Abstract
In the present work, we have introduced a weighted statistical approximation theorem for sequences of positive linear operators defined on the space of all real-valued B-continuous functions on a compact subset of \( \mathbb {R} ^{2}= \mathbb {R} \times \mathbb {R} \). Furthermore, we display an application which shows that our new result is stronger than its classical version.
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Keywords
- Weighted uniform convergence
- Double sequences
- Statistical convergence
- Korovkin-type approximation theorem
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:
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 \(\varepsilon >0\),
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 \(\varepsilon >0\),
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
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 \(\varepsilon >0,\)
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
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
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
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
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
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:
and
Then, for all \(f\in C_{b}\left( I\right) \),we have
Proof
Let \((x,y)\in I\) and \(f\in C_{b}\left( I\right) \) be fixed. Taking
we obtain from (2.1) that
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
holds for every \((u,v)\in I\). Also, by (2.12), see that
holds for all \((j,k)\in A.\) We can write for all \((m,n)\in A\) from (2.6) and (2.7),
where \(\alpha (\varepsilon )=\max \{\alpha _{1}(\varepsilon ),\alpha _{2}(\varepsilon )\}.\) It follows from the last inequality that
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
Because of \(\varepsilon \) is arbitrary, we obtain
Hence,
Now for a given \(r>0\), consider the following sets:
Hence, inequality (2.9) yields that
which gives,
Letting \(m,n\rightarrow \infty \) (in any manner) and also using (2.13), we see from (2.10) that
Furthermore, if we use the inequality
and if we take limit as \(m,n\rightarrow \infty \), then it follows from (2.5) and (2.11) that
which means
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:
and
Then, for all \(f\in C_{b}(I),\) we have
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
on \(C_{b}\left( I\right) .\)
(a) Using these polynomials, we introduce the following positive linear operators on \(C_{b}\left( I\right) :\)
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
Since \(st^{2}-\lim \alpha _{mn}=0\), we conclude that
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 \(C_{b}\left( I\right) \):
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
Since \(st_{\overline{N_{2}}}-\lim \beta _{mn}=0\), we conclude that
So, by Theorem 1, we have
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].
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Dirik, F. (2018). Weighted Statistical Convergence of Bögel Continuous Functions by Positive Linear Operator. In: Mohiuddine, S., Acar, T. (eds) Advances in Summability and Approximation Theory. Springer, Singapore. https://doi.org/10.1007/978-981-13-3077-3_11
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