On a mean method of summability

Year 2021, Volume: 3 Issue: 1, 15 - 19, 29.04.2021

İbrahim Çanak

https://doi.org/10.47087/mjm.896657

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.

Keywords

summability by the weighted mean method, Tauberian conditions and theorems, slow decrease and oscillation with respect to a weight function

References

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