Keywords

Mathematics Subject Classification

1 Introduction

The classical Korovkin theory is mostly connected with the approximation to continuous functions by means of positive linear operators (see, for instance, [1, 17]). In order to work up the classical Korovkin theory, the space of Bögel-type continuous (or, simply, B-continuous) functions instead of the classical theory has been studied in [2,3,4]. The concept of statistical convergence for sequences of real numbers was introduced by Fast [14] and Steinhaus [21] independently in the same year 1951. Some Korovkin-type theorems in the setting of a statistical convergence were given by [5, 6, 10,11,12,13, 22].

Now we recall some notations and definitions.

A double sequence \(x=(x_{mn}),\)\(m,n\in \mathbb {N},\) is convergent in Pringsheim’s sense if, for every \(\varepsilon \) \(>0,\) there exists \(N=N(\varepsilon )\in \mathbb {N}\) such that \(\left| x_{mn}- L \right| <\varepsilon \) whenever \(m,n>N\), then \( L \) is called the Pringsheim limit of x and is denoted by \(P-\lim x= L \) (see [20]). Also, if there exists a positive number M such that \( \left| x_{mn}\right| \le M\) for all \((m,n)\in \mathbb {N} ^{2}= \mathbb {N} \times \mathbb {N},\) then \(x=\left( x_{mn}\right) \) is said to be bounded. Note that in contrast to the case for single sequences, a convergent double sequence need not to be bounded.

Definition 1

([19]) Let \(K\subset \mathbb {N}^{2}=\mathbb {N\times N}\). Then density of K, denoted by \(\delta ^{2}(K)\), is given by:

$$\begin{aligned} \delta ^{2}(K):=P-\lim _{m,n}\frac{\left| \left\{ j\le m,k\le n:\left( j,k\right) \in K\right\} \right| }{mn} \end{aligned}$$

provided that the limit on the right-hand side exists in the Pringsheim sense by |B| we mean the cardinality of the set \(B\subset \mathbb {N} ^{2}= \mathbb {N} \times \mathbb {N} \). A real double sequence \(x=(x_{mn})\) is said to be statistically convergent to L if, for every \(\varepsilon >0\),

$$\begin{aligned} \delta ^{2}\left( \left\{ (m,n)\in \mathbb {N} ^{2}:\left| x_{mn}- L \right| \ge \varepsilon \right\} \right) =0. \end{aligned}$$

In this case, we write \(st^{2}-\lim x=L\).

The concept of weighted statistical convergence was defined by Karakaya and Chishti [16]. Recently, Mursaleen et al. [18] modified the definition of weighted statistical convergence. In [15], Ghosal showed that both definitions of weighted statistical convergence are not well defined in general. So Ghosal modified the definition of weighted statistical convergence as follows:

Definition 2

Let \(\{p_{j}\}\), \(\{q_{k}\}\), \(j,k\in \mathbb {N} \) be sequences of nonnegative real numbers such that \(p_{1}\) \(>0\), \(\underset{j\rightarrow \infty }{\lim \inf }p_{j}>0,\) \(q_{1}\) \(>0\), \(\underset{ k\rightarrow \infty }{\lim \inf }q_{k}>0\) and \(P_{m}=\mathop {\displaystyle \sum }\limits _{j=1}^{m}p_{j}\) and \(Q_{n}=\mathop {\displaystyle \sum }\limits _{k=1}^{n}q_{k}\) where \( n,m\in \mathbb {N} \), \(P_{m}\) \(\rightarrow \infty \) as \(m\rightarrow \infty ,\) \( Q_{n}\rightarrow \infty \) as \(n\rightarrow \infty \). The double sequence \( x=(x_{jk})\) is said to be weighted statistical convergent (or \(S_{_{ \overline{_{N_{2}}}}}\)-convergent) to L if for every \(\varepsilon >0\),

$$\begin{aligned} P-\underset{m,n}{\lim }\frac{1}{P_{m}Q_{n}}\ |\{j\le P_{m},k\le Q_{n}:p_{j}q_{k}|x_{jk}-L|\ge \varepsilon \}|=0. \end{aligned}$$

In this case, we write \(st_{\overline{_{N_{2}}}}-\lim x=L\) and we denote the set of all weighted statistical convergent sequences by \(S_{_{\overline{ _{N_{2}}}}}\).

