0xf(t)dt and τq(0)(x):=s(x),τq(m)(x):=1Q(x)∫0xτq(m−1)(t)q(t)dt (x>0,m=1,2,…), provided that Q(x) > 0.We say that ∫0∞f(x)dx is summable to L by the m-th iteration of weighted mean method determined by the function q(x), or for short, (N¯, q,m) integrable to a finite number L if limx→∞τq(m)(x)=L.In this case, we write s(x)→L(N¯, q,m).It is known that if the limit limx→∞s(x)=L exists, then limx→∞τq(m)(x)=L also exists. However, the converse of this implication is not always true. Some suitable conditions together with the existence of the limx→∞τq(m)(x), which is so called Tauberian conditions, may imply convergence of limx→∞s(x).In this paper, one- and two-sided Tauberian conditions in terms of the generating function and its generalizations for (N¯, q, m) summable integrals of real- or complex-valued functions have been obtained. Some classical type Tauberian theorems given for Cesàro summability (C, 1) and weighted mean method of summability (N¯, q) have been extended and generalized.