On a mean method of summability

Abstract

Let $p(x)$ be a nondecreasing real-valued continuous function
 on $R_+:=[0,\infty)$ such that $p(0)=0$ and $p(x) \to \infty$ as $x \to \infty$.
 Given a real or complex-valued integrable function $f$ in Lebesgue's sense on every bounded interval $(0,x)$
 for $x>0$, in symbol $f \in L^1_{loc} (R_+)$, we set
 $$
 s(x)=\int _{0}^{x}f(u)du
 $$
 and
 $$
 \sigma _{p}(s(x))=\frac{1}{p(x)}\int_{0}^{x}s(u)dp(u),\,\,\,\,x>0
 $$
 provided that $p(x)>0$.
 
 A function $s(x)$
 is said to be summable to $l$ by the weighted mean method determined
 by the function $p(x)$, in short, $(\overline{N},p)$ summable to $l$,
 if
 $$
 \lim_{x \to \infty}\sigma _{p}(s(x))=l.
 $$
 
 If the limit $\lim _{x \to \infty} s(x)=l$
 exists, then $\lim _{x \to \infty} \sigma _{p}(s(x))=l$ also exists. However, the converse is not true in general.
 In this paper, we give an alternative proof a Tauberian theorem stating that convergence follows from summability by weighted mean method on $R_+:=[0,\infty)$ and a Tauberian condition of slowly decreasing type with respect to the weight function due to Karamata. These Tauberian conditions are one-sided or two-sided if $f(x)$ is a real or complex-valued function, respectively. Alternative proofs of some well-known Tauberian theorems given for several important summability methods can be obtained by choosing some particular weight functions.