Remark 1

If \(p_{j}=1\), \(q_{k}=1\) for all j, k, then weighted statistical convergence is reduced to statistical convergence for double sequences.

Example 1

Let \(x=(x_{mn})\) is a sequence defined by

$$\begin{aligned} x_{mn}:=\left\{ \begin{array}{cc} mn, &{} m\text { and }n\text { are squares,} \\ 0, &{} \text {otherwise,} \end{array} \right. . \end{aligned}$$

Let \(p_{j}=j\), \(q_{k}=k\) for all j, k. Then \(P_{m}=\frac{m(m+1)}{2}\) and \( Q_{n}=\frac{n\left( n+1\right) }{2}\) . Since, for every \(\varepsilon >0,\)

$$\begin{aligned}&P-\underset{m,n}{\lim }\frac{|\{j\le P_{m},k\le Q_{n}:p_{j}q_{k}|x_{jk}-0|\ge \varepsilon \}|}{P_{m}Q_{n}}\ \\\le & {} P-\underset{m,n}{\lim }\frac{|\{j\le P_{m},k\le Q_{n}:p_{j}q_{k}|x_{jk}|\ne 0\}|}{P_{m}Q_{n}} \\\le & {} P-\underset{m,n}{\lim }\frac{\sqrt{P_{m}}\sqrt{Q_{n}}}{P_{m}Q_{n}}=0 \end{aligned}$$

So \(x=(x_{mn})\) is weighted statistical convergent to 0 but not Pringsheim’s sense convergent.

In [15], Ghosal showed that both convergences which are weighted statistical convergence and statistical convergence do not imply each other in general.

In the work, using the Definition 2, we prove Korovkin-type approximation theorem for double sequences of B-continuous functions defined on a compact subset of the real two-dimensional space. Finally, we give an application which shows that our new result is stronger than its classical version.

2 A Korovkin-Type Approximation Theorem

Bögel introduced the definition of B-continuity [7,8,9] as follows:

Let \(I\ \) be a compact subset of \( \mathbb {R} ^{2}= \mathbb {R} \times \mathbb {R}.\) Then, a function \(f:I\rightarrow \) \( \mathbb {R} \) is called a B-continuous at a point \(\left( x,y\right) \in I\) if, for every \(\varepsilon >0\), there exists a positive number \(\delta =\delta (\varepsilon )\) such that

$$\begin{aligned} \left| \Delta _{xy}\left[ f\left( u,v\right) \right] \right| <\varepsilon , \end{aligned}$$

for any \(\left( u,v\right) \in I\) with \(\left| u-x\right| <\delta \) and \(\left| v-y\right| <\delta \), where the symbol \(\Delta _{xy} \left[ f\left( u,v\right) \right] \) denotes the mixed difference of f defined by

$$\begin{aligned} \Delta _{xy}\left[ f\left( u,v\right) \right] =f(u,v)-f(u,y)-f(x,v)+f(x,y). \end{aligned}$$

By \(C_{b}(I)\), we denote the space of all B-continuous functions on I. Recall that C(I) and B(I) denote the space of all continuous (in the usual sense) functions on I and the space of all bounded functions on I,  respectively. Then, notice that \(C(I)\subset C_{b}(I).\) Moreover, one can find an unbounded B-continuous function, which follows from the fact that, for any function of the type \(f(u,v)=g(u)+h(v),\) we have \(\Delta _{xy}\left[ f\left( u,v\right) \right] =0\) for all \((x,y),(u,v)\in I\). \(\left\| f\right\| \) denotes the supremum norm of f in B(I).

