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 e L~1_(loc)(R_+) such that Q(x) := f_0~x q(t)dt ± 0 for each x > 0, Q(0) = 0 and Q(x) → ∞ as x → ∞. Given a real or complex-valued function f ∈ L~1_(loc)(R_+), we define s(x) := f_0~x f(t)dt and {formula}, provided that Q(x) > 0. We say that f~∞_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 {formula}. In this case, we write s(x) → L(N, q, m). It is known that if the limit lim_(x→∞) s(x) = L exists, then limx→∞ τ~((m))_q(x), = L also exists. However, the converse of this implication is not always true. Some suitable conditions together with the existence of the limit lim_(x→∞ τ(m)q(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.
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