\documentclass[reqno,b5paper]{amsart}
\usepackage{amsfonts}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{enumerate}
\usepackage[mathscr]{eucal}

\setcounter{MaxMatrixCols}{12}
%TCIDATA{OutputFilter=Latex.dll}
%TCIDATA{Version=5.00.0.2606}
%TCIDATA{<META NAME="SaveForMode" CONTENT="1">}
%TCIDATA{BibliographyScheme=Manual}
%TCIDATA{LastRevised=Friday, August 02, 2013 13:31:16}
%TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}

\setlength{\textwidth}{142mm} \setlength{\textheight}{183mm}
\newtheorem{Definition}{Definition}[section]
\newtheorem{Theorem}{Theorem}[section]
\newtheorem{Lemma}{Lemma}[section]
\newtheorem{Proposition}{Proposition}[section]
\newtheorem{Corollary}{Corollary}[section]
\newtheorem{Example}{Example}[section]
\newtheorem{Remark}{Remark}[section]
\newtheorem{Note}{Note}[section]
\newtheorem{Proof}{Proof}
{}
\newcommand{\lb}{\overline{\mathcal{L}}}
\newcommand{\rb}{\overline{\mathcal{R}}}
\newcommand{\hb}{\overline{\mathcal{H}}}
\newcommand{\jb}{\overline{\mathcal{J}}}
\newcommand{\lc}{\mathcal{L}}
\newcommand{\rc}{\mathcal{R}}
\newcommand{\hc}{\mathcal{H}}
\newcommand{\jc}{\mathcal{J}}
\newcommand{\cb}{\mathcal{B}}
\newcommand{\limxmn}{lim_{m \rightarrow \infty \atop n \rightarrow \infty} x_{mn}}

%\input{tcilatex}

\begin{document}
\title[Korovkin type approximation on an infinite interval]{Korovkin type approximation on an infinite interval via generalized matrix summability method using ideal}
\author[S. Dutta, R. Ghosh]{Sudipta Dutta$^{*}$, Rima Ghosh$^{**}$}
%\address{\llap{*}Department of Mathematics\\
%Jadavpur University\\
%Jadavpur, Kol-700032\
%West Bengal\\
%India}
\address{\llap{**}
Assistant Professor\\
Department of Mathematics\\
Govt. General Degree College At Manbazar-II\\
Purulia\\
Pin-723131\\
West Bengal\\
India}
\email{drsudipta.prof@gmail.com}
\address{\llap{**}Assistant Teacher\\
Garfa D.N.M. Girls High School\\
Kolkata-700075\\
West Bengal\\
India}
\email{rimag944@gmail.com}
\email{}
\subjclass[2010]{Primary:40A35, Secondary:47B38,41A25,41A36}
\date{}
\maketitle

\begin{abstract}
Following the notion of $A^\mathcal{I}$-summability method for
real sequences \cite{espdsd2} we establish a Korovkin type approximation theorem for positive linear operators on $UC_{*}[0,\infty)$, the Banach space of all real valued uniform continuous functions on $ [0,\infty)$ with the property that $\displaystyle{\lim_{x\rightarrow \infty}f(x)}$ exists finitely for any $f\in UC_{*}[0,\infty)$.  In the last section, we extend the Korovkin type approximation theorem for positive linear operators on $UC_{*}\left([0,\infty)\times[0,\infty)\right)$. We then construct an example which shows that our new result is stronger than its classical version.
\end{abstract}

\setcounter {page}{1}


\textbf{\ Key words:}  Positive linear
operator, Korovkin type approximation theorem, Ideal, $A^\mathcal {I}$-summable, $A^{\mathcal {I}}_2$-summable.\newline

%\textbf {AMS subject classification (2010) :} Primary:40A35, Secondary:47B38,41A25,41A36.  \\

\section{\textbf{Introduction and Background}}

% \small\\\\
Throughout the paper $\mathbb{N}$ will denote the set of all positive
integers. For a sequence $\{L_n\}_{n \in \mathbb{N}}$ of positive linear operators on $C(X)$%
, the space of real valued continuous functions on a compact subset $X$ of
real numbers, Korovkin \cite{korov} first established the necessary and
sufficient conditions for the uniform convergence of $\{L_n(f)\}_{n\in
\mathbb{N}}$ to a function $f$ by using the test functions $%
e_1=1,~e_2=x,~e_3=x^2$ \cite{alipran}. The study of the Korovkin type
approximation theory has a long history and is a well-established area of
research. In recent years,
using the concept of uniform statistical convergence various statistical
approximation results have been proved (\cite{duman2}). Erku\c{s} and
Duman \cite{erkus} studied a Korovkin type approximation theorem via $A$%
-statistical convergence in the space $H_w(I^2)$ where $I^2=[0,
\infty)\times [0, \infty)$ which was extended for double sequences of
positive linear operators of two variables in $A$-statistical sense by
Demirci and Dirik in \cite{demirci2,demirci3}. Further it was extended for double sequences of
positive linear operators of two variables in $A^\mathcal {I}_2$-statistical sense and in the sense of $A^\mathcal {I}_2$-summability method, by Dutta et. al. \cite{sdpd,sdsevdapd}.



Our primary interest, in this paper, is to obtain general Korovkin type approximation theorem for positive linear operators on the space $ UC_{*}(D)$, the Banach space of all real valued uniform continuous functions on $ D := [0,\infty) $ with the property that $ \lim_{x\rightarrow \infty}~f(x) $ exists and finite, endowed with the supremum norm $ \Vert f \Vert_* = \sup_{x \in D} \mid f(x) \mid  $ for $ f \in UC_{*}(D) $, using the concept of $ A^{^{\mathcal{I}}}$-summability method for real sequences and test functions $ 1,~e^{-x} ,~ e^{-y}$. In the last section, we extend the Korovkin-type approximation theorem for double sequence of positive linear operators on $UC_{*}\left([0,\infty)\times[0,\infty)\right)$. We also construct an example which shows that our new result is stronger than its classical version.

The concept of convergence of a sequence of real numbers was extended to
statistical convergence by Fast \cite{fast}. Further investigations started
in this area after the pioneering works of \v{S}al\'{a}t \cite{salat} and
Fridy \cite{fridy}. The notion of $\mathcal{I}$-convergence of real
sequences was introduced by Kostyrko et. al. \cite{kos} as a generalization
of statistical convergence using the notion of ideals. On
the other hand statistical convergence was generalized to $A$-statistical
convergence by Kolk (\cite{kolk1}). Later a lot of works have been
done on matrix summability and $A$-statistical convergence (see \cite{Belenmursaleen,connor1,demirci1,Edely,kolk1,Mursaleenalotaibi1,espd}).
In particular, in \cite{espdsd1,espdsd2} the very general notion of $A^%
\mathcal{I}$-statistical convergence and $A^%
\mathcal{I}$-summability was introduced and studied.