Let L be a linear operator from \(C_{b}\left( I\right) \) into \(B\left( I\right) \). Then, as usual, we say that L is positive linear operator provided that \(f\ge 0\) implies \(L\left( f\right) \ge 0\). Also, we denote the value of \(L\left( f\right) \) at a point \((x,y)\in I\) by L(f(u, v); x, y) or, briefly, L(f; x, y). Since

$$\begin{aligned} \Delta _{xy}\left[ f(u,y)+f(x,v)-f(u,v)\right] =-\Delta _{xy}\left[ f(u,v) \right] \end{aligned}$$

holds for all (x, y),  \((u,v)\in I,\) the B-continuity of f implies the B-continuity of \(F_{xy}(u,v):=f(u,y)+f(x,v)-f(u,v)\) for every fixed \( (x,y)\in I.\) We also use the following test functions

$$\begin{aligned} f_{0}(x,y)=1, \quad f_{1}(x,y)=x, \quad f_{2}(x,y)=y\text { and } f_{3}(x,y)=x^{2}+y^{2}. \end{aligned}$$

We recall that the following lemma for B-continuous functions was proved by Badea et al. [3].

Lemma 1

([3]) If \(f\in C_{b}(I),\) then, for every \(\varepsilon >0,\) there are two positive numbers \(\alpha _{1}(\varepsilon )=\alpha _{1}(\varepsilon ,f)\) and \( \alpha _{2}(\varepsilon )=\alpha _{2}(\varepsilon ,f)\) such that

$$\begin{aligned} \Delta _{xy}\left[ f\left( u,v\right) \right] \le \frac{\varepsilon }{3} +\alpha _{1}(\varepsilon )(u-x)^{2}+\alpha _{2}(\varepsilon )(v-y)^{2} \end{aligned}$$

holds for all (x, y), \((u,v)\in I.\)

Now we have the following main result.

Theorem 1

Let \(\left( L_{mn}\right) \) be a double sequence of positive linear operators acting from \(C_{b}\left( I\right) \) into \(B\left( I\right) \). Assume that the following conditions hold:

$$\begin{aligned} P-\underset{m,n}{\lim }\frac{1}{P_{m}Q_{n}}\ |\{j\le P_{m},k\le Q_{n}:L_{jk}(f_{0};x,y)=f_{0}(x,y)\text { for all }(x,y)\in I\}|=1 \end{aligned}$$
(2.1)

and

$$\begin{aligned} st_{\overline{N_{2}}}-\lim \left\| L_{mn}\left( f_{i};x,y\right) -f_{i}(x,y)\right\| =0, i=1,2,3. \end{aligned}$$
(2.2)

Then, for all \(f\in C_{b}\left( I\right) \),we have

$$\begin{aligned} st_{\overline{N_{2}}}-\lim \left\| L_{mn}\left( f(u,y)+f(x,v)-f(u,v);x,y\right) -f(x,y)\right\| =0. \end{aligned}$$
(2.3)

Proof

Let \((x,y)\in I\) and \(f\in C_{b}\left( I\right) \) be fixed. Taking

$$\begin{aligned} A:=\left\{ j\le P_{m},k\le Q_{n}:L_{jk}(f_{0};x,y)=f_{0}(x,y)=1\text { for all }(x,y)\in I\right\} , \end{aligned}$$
(2.4)

we obtain from (2.1) that

$$\begin{aligned} P-\underset{m,n}{\lim }\frac{1}{P_{m}Q_{n}}\ |\{j\le P_{m},k\le Q_{n}:L_{jk}(f_{0};x,y)\ne f_{0}(x,y)\text { for all }(x,y)\in I\}|=0. \end{aligned}$$
(2.5)

Using the B-continuity of the function \(F_{xy}(u,v):=f(u,y)+f(x,v)-f(u,v)\), Lemma 1 implies that, for every \(\varepsilon >0,\) there exist two positive numbers \(\alpha _{1}(\varepsilon )\) and \(\alpha _{2}(\varepsilon )\) such that

$$\begin{aligned} \left| \Delta _{xy}\left[ f(u,y)+f(x,v)-f(u,v)\right] \right| \le \frac{\varepsilon }{3}+\alpha _{1}(\varepsilon )(u-x)^{2}+\alpha _{2}(\varepsilon )(v-y)^{2} \end{aligned}$$
(2.6)

holds for every \((u,v)\in I\). Also, by (2.12), see that

$$\begin{aligned} L_{jk}\left( f(u,y)+f(x,v)-f(u,v);x,y\right) -f(x,y)=L_{jk}\left( \Delta _{xy}\left[ f(u,y)+f(x,v)-f(u,v)\right] ;x,y\right) \end{aligned}$$
(2.7)

