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.
summability by the weighted mean method Tauberian conditions and theorems slow decrease and oscillation with respect to a weight function
Birincil Dil | İngilizce |
---|---|
Konular | Matematik |
Bölüm | Articles |
Yazarlar | |
Yayımlanma Tarihi | 29 Nisan 2021 |
Kabul Tarihi | 22 Mart 2021 |
Yayımlandığı Sayı | Yıl 2021 Cilt: 3 Sayı: 1 |
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ISSN 2667-7660