Recall that a real double sequence $\{x_{mn}\}_{m,n\in \mathbb{N}}$ is said to be
convergent to $L$ in Pringsheim's sense if for every $\varepsilon >0$ there
exists $N(\varepsilon )\in \mathbb{N}$ such that $|x_{mn}-L|<\varepsilon $
for all $m,n>N(\varepsilon )$ and denoted by $\displaystyle%
\lim_{m,n}~x_{mn}=L$. A double sequence is called
bounded if there exists a positive number $M$ such that $|x_{mn}|\leq M$ for
all $(m,n)\in \mathbb{N}\times \mathbb{N}$. A real double sequence $%
\{x_{mn}\}_{m,n\in \mathbb{N}}$ is statistically convergent to $L$ if for
every $\varepsilon >0$,
\begin{center}
$\displaystyle\lim_{j,k}\frac{\left\vert \{m\leq j,n\leq k:|x_{mn}-L|\geq \varepsilon \}\right\vert }{jk}=0$\cite{mursaleen}.
\end{center}
Recall that a family $\mathcal{I}\subset 2^Y$ of subsets of a nonempty set $Y$ is said to be an ideal in $Y$ if $(i)A,B \in \mathcal{I}$ implies $A\cup B
\in \mathcal{I};(ii)A\in \mathcal{I},B\subset A$ implies $B\in \mathcal{I}$,
while an admissible ideal $\mathcal{I}$ of $Y$ further satisfies $\{x\}\in
\mathcal{I}$ for each $x\in Y$. If $\mathcal{I}$ is a non-trivial proper
ideal in $Y$ (i.e. $Y \notin \mathcal{I},\mathcal{I}\neq \{\emptyset\}$)
then the family of sets $F(\mathcal{I})= \{M\subset Y:$ there exists $A \in
\mathcal{I}: M=Y\setminus A\}$ is a filter in $Y$. It is called the filter
associated with the ideal $\mathcal{I}$. A non-trivial ideal $\mathcal{I}$
of $\mathbb{N}\times \mathbb{N}$ is called strongly admissible if $%
\{i\}\times \mathbb{N}$ and $\mathbb{N}\times \{i\}$ belong to $\mathcal{I}$
for each $i \in \mathbb{N}$. It is evident that a strongly admissible ideal
is admissible also. Let $\mathcal{I}_0=\{A \subset \mathbb{N}\times \mathbb{N%
}:\mbox{there is}~ m(A)\in \mathbb{N} ~\mbox{such that}~i,j \geq m(A)\Longrightarrow
(i,j)\notin A\}$. Then $\mathcal{I}_0$ is a non-trivial strongly admissible
ideal \cite{pdpkwwpm}.


\vspace{8mm}
\section{\textbf{A Korovkin type approximation for a sequence of positive linear operators of single variable}}
Throughout this section $\mathcal{I}$ denotes the non-trivial admissible ideal on $\mathbb{N}.$ If $\left\{ x_{k}\right\} _{k\in
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
}$ is a sequence of real numbers and $A=(a_{nk})_{n,k=1}^{\infty }$ is an
infinite matrix, then $Ax$ is the sequence whose n-th term is given by
\begin{equation*}
A_{n}(x)=\sum\limits_{k=1}^{\infty }a_{nk}x_{k}.
\end{equation*}%
A matrix $A$ is called
regular if $A\in \left( c,c\right) $ and $\underset{k\rightarrow \infty }{\lim }A_{k}\left( x\right) =\underset{k\rightarrow \infty }{\lim
}x_{k}$ for all $x=\{x_{k}\}_{k\in\mathbb{N}
}\in c$ when $c$, as usual, stands for the set of all convergent
sequences. It is well-known that the necessary and sufficient
conditions for $A$ to be regular are
\begin{eqnarray*}
 &R1)&~||A||=\displaystyle{\sup_{n}{\sum_{k}}|a_{nk}|}<\infty;\\
 &R2)&~\displaystyle{\lim_{n}a_{nk}}=0,~\mbox{for each}~k;\\
 &R3)&~\displaystyle{\lim_{n}{\sum_{k}}a_{nk}}=1.
\end{eqnarray*}



We first recall the following definition

\begin{Definition}[\cite{espdsd1}]
Let $A=(a_{nk})$ be a non-negative regular summability matrix. Then a
real sequence $x=\{x_{k}\}_{k \in \mathbb{N}}$ is said to be $A^%
\mathcal{I}$-summable to a number $L$ if for every $\varepsilon>0$,
$\left\{n \in \mathbb{N}: |A_{n}(x)-L|\geq\varepsilon\right\} \in
\mathcal{I}$ where $A_{n}(x)=\displaystyle\sum_{k=1}^\infty a_{nk}x_k$.

Thus $x=\{x_{k}\}_{k \in \mathbb{N}}$ is $A^\mathcal{I}$-summable to a
number $L$ if and only if $\{A_{n}(x)\}_{n\in\mathbb{N}}$ is $\mathcal{I}$-convergent to $L$.
In this case, we write $\mathcal{I}\mbox{-}\displaystyle{\lim_{n}}%
\displaystyle{\sum_{k\in \mathbb{N}}}a_{nk}x_{k}=L$.
\end{Definition}

It should be noted that for $\mathcal{I}=\mathcal{I}_{d}$, the set of
all subsets of $\mathbb{N}$ with natural density zero, $A^\mathcal{I}$-summability reduces to statistical $A$-summability \cite{Edely}.

We now establish a Korovkin type approximation theorem for positive linear operators on $UC_{*}[0,\infty)$, the Banach space of all real valued uniform continuous functions on $ [0,\infty)$ with the property that $\displaystyle{\lim_{x\rightarrow \infty}f(x)}$ exists finitely for any $f\in UC_{*}[0,\infty)$.
If $L$ be a positive linear operator then $L(f)\geq 0$ for any positive function $f.$ Also we denote the value of $L(f)$ at a point $x\in [0,\infty)$ by $L(f;x).$
\begin{Theorem}
Let $\{L_{n}\}$ be a sequence of positive linear operators from $  UC_{*}[0,\infty)$ into itself and let, $ A=(a_{jn}) $ be a non-negative regular summability matrix then for all $ f \in UC_{*}[0,\infty) $
$$ \mathcal{I}\mbox{-}\displaystyle\lim_{n} \Vert\displaystyle\sum_{k=1}^\infty a_{nk} L_{k}(f) - f \Vert_{*} = 0 $$
if and only if the following statements hold
$$ \mathcal{I}\mbox{-}\displaystyle\lim_{n} \Vert \displaystyle\sum_{k=1}^\infty a_{nk} L_{k}( e^{-pt} ) - e^{-px} \Vert_{*}=0 , p=0,1,2. $$

