Tauberian theorems for iterations of weighted mean summable integrals

Abstract

Let p be a positive weight function on $$A:=[1, \infty )$$ which is integrable in Lebesgue’s sense over every finite interval (1, x) for $$1<x<\infty $$ , in symbol: $$p \in L^{1}_{loc} (A)$$ such that $$P(x)=\int _{1}^{x} p(t) dt\ne 0$$ for each $$x>1$$ , $$P(1)=0$$ and $$P(x) \rightarrow \infty $$ as $$x \rightarrow \infty $$ . For a real-valued function $$f \in L^{1}_{loc} (A)$$ , we set $$s(x):=\int _{1}^{x}f(t)dt$$ and denote $$\begin{aligned} \sigma ^{(0)}_p(x):=s(x), \sigma ^{(k)}_p(x):=\frac{1}{P(x)}\int _1^x \sigma ^{(k-1)}_p(t) p(t)dt\,\,\, (x>1, k=1,2,\ldots ), \end{aligned}$$ provided that $$P(x)>0$$ . If $$\begin{aligned} \lim _{x\rightarrow \infty }\sigma ^{(k)}_p(x)=L, \end{aligned}$$ we say that $$\int _{1}^{\infty }f(x)dx$$ is summable to L by the k-th iteration of weighted mean method determined by the function p(x), or for short, $$({\overline{N}},p,k)$$ integrable to a finite number L and we write $$s(x)\rightarrow L({\overline{N}},p,k)$$ . It is well-known that the existence of the limit $$\lim _{x \rightarrow \infty } s(x)=L$$ implies that of $$\lim _{x \rightarrow \infty } \sigma ^{(k)}_p(x)=L$$ . But the converse of this implication is not true in general. In this paper, we obtain some Tauberian theorems for the weighted mean method of integrals in order that the converse implication holds true. Our results extend and generalize some classical type Tauberian theorems given for Cesaro and logarithmic summability methods of integrals.