On a mean method of summability

Authors : İbrahim ÇANAK

Pages : 15-19

Doi:10.47087/mjm.896657

View : 11 | Download : 10

Publication Date : 2021-04-29

Article Type : Research Paper

Abstract :Let $pinsert ignore into journalissuearticles values(x);$ be a nondecreasing real-valued continuous function on $R_+:=[0,\infty);$ such that $pinsert ignore into journalissuearticles values(0);=0$ and $pinsert ignore into journalissuearticles values(x); \to \infty$ as $x \to \infty$. Given a real or complex-valued integrable function $f$ in Lebesgue`s sense on every bounded interval $insert ignore into journalissuearticles values(0,x);$ for $x>0$, in symbol $f \in L^1_{loc} insert ignore into journalissuearticles values(R_+);$, we set $$ sinsert ignore into journalissuearticles values(x);=\int _{0}^{x}finsert ignore into journalissuearticles values(u);du $$ and $$ \sigma _{p}insert ignore into journalissuearticles values(sinsert ignore into journalissuearticles values(x););=\frac{1}{pinsert ignore into journalissuearticles values(x);}\int_{0}^{x}sinsert ignore into journalissuearticles values(u);dpinsert ignore into journalissuearticles values(u);,\,\,\,\,x>0 $$ provided that $pinsert ignore into journalissuearticles values(x);>0$. A function $sinsert ignore into journalissuearticles values(x);$ is said to be summable to $l$ by the weighted mean method determined by the function $pinsert ignore into journalissuearticles values(x);$, in short, $insert ignore into journalissuearticles values(\overline{N},p);$ summable to $l$, if $$ \lim_{x \to \infty}\sigma _{p}insert ignore into journalissuearticles values(sinsert ignore into journalissuearticles values(x););=l. $$ If the limit $\lim _{x \to \infty} sinsert ignore into journalissuearticles values(x);=l$ exists, then $\lim _{x \to \infty} \sigma _{p}insert ignore into journalissuearticles values(sinsert ignore into journalissuearticles values(x););=l$ 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 $finsert ignore into journalissuearticles values(x);$ 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