\end{Theorem}

 \begin{proof}
 Since the necessity is clear, then it is enough to proof sufficiency.
 Our objective is to show that for given $ \varepsilon > 0 $ there exist constants $ C_{0} $ , $ C_{1} $ , $ C_{2} $
(depending on $ \varepsilon > 0$) such that
\begin{eqnarray*}
\Vert\displaystyle\sum_{k=1}^\infty a_{nk} L_{k}(f) - f \Vert_{*} \leq \varepsilon +  C_{2} \Vert\displaystyle\sum_{k=1}^\infty a_{nk} L_{k}( e^{-2t} ) - e^{-2x} \Vert_{*} + C_{1} \Vert\displaystyle\sum_{k=1}^\infty a_{nk} L_{k}( e^{-t} ) - e^{-x} \Vert_{*}\\
 + C_{0} \Vert \displaystyle\sum_{k=1}^\infty a_{nk}L_{k}( 1 ) - 1 \Vert_{*}.
 \end{eqnarray*}
If this is done then our hypothesis implies that for any $ \varepsilon > 0 $ ,  $$ \lbrace n \in\mathbb{N} : \Vert\displaystyle\sum_{k=1}^\infty a_{nk}L_{k}(f)-f\Vert \geq \varepsilon \rbrace \in \mathcal{I}. $$
Let $ f \in UC_{*}[0,\infty) $ then $ \exists $ a constant $ M $ such that $ \mid f(x) \mid \leq M $
for each $ x \in [0,\infty) $. Let $ \varepsilon $ be an arbitrary positive number. By hypothesis we may find $ \delta := \delta (\varepsilon) > 0 $ such that for every $ t,x \in [0,\infty)$, $\mid e^{-t} - e^{-x} \mid < \delta $ implies $ \mid f(t)- f(x) \mid < \varepsilon. $
We can write $ \mid f(t)- f(x) \mid < 2M ~\forall~ t,x \in [0,\infty).$
Also if $ \mid e^{-t} - e^{-x} \mid \geq \delta $ then
$$ \mid f(t)- f(x) \mid < \frac{2M}{\delta^{2}} ( e^{-t} - e^{-x} )^{2}. $$
Then for all $ t,x \in [0,\infty),$
$$ \mid f(t)- f(x) \mid < \varepsilon + \frac{2M}{\delta^{2}} ( e^{-t} - e^{-x} )^{2}. $$
Then for  $ n \in\mathbb{N},$ using the linearity and the positivity of the operators $L_{n}$,
\begin{eqnarray*}
\mid\displaystyle\sum_{k=1}^\infty a_{nk}L_{k}(f(t);x)-f(x)\mid
&\leq&\displaystyle\sum_{k=1}^\infty a_{nk}L_{k}(\mid f(t)-f(x)\mid ;x)\\
&&+ \mid f(x) \mid \mid \displaystyle\sum_{k=1}^\infty a_{nk}L_{k}(1;x)-1 \mid\\
&\leq&\displaystyle\sum_{k=1}^\infty a_{nk}L_{k}(\varepsilon + \frac{2M}{\delta^{2}} (e^{-t}-e^{-x})^{2};x)\\
&&+ \mid f(x) \mid \mid \displaystyle\sum_{k=1}^\infty a_{nk}L_{k}(1;x)-1\mid \\
&\leq& \varepsilon + (\varepsilon + M )\mid \displaystyle\sum_{k=1}^\infty a_{nk}L_{k}(1;x)-1\mid\\
&&+ \frac{2M}{\delta^{2}}\displaystyle\sum_{k=1}^\infty a_{nk} L_{k}((e^{-t}-e^{-x} )^{2};x)\\
&\leq & \varepsilon+(\varepsilon + M )\mid \displaystyle\sum_{k=1}^\infty a_{nk}L_{k}(1;x)-1 \mid\\
&&+ \frac{2M}{\delta^{2}}\mid e^{-2x} \mid \mid \displaystyle\sum_{k=1}^\infty a_{nk}L_{k}(1;x)-1\mid\\
&&+\frac{2M}{\delta^{2}}\mid \displaystyle\sum_{k=1}^\infty a_{nk}L_{k}(e^{-2t};x)-e^{-2x}\mid\\
&&+ \frac{4M}{\delta^{2}}\mid e^{-x} \mid \mid \displaystyle\sum_{k=1}^\infty a_{nk}L_{k}(e^{-t};x)-e^{-x}\mid
\end{eqnarray*}
where $ \mid e^{-kt}\mid\leq 1 ~\forall~ t \in [0,\infty) ~\text{and} ~k \in\mathbb{N}. $\\
Then taking supremum over $ x \in [0,\infty) $ we have
\begin{eqnarray*}
 \Vert \displaystyle\sum_{k=1}^\infty a_{nk}L_{k}(f)-f\Vert_{*} \leq \varepsilon + K \lbrace \Vert \displaystyle\sum_{k=1}^\infty a_{nk}L_{k}(1)-1 \Vert_{*}+ \Vert \displaystyle\sum_{k=1}^\infty a_{nk}L_{k}(e^{-t})-e^{-x}\Vert_{*}\\ + \Vert \displaystyle\sum_{k=1}^\infty a_{nk}L_{k}(e^{-2t})-e^{-2x}\Vert_{*}\rbrace
\end{eqnarray*}
where $ K= \max \lbrace \varepsilon+M+\frac{2M}{\delta^{2}} ,\frac{2M}{\delta^{2}} , \frac{4M}{\delta^{2}}\rbrace $.
For a given $ r > 0 $ choose $ \varepsilon > 0 $ such that $ \varepsilon < r $
let us define the following sets
$$ D = \lbrace n \in\mathbb{N} : \Vert \displaystyle\sum_{k=1}^\infty a_{nk}L_{k}(f) - f \Vert_{*} \geq r \rbrace $$
$$ D_{1}= \lbrace n \in\mathbb{N} : \Vert \displaystyle\sum_{k=1}^\infty a_{nk}L_{k}(1) - 1 \Vert_{*} \geq \frac{r - \varepsilon}{3K} \rbrace $$
$$ D_{2}= \lbrace n \in\mathbb{N} : \Vert \displaystyle\sum_{k=1}^\infty a_{nk}L_{k}(e^{-t}) - e^{-x} \Vert_{*} \geq \frac{r - \varepsilon}{3K} \rbrace $$
$$ D_{3}= \lbrace n \in\mathbb{N} : \Vert \displaystyle\sum_{k=1}^\infty a_{nk}L_{k}(e^{-2t}) - e^{-2x} \Vert_{*} \geq \frac{r - \varepsilon}{3K} \rbrace. $$
It follows that $ D \subset D_{1} \cup D_{2} \cup D_{3}.$
Since from hypotheses $D_1, ~D_2,~D_3$  are belong to $\mathcal {I}$ so $D\in\mathcal{I}$ i.e.  $$ \lbrace n \in\mathbb{N} : \Vert\displaystyle\sum_{k=1}^\infty a_{nk}L_{k}(f)-f\Vert \geq \varepsilon \rbrace \in \mathcal{I} $$
and this completes the proof.
 \end{proof}