holds for all \((j,k)\in A.\) We can write for all \((m,n)\in A\) from (2.6) and (2.7),

$$\begin{aligned} \left| L_{jk}\left( f(u,y)+f(x,v)-f(u,v);x,y\right) -f(x,y)\right|= & {} \left| L_{jk}\left( \Delta _{xy}\left[ f(u,y)+f(x,v)-f(u,v)\right] ;x,y\right) \right| \\\le & {} L_{jk}\left( \left| \Delta _{xy}\left[ f(u,y)+f(x,v)-f(u,v)\right] \right| ;x,y\right) \\\le & {} \frac{\varepsilon }{3}+\alpha _{1}(\varepsilon )L_{jk}\left( (u-x)^{2};x,y\right) \\&+~\alpha _{2}(\varepsilon )L_{jk}\left( (v-y)^{2};x,y\right) \\\le & {} \frac{\varepsilon }{3}+\alpha (\varepsilon )\{x^{2}+y^{2}+L_{jk}(f_{3};x,y) \\&-2xL_{jk}(f_{1};x,y)-2yL_{jk}(f_{2};x,y)\}, \end{aligned}$$

where \(\alpha (\varepsilon )=\max \{\alpha _{1}(\varepsilon ),\alpha _{2}(\varepsilon )\}.\) It follows from the last inequality that

$$\begin{aligned} \left| L_{jk}\left( f(u,y)+f(x,v)-f(u,v);x,y\right) -f(x,y)\right| \le \frac{\varepsilon }{3}+\alpha (\varepsilon )\sum _{i=1}^{3}\left| L_{jk}\left( f_{i};x,y\right) -f_{i}(x,y)\right| \end{aligned}$$
(2.8)

holds for all \((j,k)\in A\). Taking supremum over \((x,y)\in I\) on both sides of inequality (2.8), we obtain, for all \((j,k)\in I\), that

$$\begin{aligned} \left\| L_{jk}\left( f(u,y)+f(x,v)-f(u,v);x,y\right) -f(x,y)\right\| \le \frac{\varepsilon }{3}+\alpha (\varepsilon )\sum _{i=1}^{3}\left\| L_{jk}\left( f_{i};x,y\right) -f_{i}(x,y)\right\| . \end{aligned}$$

Because of \(\varepsilon \) is arbitrary, we obtain

$$\begin{aligned} \left\| L_{jk}\left( f(u,y)+f(x,v)-f(u,v);x,y\right) -f(x,y)\right\| \le \alpha (\varepsilon )\sum _{i=1}^{3}\left\| L_{jk}\left( f_{i};x,y\right) -f_{i}(x,y)\right\| . \end{aligned}$$

Hence,

$$\begin{aligned} p_{j}q_{k}\left\| L_{jk}\left( f(u,y)+f(x,v)-f(u,v);x,y\right) -f(x,y)\right\| \le \alpha (\varepsilon )\sum _{i=1}^{3}p_{j}q_{k}\left\| L_{jk}\left( f_{i};x,y\right) -f_{i}(x,y)\right\| . \end{aligned}$$
(2.9)

Now for a given \(r>0\), consider the following sets:

$$\begin{aligned} U&:&=\left\{ j\le P_{m},k\le Q_{n}:p_{j}q_{k}\left\| L_{jk}\left( f(u,y)+f(x,v)-f(u,v);x,y\right) -f(x,y)\right\| \ge r\right\} , \\ U_{i}&:&=\left\{ j\le P_{m},k\le Q_{n}:p_{j}q_{k}\left\| L_{jk}\left( f_{i};x,y\right) -f_{i}(x,y)\right\| \ge \frac{r}{3\alpha (\varepsilon )} \right\} , \quad i=1,2,3, \end{aligned}$$

