Abstract
The concept of \(\alpha \beta \)-statistical convergence was introduced and studied by Aktuğlu (Korovkin type approximation theorems proved via \(\alpha \beta \)-statistical convergence, J Comput Appl Math 259:174–181, 2014). In this work, we generalize the concept of \(\alpha \beta \)-statistical convergence and introduce the concept of weighted \(\alpha \beta \)-statistical convergence of order \(\gamma \), weighted \(\alpha \beta \)-summability of order \(\gamma \), and strongly weighted \(\alpha \beta \)-summable sequences of order \(\gamma \). We also establish some inclusion relation, and some related results for these new summability methods. Furthermore, we prove Korovkin type approximation theorems through weighted \(\alpha \beta \)-statistical convergence and apply the classical Bernstein operator to construct an example in support of our result.
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1 Introduction, notations and known results
Let \(K\) be a subset of \(\mathbb {N}\), the set of natural numbers and \(K_{n}=\left\{ k\le n: k\in K \right\} \). The natural density of \(K\) is defined by \(\delta (K)=\lim _{n}\frac{1}{n}|K_{n}|\) provided it exists, where \(|K_{n}|\) denotes the cardinality of set \(K_{n}\). A sequence \(x=(x_{k})\) is called statistically convergent (\(st\)-convergent) to the number \(\ell \), denoted by \(st-\lim x=\ell \), for each \(\epsilon >0,\) the set \(K_{\epsilon }=\left\{ k\in \mathbb {N}:|x_{k}-\ell |\ge \epsilon \right\} \) has natural density zero, that is
The concept of statistical convergence has been defined by Fast [5] and studied by many other authors. It is well know that every statistically convergent sequence is ordinary convergent, but the converse is not true. For example, \(y=(y_{k})\) is defined as follows;
It is clear that the sequence \(y=(y_{k})\) is statistical convergent to zero but not convergent. The idea \(\alpha \beta \)-statistical convergence was introduced by Aktuğlu in [7] as follows:
Let \(\alpha (n)\) and \(\beta (n)\) be two sequences positive number which satisfy the following conditions
-
(i)
\(\alpha \) and \(\beta \) are both non-decreasing,
-
(ii)
\(\beta (n)\ge \alpha (n)\),
-
(iii)
\(\beta (n)-\alpha (n)\longrightarrow \infty \) as \(n\longrightarrow \infty \)
and let \(\Lambda \) denote the set of pairs \((\alpha ,\beta )\) satisfying (i)–(iii). For each pair \((\alpha ,\beta )\in \Lambda \), \(0<\gamma \le 1\) and \(K\subset \mathbb {N}\), we define \(\delta ^{\alpha ,\beta }(K,\gamma )\) in the following way
where \(P^{\alpha ,\beta }_{n}\) in the closed interval \([\alpha (n),\beta (n)]\). A sequence \(x=(x_{k})\) is said to be \(\alpha \beta \)-statistically convergent of order \(\gamma \) to \(\ell \) or \(S^{\gamma }_{\alpha \beta }\)-convergent, if
In this paper generalizing above idea, we define the weighted \(\alpha \beta \)-statistical convergence of order \(\gamma \), the weighted \(\alpha \beta \)-summability of order \(\gamma \) and the weighted \(\alpha \beta \)-summability.
2 The weighted \(\alpha \beta \)-summability
Let \(s=(s_{k})\) be a sequence of non-negative real numbers such that \(s_{0}>0\) and
Definition 2.1
-
(a)
A sequence \(x=(x_{k})\) is said to be strongly weighted \(\alpha \beta \)-summable of order \(\gamma \) to a number \(\ell \) if
$$\begin{aligned} \lim _{n \rightarrow \infty }\frac{1}{S_{n}^{\gamma }}\sum _{k\in P^{\alpha ,\beta }_{n}}s_{k}|x_{k}-\ell |\longrightarrow 0 \quad \text {as}\,\,n\longrightarrow \infty . \end{aligned}$$We denote it by \(x_{k}\longrightarrow \ell [\overline{N}^{\gamma }_{\alpha \beta },s]\). Similarly, for \(\gamma =1\) the sequences \(x=(x_{k})\) is said to be strongly weighted \(\alpha \beta \)-summable to \(\ell \). The set of all strongly weighted \(\alpha \beta \)-summable of order \(\gamma \) and strongly weighted \(\alpha \beta \)-summable sequences will be denoted \([\overline{N}^{\gamma }_{\alpha \beta },s]\) and \([\overline{N}_{\alpha \beta },s]\), respectively.