 %\begin{Remark}
 %We now exhibit a sequence of positive linear operator $\{L_n\}$ s.t.
 %$A^{\mathcal{I}}-st-\lim_{n}  \Vert L_{n}(f)-f \Vert _{*}  = 0$ \\ but \\
  %   $st_{A}-\lim_{n} \Vert L_{n}(f)-f \Vert _{*} \neq 0.$  \\

   %  We consider the following  Baskakov operators $B_n:UC_{*}[0,\infty)\rightarrow UC_{*}[0,\infty)$ defined by
    % $$B_{n}f(x)= \sum_{k=0}^{\infty}  \binom{n-1+k}{k} x^{k}(1+x)^{-n-k}f({\frac{k}{n}}) $$
%Thus
 %$$ B_{n}(e^{-u},x)= \sum_{k=0}^{\infty} x^{k}e^{\frac{-k}{n}} \binom{n-1+k}{k} (1+x)^{-n-k} = (1+x-xe^{\frac{-1}{n}} )^{-n} $$
 %$$ B_{n}(1,x)=1 $$
 %$$ B_{n}(e^{-2u},x)= (1+x-xe^{\frac{-2}{n}} )^{-n}$$
 %where $ x \in [0,\infty) $\\
 %Let,
%\begin{equation*}
%\alpha_{n}= \begin{cases}
 %            1, & \text{for} ~n~ \text{even} \\
  %           0, & \text{otherwise}
   %          \end{cases}
%\end{equation*}

%Let us define $ L_{n}(f,x)= (1+\alpha_{n}) B_{n}(f,x)$ for any $ f \in UC_{*}[0,\infty).$
%By previous theorem,
%$$ A^{\mathcal{I}}-st-lim\Vert L_{n}(f)-f\Vert_{*}=0 $$ where $ A $ is a regular summability matrix with $ \max{a_{jn}}=0 $
%\\ But, as $ st_{A}-lim~ \alpha_{n} \neq 0 $
%\\so, $ st_{A}-lim~ \Vert L_{n}(f)-f \Vert_{*}\neq 0 $
 %\end{Remark}
\vspace{8mm}
\section{\textbf{A Korovkin type approximation for a sequence of positive linear operators of two variables}}
Throughout this section $\mathcal{I}$ denotes the non-trivial strongly admissible ideal on $\mathbb{N}\times\mathbb{N}.$ Let $A=(a_{jkmn})$ be a four dimensional summability matrix. For a given double sequence $\{x_{mn}\}_{{m,n} \in \mathbb{N}}$, the $A$-transform of $x$%
, denoted by $Ax:=((Ax)_{jk})$, is given by
\begin{center}
$(Ax)_{jk}=\displaystyle{\sum_{(m,n)\in \mathbb{N}^2}}a_{jkmn}x_{mn}$
\end{center}
provided the double series converges in Pringsheim sense for every $(j,k)\in
\mathbb{N}^2$. In 1926, Robison \cite{Robison} presented a four dimensional
analog of the regularity by considering an additional assumption of
boundedness. This assumption was made because a convergent double sequence
is not necessarily bounded.

Recall that a four dimensional matrix $A=(a_{jkmn})$ is said to be
RH-regular if it maps every bounded convergent double sequence into a
convergent double sequence with the same limit. The Robison-Hamilton
conditions state that a four dimensional matrix $A=(a_{jkmn})$ is RH-regular
if and only if
\begin{eqnarray*}
&(i)& \displaystyle{\lim_{j,k}}~a_{jkmn}=0~\mbox{for each}~ (m,n)\in \mathbb{%
N}^2, \\
&(ii)& \displaystyle{\lim_{j,k}}\displaystyle{\sum_{(m,n) \in \mathbb{N}^2}}%
a_{jkmn}=1, \\
&(iii)&\displaystyle{\lim_{j,k}}\displaystyle{\sum_{m \in \mathbb{N}}}%
|a_{jkmn}|=0~\mbox{for each}~n \in \mathbb{N}, \\
&(iv)&\displaystyle{\lim_{j,k}}\displaystyle{\sum_{n \in \mathbb{N}}}%
|a_{jkmn}|=0~\mbox{for each}~m \in \mathbb{N}, \\
&(v)&\displaystyle{\sum_{(m,n) \in \mathbb{N}^2}}|a_{jkmn}| ~\mbox{is
convergent for each}~(j,k)\in \mathbb{N}^2, \\
&(vi)& ~\mbox{there exist finite positive integers}~ M_0 ~\mbox{and}~ N_0~ %
\mbox{such that} ~\displaystyle{\sum_{m,n > N_0}}|a_{jkmn}|<M_0
\end{eqnarray*}
holds for every $(j,k)\in \mathbb{N}^2$.

Let $A=(a_{jkmn})$ be a nonnegative RH-regular summability matrix and let $%
K\subset \mathbb{N}^2$. Then the $A$-density of $K$ is given by

\begin{center}
$\delta^{(2)}_A\{K\}=\displaystyle{\lim_{j,k}}\displaystyle{\sum_{(m,n) \in
K}}a_{jkmn}.$
\end{center}

Recall the following definition
\begin{Definition}[\cite{sdsevdapd}]
Let $A=(a_{jkmn})$ be a nonnegative RH-regular summability matrix. Then a
real double sequence $x=\{x_{mn}\}_{m,n \in \mathbb{N}}$ is said to be $A^%
\mathcal{I}_2$-summable to a number $L$ if for every $\varepsilon>0$,
$\left\{(j,k) \in \mathbb{N}^2: |(Ax)_{j,k}-L|\geq\varepsilon\right\} \in
\mathcal{I}$.

Thus $x=\{x_{mn}\}_{m,n \in \mathbb{N}}$ is $A^\mathcal{I}_2$-summable to a
number $L$ if and only if $(Ax)_{j,k}$ is $\mathcal{I}$-convergent to $L$.
In this case, we write $\mathcal{I}_2\mbox{-}\displaystyle{\lim_{j,k}}%
\displaystyle{\sum_{(m,n)\in \mathbb{N}^2}}a_{jkmn}x_{mn}=L$.
\end{Definition}

It should be noted that, if we take $\mathcal{I}=\mathcal{I}_d$, the set of
all subsets of $\mathbb{N}\times\mathbb{N}$ with natural density zero, then $%
A^\mathcal{I}_2$-summability reduces to the notion of statistical $A$%
-summability for double sequence\cite{Belenmursaleen}.