Hence, inequality (2.9) yields that

$$\begin{aligned} \frac{\left| U\cap A\right| }{P_{m}Q_{n}}\le \frac{\left| U_{1}\cap A\right| }{P_{m}Q_{n}}+\frac{\left| U_{2}\cap A\right| }{P_{m}Q_{n}}+\frac{\left| U_{3}\cap A\right| }{P_{m}Q_{n}}, \end{aligned}$$

which gives,

$$\begin{aligned} P-\lim \frac{\left| U\cap A\right| }{P_{m}Q_{n}}\le \sum _{i=1}^{3}\left\{ P-\lim \frac{\left| U_{i}\cap A\right| }{ P_{m}Q_{n}}\right\} \le \sum _{i=1}^{3}\left\{ P-\lim \frac{\left| U_{i}\right| }{P_{m}Q_{n}}\right\} \end{aligned}$$
(2.10)

Letting \(m,n\rightarrow \infty \) (in any manner) and also using (2.13), we see from (2.10) that

$$\begin{aligned} P-\lim \frac{\left| U\cap A\right| }{P_{m}Q_{n}}=0. \end{aligned}$$
(2.11)

Furthermore, if we use the inequality

$$\begin{aligned} \frac{\left| U\right| }{P_{m}Q_{n}}= & {} \frac{\left| U\cap A\right| }{P_{m}Q_{n}}+\frac{\left| U\cap \left( \mathbb {N} ^{2}\backslash A\right) \right| }{P_{m}Q_{n}} \\\le & {} \frac{\left| U\cap A\right| }{P_{m}Q_{n}}+\frac{\left| \mathbb {N}^{2}\backslash A\right| }{P_{m}Q_{n}} \end{aligned}$$

and if we take limit as \(m,n\rightarrow \infty \), then it follows from (2.5) and (2.11) that

$$\begin{aligned} P-\lim \frac{\left| U\right| }{P_{m}Q_{n}}=0, \end{aligned}$$

which means

$$\begin{aligned} st_{\overline{N_{2}}}-\lim \left\| L_{mn}\left( f(u,y)+f(x,v)-f(u,v);x,y\right) -f(x,y)\right\| =0=0. \end{aligned}$$

This completes the proof.    \(\square \)

If \(p_{j}=1\) and \(q_{k}=1\) with \(j,k\in \mathbb {N} \), then we obtain the statistical case of the Korovkin-type result for a double sequences on \(C_{b}\left( I\right) \) introduced in [13],

Theorem 2

([13]) Let \(\left( L_{mn}\right) \) be a sequence of positive linear operators acting from \(C_{b}\left( I\right) \) into \(B\left( I\right) \). Assume that the following conditions hold:

$$\begin{aligned} \delta ^{2}\left\{ (m,n)\in \mathbb {N}^{2}:L_{mn}(f_{0};x,y)=1\text { for all }(x,y)\in I\right\} =1 \end{aligned}$$
(2.12)

and

$$\begin{aligned} st^{2}-\lim _{m,n}\left\| L_{mn}\left( f_{i};x,y\right) -f_{i}(x,y)\right\| =0\text { for }i=1,2,3. \end{aligned}$$
(2.13)

Then, for all \(f\in C_{b}(I),\) we have

$$\begin{aligned} st^{2}-\lim _{m,n}\left\| L_{mn}(f(u,y)+f(x,v)-f(u,v);x,y)-f(x,y)\right\| =0. \end{aligned}$$

Now we present an example for double sequences of positive linear operators. The first one shows that Theorem 1 does not work but Theorem 2 works. The second one gives that our approximation theorem and Theorem 2 work.

Example 2

Let \(I=\left[ 0,1\right] \times \left[ 0,1\right] \). Consider the double Bernstein polynomials

$$\begin{aligned} B_{mn}(f;x,y)=\sum _{s=0}^{m}\sum _{t=0}^{n}f\left( \frac{s}{m},\frac{t}{n} \right) x^{s}y^{t}\left( 1-x\right) ^{m-s}\left( 1-y\right) ^{n-t} \end{aligned}$$

on \(C_{b}\left( I\right) .\)

