MMN-2136
On the Cesaro summability for functions of two variables
Ü. Totur; I. Canak;Abstract
For a continuous function $f(T,S)$ on $\mathbb{R}_{+}^{2}=[0,\infty)\times [0,\infty)$, we define its integral on $\mathbb{R}_{+}^{2}$ by
\begin{equation*}
F(T,S)=\int _{0}^{T}\int _{0}^{S} f(t,s)dt ds,
\end{equation*}
and its $(C,\alpha,\beta)$ mean by
\begin{equation*}
\sigma _{\alpha,\beta}(T,S)= \int _{0}^{T} \int _{0}^{S}\left(1-\frac{t}{T}\right)^{\alpha}\left(1-\frac{s}{S}\right)^{\beta}f(t,s)dtds,
\end{equation*}
where $\alpha >-1$, and $\beta >-1$.
We say that $\int _{0}^{\infty}\int _{0}^{\infty} f(t,s)dt ds$
is $(C, \alpha, \beta)$ integrable to $L$
if $\lim _{T,S \to \infty} \sigma _{\alpha,\beta} (T,S)=L$ exists.
We prove that if $\lim _{T,S \to \infty} \sigma _{\alpha,\beta} (T,S)=L$ exists for some $\alpha >-1$ and $\beta >-1$, then $\lim _{T,S \to \infty} \sigma _{\alpha+h,\beta +k} (T,S)=L$ exists for all $h>0$ and $k>0$.
Next, we prove that if $\int _{0}^{\infty}\int _{0}^{\infty} f(t,s)dt ds$ is $(C, 1, 1)$ integrable to $L$ and
\begin{equation*}
T\int_{0}^{S}f(T,s)ds=O(1)
\end{equation*}
and
\begin{equation*}
S\int_{0}^{T}f(t,S)ds=O(1)
\end{equation*}
then $\lim _{T,S \to \infty} F(T,S)=L$ exists.
Vol. 19 (2018), No. 2, pp. 1203-1215