We now establish the Korovkin type approximation theorem for a double sequence of positive linear operators on $UC_{*}\left([0,\infty)\times[0,\infty)\right)$, the Banach space of all real valued uniform continuous functions defined on $ [0,\infty)\times [0,\infty)$ with the property that $\displaystyle{\lim_{(x,y)\rightarrow (\infty,\infty)}f(x)}$ exists finitely for any $f \in UC_{*}\left([0,\infty)\times[0,\infty)\right)$ endowed with the supremum norm $||f||_{*}=\displaystyle\sup_{x,y\in [0,\infty)} |f(x,y)|$, in $A^\mathcal{I}_2$-summability method. If $L$ be a positive linear operator then $L(f)\geq 0$ for any positive function $f.$ Also we denote the value of $L(f)$ at a point $(x,y)\in [0,\infty)\times [0,\infty)$ by $L(f;x,y).$

\begin{Theorem}
Assume $\mathcal{K}:=[0,\infty)\times[0,\infty)$ and let $\{L_{mn}\}_{{m,n}\in\mathbb{ N}} $ be a sequence of positive linear operators on $UC_{*}\left(\mathcal{K}\right)$, the Banach space of all real valued uniform continuous functions defined on $ \mathcal{K}$ with the property that $\displaystyle{\lim_{(x,y)\rightarrow (\infty,\infty)}f(x)}$ exists finitely for any $f \in UC_{*}\left(\mathcal{K}\right)$ and let
$A=(a_{jkmn})$ be a non-negative RH-regular summability matrix. Then for any $f \in UC_{*}\left(\mathcal{K}\right)$,
\begin{eqnarray*}
\mathcal{I}_2\mbox{-}\displaystyle{\lim_{j,k}}~\Vert\displaystyle\sum_{(m,n)\in\mathbb{N}^2} a_{jkmn}L_{mn}(f)-f\Vert_*=0
\end{eqnarray*}
is satisfied if the following hold
\begin{eqnarray}
\mathcal{I}_2\mbox{-}\displaystyle{\lim_{j,k}}\Vert\displaystyle\sum_{(m,n)\in\mathbb{N}^2}a_{jkmn}L_{mn}(f_{i})-f_i\Vert_*=0,~i=0,1,2,3
\end{eqnarray}
where
$f_0=1,~f_1=e^{-x},~f_2=e^{-y},f_3=e^{-2x}+e^{-2y}.$
\end{Theorem}

\begin{proof}
Assume that $(1)$ holds. Let $f \in UC_{*}\left(\mathcal{K}\right)$. Our objective is to show that for given $\varepsilon >0$ there exist constants
$C_0,~C_1,~C_2,~C_3$ (depending on $\varepsilon >0$) such that
\begin{eqnarray*}
||\displaystyle\sum_{(m,n)\in\mathbb{N}^2} a_{jkmn}L_{mn}(f)-f||_{*}\leq \varepsilon +\displaystyle{\sum_{i=0}^3}C_i||\displaystyle\sum_{(m,n)\in\mathbb{N}^2} a_{jkmn}L_{mn}(f_{i})-f_{i}||_{*}.
\end{eqnarray*}
If this is done then our hypothesis implies that for any  $ \varepsilon > 0 $ ,  $$ \lbrace (j,k)\in\mathbb{N}^2 : \Vert\displaystyle\sum_{(m,n)\in\mathbb{N}^2} a_{jkmn}L_{mn}(f)-f\Vert_{*} \geq \varepsilon \rbrace \in \mathcal{I}. $$
To this end, start by observing that for each $(u,v) \in \mathcal{K}$ the function $0\leq g_{uv} \in UC_{*}\left(\mathcal{K}\right)$  defined by
$$g_{uv}(s,t)={(e^{-s}-e^{-u})^2+((e^{-t}-(e^{-v})^2}$$ satisfies
$g_{uv}=(e^{-x})^2+(e^{-y})^2-2e^{-u}e^{-x}-2e^{-v}e^{-y}+(e^{-u})^2+(e^{-v})^2$.
Since each $L_{mn}$ is a positive operator, $L_{mn}g_{uv}$ is a positive function. In particular, we have for each $(u,v)\in \mathcal{K}$,\\
$0 \leq \displaystyle\sum_{(m,n)\in\mathbb{N}^2} a_{jkmn}L_{mn}(g_{uv})(u,v)$\\
$\mbox{~~}=\left[\displaystyle\sum_{(m,n)\in\mathbb{N}^2} a_{jkmn}L_{mn}\left(\left(e^{-x}\right)^2+\left(e^{-y}\right)^2-2e^{-u}e^{-x}-2e^{-v}e^{-y}+\left(e^{-u}\right)^2
+\left(e^{-v}\right)^2;u,v\right)\right]$\\
$\mbox{~}=\left[\displaystyle\sum_{(m,n)\in\mathbb{N}^2} a_{jkmn}L_{mn}\left(\left(e^{-x}\right)^2+\left(e^{-y}\right)^2;u,v\right)-\left(e^{-u}\right)^2-\left(e^{-v}\right)^2\right]$\\
$\mbox{~~~~}-2e^{-u}\left[\displaystyle\sum_{(m,n)\in\mathbb{N}^2} a_{jkmn}L_{mn}\left(e^{-x};u,v\right)-e^{-u}\right]-2e^{-v}\left[\displaystyle\sum_{(m,n)\in\mathbb{N}^2} a_{jkmn}L_{mn}\left(e^{-y};u,v\right)-e^{-v}\right]$\\
$\mbox{~~~~}+\left\{\left(e^{-u}\right)^2+\left(e^{-v}\right)^2\right\}\left[\displaystyle\sum_{(m,n)\in\mathbb{N}^2} a_{jkmn}L_{mn}(f_{0})-f_0\right]$\\
$\leq ||\displaystyle\sum_{(m,n)\in\mathbb{N}^2} a_{jkmn}L_{mn} (f_{3})-f_3||_{*}+2e^{-u}||\displaystyle\sum_{(m,n)\in\mathbb{N}^2} a_{jkmn}L_{mn}(f_{1})-f_1||_{*}$\\
$\mbox{~~}+2e^{-v}||\displaystyle\sum_{(m,n)\in\mathbb{N}^2} a_{jkmn}L_{mn}(f_{2})-f_2||_{*}+\left\{\left(e^{-u}\right)^2+\left(e^{-v}\right)^2\right\}||\displaystyle\sum_{(m,n)\in\mathbb{N}^2} a_{jkmn}L_{mn}(f_{0})-f_0||_{*}.$