(a) Using these polynomials, we introduce the following positive linear operators on \(C_{b}\left( I\right) :\)

$$\begin{aligned} P_{mn}(f;x,y)=(1+\alpha _{mn})B_{mn}(f;x,y),\,\,\,(\,x,y)\in I\text { and } f\in C_{b}\left( I\right) , \end{aligned}$$
(2.14)

where \(\alpha :=(\alpha _{mn})\) is given by \(\alpha _{mn}:=\left\{ \begin{array}{cc} 1 &{} m,n\text { are squares,} \\ \frac{1}{\sqrt{mn}} &{} \text {otherwise,} \end{array} \right. \). Let \(p_{j}=2j+1\), \(q_{k}=k\) for all j, k. Then \(P_{m}=m^{2}\) and \(Q_{n}=\frac{n\left( n+1\right) }{2}\). Note that \(\alpha =(\alpha _{mn}) \) statistical convergent to 0 but it is not convergent and weighted statistical convergent to 0. Then, observe that

$$\begin{aligned} P_{mn}(f_{0};x,y)= & {} (1+\alpha _{mn})f_{0}(x,y), \\ P_{mn}(f_{1};x,y)= & {} (1+\alpha _{mn})f_{1}(x,y), \\ P_{mn}(f_{2};x,y)= & {} (1+\alpha _{mn})f_{2}(x,y), \\ P_{mn}(f_{3};x,y)= & {} (1+\alpha _{mn})\left[ f_{3}(x,y)+\frac{x-x^{2}}{m}+ \frac{y-y^{2}}{n}\right] . \end{aligned}$$

Since \(st^{2}-\lim \alpha _{mn}=0\), we conclude that

$$\begin{aligned} st^{2}-\lim \left\| P_{mn}\left( f_{i};x,y\right) -f_{i}(x,y)\right\| =0\text { for each }i=0,1,2. \end{aligned}$$

However, since \(\alpha \) is statistically convergent, the sequence \(\left\{ P_{mn}(f;x,y)\right\} \) given by (2.14) does satisfy the Theorem 2 for all \(f\in C_{b}\left( I\right) \). But Theorem 1 does not work since \(\alpha =(\alpha _{mn})\) is not weighted statistical convergent to 0.

\(\left( b\right) \) Now we consider the following positive linear operators on \(C_{b}\left( I\right) \):

$$\begin{aligned} T_{mn}(f;x,y)=(1+\beta _{mn})B_{mn}(f;x,y),\,\,\,(\,x,y)\in I\text { and } f\in C_{b}\left( I\right) \!, \end{aligned}$$
(2.15)

where \(\beta :=(\beta _{mn})\) is given by \(\beta _{mn}:=\left\{ \begin{array}{cc} mn &{} m,n\text { are squares,} \\ 0 &{} \text {otherwise,} \end{array} \right. \). Let \(p_{j}=j\), \(q_{k}=k\) for all j, k. Then \(P_{m}=\frac{m(m+1) }{2}\) and \(Q_{n}=\frac{n\left( n+1\right) }{2}\). Note that \(\alpha =(\alpha _{mn})\) statistical and weighted statistical convergent to 0 but it is not convergent to 0. Then, observe that

$$\begin{aligned} T_{mn}(f_{0};x,y)= & {} (1+\beta _{mn})f_{0}(x,y), \\ T_{mn}(f_{1};x,y)= & {} (1+\beta _{mn})f_{1}(x,y), \\ T_{mn}(f_{2};x,y)= & {} (1+\beta _{mn})f_{2}(x,y), \\ T_{mn}(f_{3};x,y)= & {} (1+\beta _{mn})\left[ f_{3}(x,y)+\frac{x-x^{2}}{m}+ \frac{y-y^{2}}{n}\right] . \end{aligned}$$

Since \(st_{\overline{N_{2}}}-\lim \beta _{mn}=0\), we conclude that

$$\begin{aligned} st_{\overline{N_{2}}}-\lim \left\| T_{mn}\left( f_{i};x,y\right) -f_{i}(x,y)\right\| =0\text { for each }i=1,2,3. \end{aligned}$$

So, by Theorem 1, we have

$$\begin{aligned} st_{\overline{N_{2}}}-\lim \left\| T_{mn}\left( f(u,y)+f(x,v)-f(u,v);x,y\right) -f(x,y)\right\| =0\text { for all }f\in C_{b}\left( I\right) . \end{aligned}$$

However, since \(\beta \) is weighted statistical convergent to 0,  we can say that Theorem 1 works for our operators defined by (2.15).

Therefore, this application clearly shows that our Theorem 1 is a non-trivial generalization of the classical case of the Korovkin result introduced in [3].