Let $f\in UC_{*}\left(\mathcal{K}\right)$. Then there exists a constant $M$ such that $|f(x,y)|\leq M$ for each $(x,y)\in \mathcal{K}.$ Let $\varepsilon>0$ be arbitrary.
Then by the uniform continuity of $f$ on $\mathcal{K}$ there exists a $\delta=\delta(\varepsilon) >0$ such that
if $ \mid e^{-x} - e^{-u} \mid < \delta $ and $ \mid e^{-y} - e^{-v} \mid < \delta $ then
$$ \mid f(x,y)- f(u,v) \mid < \varepsilon+\frac{2M}{\delta^{2}} \left[ \left( e^{-x} - e^{-u}\right)^2+ \left( e^{-y} - e^{-v}\right)^2\right] $$
for all $ (x,y), (u,v) \in \mathcal{K}.$\\
Since each $L_{mn}$ is positive and linear it follows that\\
%\begin{eqnarray*}
$-\varepsilon \displaystyle\sum_{(m,n)\in\mathbb{N}^2} a_{jkmn}L_{mn}(f_0)-\frac{2M}{\delta ^2}\displaystyle\sum_{(m,n)\in\mathbb{N}^2} a_{jkmn}L_{mn}(g_{uv})\\
\leq \displaystyle\sum_{(m,n)\in\mathbb{N}^2} a_{jkmn}L_{mn}(f)-f(u,v)L_{mn}(f_0)\\
\leq  \varepsilon \displaystyle\sum_{(m,n)\in\mathbb{N}^2} a_{jkmn} L_{mn}(f_0)
+\frac{2M}{\delta ^2}\displaystyle\sum_{(m,n)\in\mathbb{N}^2} a_{jkmn}L_{mn}(g_{uv}).$\\
%\end{eqnarray*}

\noindent {Therefore}\\
%\begin{eqnarray*}

$\vert\displaystyle\sum_{(m,n)\in\mathbb{N}^2} a_{jkmn}L_{mn}(f;u,v)-f(u,v)L_{mn}(f_0;u,v)\vert\\
\leq \varepsilon
+\varepsilon[\displaystyle\sum_{(m,n)\in\mathbb{N}^2} a_{jkmn}L_{mn}(f_0;u,v)-f_{0}(u,v)]
+\frac{2M}{\delta ^2}\displaystyle\sum_{(m,n)\in\mathbb{N}^2} a_{jkmn}L_{mn}(g_{uv})\\
\leq \varepsilon + \varepsilon \Vert\displaystyle\sum_{(m,n)\in\mathbb{N}^2} a_{jkmn}L_{mn}(f_{0})-f_{0}\Vert
+ \frac{2M}{\delta ^2}\displaystyle\sum_{(m,n)\in\mathbb{N}^2} a_{jkmn}L_{mn}(g_{uv}).$\\
%\end{eqnarray*}

\noindent {In particular, note that}\\

%\begin{eqnarray*}
$\vert\displaystyle\sum_{(m,n)\in\mathbb{N}^2} a_{jkmn}L_{mn}(f;u,v)-f(u,v)\vert \\
 \leq \vert\displaystyle\sum_{(m,n)\in\mathbb{N}^2} a_{jkmn}L_{mn}(f;u,v)-f(u,v)\displaystyle\sum_{(m,n)\in\mathbb{N}^2} a_{jkmn}L_{mn}(f_{0};u,v)\vert \\
\mbox{~~~~}+\vert\displaystyle\sum_{(m,n)\in\mathbb{N}^2} a_{jkmn}f(u,v)\vert \vert\displaystyle\sum_{(m,n)\in\mathbb{N}^2} a_{jkmn}L_{mn}(f_{0};u,v)-f_{0}(u,v)\vert \\
\leq  \varepsilon + (M+\varepsilon)||\displaystyle\sum_{(m,n)\in\mathbb{N}^2} a_{jkmn}L_{mn}(f_{0})-f_{0}||_{*}
+ \frac{2M}{\delta ^2}\displaystyle\sum_{(m,n)\in\mathbb{N}^2} a_{jkmn}L_{mn}(g_{uv})$\\
%\end{eqnarray*}
which implies
\begin{eqnarray*}
\Vert\displaystyle\sum_{(m,n)\in\mathbb{N}^2} a_{jkmn}L_{mn}(f)-f\Vert_{*}
 &\leq& \varepsilon + C_3\Vert\displaystyle\sum_{(m,n)\in\mathbb{N}^2} a_{jkmn}L_{mn}(f_3)-f_3\Vert_{*}\\
&&+ C_2\Vert\displaystyle\sum_{(m,n)\in\mathbb{N}^2} a_{jkmn}L_{mn}(f_2)-f_2\Vert_{*}\\
&&+C_1\Vert\displaystyle\sum_{(m,n)\in\mathbb{N}^2} a_{jkmn}L_{mn}(f_1)-f_1\Vert_{*}\\
&&+C_0\Vert\displaystyle\sum_{(m,n)\in\mathbb{N}^2} a_{jkmn}L_{mn}(f_0)-f_0\Vert_{*}
\end{eqnarray*}
where there exist such $A$ and $B$ such that
$ C_0= \left[\frac{2M}{\delta ^2}\{(e^{-A})^2+(e^{-B})^2\}+M+\varepsilon\right],~C_1=\frac{4M}{\delta ^2}e^{-A},~C_2=\frac{4M}{\delta ^2}e^{-B}~\mbox{and} ~C_3=\frac{2M}{\delta ^2}$.
i.e.
\begin{center}
 $||\displaystyle\sum_{(m,n)\in\mathbb{N}^2} a_{jkmn}L_{mn}(f)-f||_{*} \leq \varepsilon + C \displaystyle{\sum_{i=0}^3}||\displaystyle\sum_{(m,n)\in\mathbb{N}^2} a_{jkmn}L_{mn}(f_i)-f_i||_{*},~i=0,1,2,3$
\end{center}
where $C=\max \{C_0,~C_1,~C_2,~C_3\} $.

For a given $\gamma > 0$, choose $\varepsilon>0 $ such that $\varepsilon < \gamma$. Now let
\begin{center}
$U=\{(j,k)\in\mathbb{N}^2: ||\displaystyle\sum_{(m,n)\in\mathbb{N}^2} a_{jkmn}L_{mn}(f)-f||_{*}\geq \gamma\}$
\end{center}
and
\begin{center}
$U_i= \{(j,k)\in\mathbb{N}^2: ||\displaystyle\sum_{(m,n)\in\mathbb{N}^2} a_{jkmn}L_{mn}(f_i)-f_i||_{*}\geq \frac{\gamma -\varepsilon}{4C}\},~i=0,1,2,3$.
\end{center}
It follows that $U \subset \displaystyle {\bigcup_{i=0}^3} U_i $. By hypotheses each $U_{i} \in \mathcal{I}$, $i=0,1,2,3$ and consequently $U\in\mathcal{I}$
i.e. $$ \lbrace (j,k)\in\mathbb{N}^2 : \Vert\displaystyle\sum_{(m,n)\in\mathbb{N}^2} a_{jkmn}L_{mn}(f)-f\Vert_{*} \geq \gamma \rbrace \in \mathcal{I}. $$


This completes the proof of the theorem.
\end{proof}

 \begin{Remark}
We now show that our theorem is stronger than the statistical $A$-summable
version \cite{demirci4} (and so the classical version). Let $\mathcal{I}$ be
a non-trivial strongly admissible ideal of $\mathbb{N}\times \mathbb{N}$.
Choose an infinite subset $C= \{ (p_i,q_i): i\in \mathbb{N}\}~(\mbox{where}~
p_i\neq q_i,~p_1<p_2<...,~\mbox{and}~q_1<q_2<...)$ from $\mathcal{I}
\setminus \mathcal{I}_d$ where $\mathcal{I}_d$ denotes the set of all
subsets of $\mathbb{N}\times \mathbb{N}$ with natural density zero. Let $%
\{u_{mn}\}_{m,n \in\mathbb{N} }$ be given by

\begin{equation*}
u_{mn}=
\begin{cases}
1 &\mbox{if}~ m,n \mbox { are even } \\
0 & \mbox { otherwise}.%
\end{cases}%
\end{equation*}%
\newline
Let $A=(a_{jkmn})$ be given by

\begin{equation*}
a_{jkmn}=
\begin{cases}
1 & \mbox {if  } j= p_{i}, k= q_{i}, m=2p_{i}, n=2q_{i} \mbox { for some  }
i \in \mathbb{N} \\
1 & \mbox { if } (j,k)\neq (p_{i},q_i), \mbox {  for any  } i, m= 2j+1,n=2k+1
\\
0 & \mbox { otherwise. }%
\end{cases}%
\end{equation*}
Now
\begin{equation*}
y_{j,k}=\displaystyle{\sum_{(m,n)\in \mathbb{N}^2}}a_{jkmn}u_{mn}=
\begin{cases}
1 & \mbox {if  } j= p_{i}, k= q_{i} \mbox { for some  } i \in \mathbb{N} \\
0 & \mbox { if } (j,k)\neq (p_{i},q_i), \mbox {  for any  } i\in \mathbb{N}.%
\end{cases}%
\end{equation*}
Let $\varepsilon>0$ be given. Then $\{(j,k)\in\mathbb{N}^2:|y_{j,k}-0|\geq%
\varepsilon\}=C\in \mathcal{I}$. Then the sequence $\{u_{mn}\}_{m,n \in%
\mathbb{N} }$ is $A^\mathcal{I}_2$-summable to $0$. Evidently this sequence
is not statistically $A$-summable to $0$.


Let $\mathcal{K}=[0,\infty)\times [0,\infty)$. We consider the following  Baskakov operators $B_{mn}:UC_{*}(\mathcal{K})\rightarrow UC_{*}(\mathcal{K})$ defined by
 $$B_{mn}(f;x,y)= \sum_{j=0}^\infty\sum_{k=0}^{\infty}f\left(\frac{j}{n},{\frac{k}{n}}\right) \binom{m-1+j}{j} \binom{n-1+k}{k}(1+x)^{-m-j}(1+y)^{-n-k} x^j y^k.$$
We now consider the double sequence $%
\{L_{mn}\}_{{m,n}\in\mathbb{\ N}} $ of positive linear operators defined by $%
L_{mn}(f;x,y)=(1+u_{mn})B_{mn}(f;x,y)$.\\
 Then observe that
\begin{equation*}
L_{mn}(f_0;x,y)=(1+u_{mn})f_0(x,y),
\end{equation*}
\begin{equation*}
L_{mn}(f_1;x,y)=(1+u_{mn}){\left(1+x-xe^{-\frac{1}{m}}\right)}^{-m},
\end{equation*}
\begin{equation*}
L_{mn}(f_2;x,y)=(1+u_{mn}){\left(1+y-ye^{-\frac{1}{n}}\right)}^{-n},
\end{equation*}
\begin{equation*}
L_{mn}(f_3;x,y)=(1+u_{mn})\left[{\left(1+x-xe^{-\frac{1}{m}}\right)}^{-m}+{\left(1+y-ye^{-\frac{1}{n}}\right)}^{-n}\right].
\end{equation*}
Now as $A$ is a nonnegative RH-regular summability matrix and $\{u_{mn}\}_{m,n
\in\mathbb{N} }$ is $A^\mathcal{I}_2$-summable to $0$ then for any $%
\varepsilon>0$,
$$\left\{(j,k) \in \mathbb{N}^2: ||\displaystyle{\sum_{(m,n)\in \mathbb{N}^2}}%
a_{jkmn}L_{mn}(f_i)-f_i||_{*}\geq \varepsilon \right\} \in \mathcal{I},~i=0,1,2,3.$$
Therefore by previous theorem
$$\left\{(j,k) \in \mathbb{N}^2: ||\displaystyle{\sum_{(m,n)\in \mathbb{N}^2}}%
a_{jkmn}L_{mn}(f)-f||_{*}\geq \varepsilon\right\} \in \mathcal{I}.$$
But since $\{u_{mn}\}_{m,n \in \mathbb{N}}$ is not usual convergent and
statistical $A$-summable so we can say that the classical version and
statistical $A$-summable version of the previous theorem do not work for the
operator defined above.
\end{Remark}

\noindent\textbf{Acknowledgement:} The authors are indebted to Prof. Pratulananda Das, Dept of Mathematics, Jadavpur University, who support us a lot in preparing this paper.

\begin{thebibliography}{99}
\bibitem{alipran} Aliprantis C. D. and Burkinshaw O., \textit{Principles of real
analysis}, Academic Press, New York, 1998.

%\bibitem{anas} Anastassiou G A and Duman O, Towards Intelligent Modeling:
%Statistical Approximation Theory (Intelligent System Reference Library 14
%Springer-Verlag Berlin Heidelberg) (2011)

\bibitem{Belenmursaleen} Belen C., Mursaleen M., Yildirim M., \textit{Statistical $A$%
-summability of Double Sequences and A Korovkin type approximation theorem},
Bull. Korean Math. Soc., 49(2012), no. 4, 851-861.

%\bibitem{boccu1} Boccuto A, Dimitriou X and Papanastassiou N, Some versions
%of limit and Dieudonne-type theorems with respect to filter convergence for $%
%(\ell)$-group-valued measures, Cent. Eur. J. Math. 9 (6) (2011) 1298-1311

%\bibitem{boccu2} Boccuto A, Dimitriou X and Papanastassiou N,
%Brooks-Jewett-type theorems for the pointwise ideal convergence of measures
%with values in $(\ell)$-groups, Tatra Mt. Math. Publ. 49 (2011) 17-26

%\bibitem{boccu3} Boccuto A, Dimitriou X and Papanastassiou N, Basic matrix
%theorems for $\mathcal{I}$-convergence in $(\ell)$-groups, Math. Slovaca
%62 (5) (2012) 885-908

%\bibitem{boja} Bojanic R and Khan M K, Summability of Hermite-Fej\'er
%interpolation for functions of bounded variation, J. Natur. Sci. Math. 32
%(1) (1992) 5-10

\bibitem{connor1} Connor J., \textit{The Statistical and strong $p$-Cesaro
convergence of sequences}, Analysis, 8(1988), 47-63.

%\bibitem{connor2} Connor J, On strong matrix summability with respect to a
%modulus and statistical convergence, Canad. Math. Bull. 32 (1989) 194-198

\bibitem{pdpkwwpm} Das P., Kostyrko P., Wilczy\'{n}ski W., Malik P., \textit{$%
\mathcal{I}$ and $\mathcal{I^*}$-convergence of double sequences}, Math.
Slovaca, 58(2008), no. 5, 605-620.

%\bibitem{pdessg} Das P, Savas E and Ghosal S K, On generalizations of
%certain summability methods using ideals, Appl. Math. Lett. 24 (2011) 1509 -
%1514.

\bibitem{demirci1} Demirci K., \textit{Strong $A$-summabilty and $A$-statistical
convergence}, Indian J. Pure Appl. Math., 27(1996), 589-593.

\bibitem{demirci2} Demirci K., Dirik F., \textit{A Korovkin type approximation
theorem for double sequences of positive linear operators of two variables
in $A$-statistical sense}, Bull. Korean Math. Soc., 47(2010), no. 4, 825-837.

\bibitem{demirci4} Demirci K., Karaku\c{s} S., \textit{$A$-statistical Korovkin type approximation
theorem for functions of two variables on an infinite interval}, Acta Math. Univ. Comenian.(N.S.), Vol. LXXXI(2012), no. 2, 151-157.


\bibitem{demirci3} Demirci K., Karaku\c{s} S., \textit{Korovkin type approximation
theorem for double sequences of positive linear operators via statistical $A$%
-summability}, Results Math., 63(2013), 1-13.



%\bibitem{duman1} Duman O, Erku\c{s} E and Gupta V, Statistical rates on the
%multivariate approximation theory, Math. Comp. Model. 44 (9-10) (2006)
%763-770

\bibitem{duman2} Duman O., Khan M. K., Orhan C., \textit{$A$-statistical convergence
of approximating operators}, Math. Inequal. Appl., 6 (2003), no. 4, 689-699.

\bibitem{sdpd} Dutta S., Das P., \textit{Korovkin type approximation theorem in $A^{\mathcal{I}}_2$-statistical sense},
 Mat. Vesnik, 67(2015), no. 4, 288-300.

\bibitem{sdsevdapd}  Dutta S.,  Akda\v{g} S., Das P., \textit{Korovkin type approximation theorem via $A^{\mathcal{I}}_2$-summability method},
 Filomat 30(2016), no. 10, 2663-2672.


\bibitem{erkus} Erku\c{s} E. and Duman O., \textit{$A$-statistical extension of the
Korovkin type approximation theorem}, Proc. Indian Acad. Sci. Math. Sci., 115(2005), no. 4, 499-508.

\bibitem{Edely} Edely O. H. H., Mursaleen M., \textit{On statistical $A$-summability},
Math. Comp. Model., 49(2009), no. 8, 672-680

\bibitem{fast} Fast H., \textit{Sur la convergence Statistique}, Colloq. Math., 2(1951), 241-244.

\bibitem{fridy} Fridy J. A., \textit{On Statistical convergence}, Analysis, 5(1985),
301-313.

%\bibitem{freedman} Freedman A R and Sember J J, Densities and summability,
%Pacific J. Math. 95(1981) 293-305

\bibitem{korov} Korovkin P. P., \textit{Linear operators and Approximation theory},
Delhi: Hindustan Publ. Co., 1960.

\bibitem{kolk1} Kolk E., \textit{Matrix summability of Statistically convergent
sequences}, Analysis, 13(1993), 77-83.

%\bibitem{kolk2} Kolk E, The statistical convergence in Banach spaces, Tartu
%\"{U}l. Toimetised 928 (1991) 41-52

\bibitem{kos} Kostyrko P., \v{S}al\'{a}t T., Wilczy\'{n}ski W., \textit{$\mathcal{I}$%
-convergence}, Real Anal. Exchange, 26(2000/2001), no. 2, 669-685.

%\bibitem{lahiri} Lahiri B K and Das P, $\mathcal{I}$ and $\mathcal{I}^*$
%convergence  in topological spaces, Math. Bohemica 130 (2005) 153-160

%\bibitem{maddox} Maddox I J, Space of strongly summable sequence, Quart. J.
%Math. Oxford Ser. 18 (2) (1967) 345-355

%\bibitem{miller} Miller H I and Miller-Van Wieren L, A matrix
%characterization of statistical convergence of double sequences, Sarajevo
%Journal of Mathematics 4 (16) (2008) 91-95

%\bibitem{moricz} M\'{o}ricz F, Statistical convergence of multiple
%sequences, Arch. Math. (Basel) 81 (1) (2003) 82-89

\bibitem{mursaleen} Mursaleen M. and Edely O. H. H., \textit{Statistical  convergence of
double sequences}, J. Math. Anal. Appl., 288(2003), 223-331.

%%\bibitem{Mursaleenalotaibi} Mursaleen M and Alotaibi A, Statistical
%summability and approximation by de la Vall\'{e}e-poussin mean, Appl. Math. Lett.
%24 (2011) 672-680

\bibitem{Mursaleenalotaibi1} Mursaleen M., Alotaibi A., \textit{Korovkin type
approximation theorem for functions of two variables through statistical $A$%
-summability}, Advances in Difference Equations,
https://doi.org/10.1186/1687-1847-2012-65.

%\bibitem{pringsheim} Pringsheim A, Zur Theorie der zweifach unendlichen
%Zahlenfolgen, Math. Ann. 53 (1900) 289-321

\bibitem{Robison} Robison G. M., \textit{Divergent double sequences and series}, Trans.
Amer. Math. Soc., 28(1926), no. 1, 50-73.

\bibitem{salat} \v{S}al\'at T., \textit{On Statistically convergent sequences of real
numbers}, Math. Slovaca, 30(1980), 139-150.

% \bibitem{schoen} I. J. Schoenberg, The integrability of certain functions and related summability
% methods, Amer. Monthly, 66 (5) (1959), 362-375.

\bibitem{espd} Savas E., Das P., \textit{A generalized statistical convergence via
ideals}, Appl. Math. Lett., 24(2011), 826-830.

\bibitem{espdsd1} Savas E., Das P., Dutta S, \textit{A note on some generalized
summability methods}, Acta Math. Univ. Comenian.(N.S.), 82(2013), no. 2, 297-304.

\bibitem{espdsd2} Savas E., Das P., Dutta S., \textit{A note on strong matrix
summability via ideals}, Appl. Math. Lett., 25(2012), 733-738.
\end{thebibliography}

\end{document}