-
(b)
A sequence \(x=(x_{k})\) is said to be weighted \(\alpha \beta \)-summable of order \(\gamma \) to \(\ell \), if \(z^{\gamma }_{n}(x)\longrightarrow \ell \) as \(n\longrightarrow \infty \). Similarly, for \(\gamma =1\) the sequence \(x=(x_{k})\) is said to be weighted \(\alpha \beta \)-summable to \(\ell \), if \(z_{n}(x)\longrightarrow \ell \) as \(n\longrightarrow \infty \). The set of all weighted \(\alpha \beta \)-summable of order \(\gamma \) and weighted \(\alpha \beta \)-summable sequences will be denoted \((\overline{N}^{\gamma }_{\alpha \beta },s)\) and \((\overline{N}_{\alpha \beta },s)\), respectively.
This definition includes the following special cases:
-
(i)
If \(\gamma =1\), \(\alpha (n)=0\) and \(\beta (n)=n\), weighted \(\alpha \beta \)-summable is reduced to weighted mean summable, and \([\overline{N}_{\alpha \beta },s]\) summable sequences are reduced to \((\overline{N},p_{n})\) summable sequences introduced in [8, 14].
-
(ii)
Let \(\lambda _{n}\) be a none-decreasing sequence of positive numbers tending to \(\infty \) such that \(\lambda _{n}\le \lambda _{n}+1\), \(\lambda _{1}=1\). If we take \(\gamma =1\), \(\alpha (n)=n-\lambda _{n}+1\) and \(\beta (n)=n\) then weighted \(\alpha \beta \)-summability is reduced to \((\overline{N}_{\lambda }; p)\)-summability and \([\overline{N}_{\alpha \beta },s]\) summable sequences are reduced to \([\overline{N}_{\lambda }; p]\)-summable sequences introduced in [2].
-
(iii)
If we take \(\gamma =1\), \(\alpha (n)=n-\lambda _{n}+1\), \(\beta (n)=n\) and \(s_{k}=1\) for all \(k\) then \(\alpha \beta \)-summability is reduced to \((V; \lambda )\)-summability introduced in [10] and \([\overline{N}_{\alpha \beta },s]\)-summable sequences are reduced to \([V,\lambda ]\)-summable sequences introduced in [12].
-
(iv)
Recall that a lacunary sequence \(\theta =\{k_{r}\}\) is an increasing integer sequence such that \(k_{0}=0\) and \(h_{r}:=k_{r}-k_{r-1}\). If we take \(\gamma =1\), \(\alpha (r)=k_{r-1}+1\) and \(\beta (r)=k_{r}\); then \(P^{\alpha ,\beta }(r)=[k_{r-1}+1,k_{r}]\). But because of \([k_{r-1}+1,k_{r}]\cap \mathbb {N}=(k_{r-1},k_{r}]\cap \mathbb {N}\), we have
$$\begin{aligned} \sum _{k\in [\alpha (n), \beta (n)]}s_{k}x_{k}=\sum _{k\in [k_{r-1}+1, k_{r}]}s_{k}x_{k}=\sum _{k\in (k_{r-1}, k_{r}]}s_{k}x_{k}. \end{aligned}$$This means that \([\overline{N}_{\alpha \beta },s]\)-summable sequences are reduced to weighted lacunary summable, and \([\overline{N}_{\alpha \beta },s]\)-summable sequence are reduced to strong weighted lacunary summable sequences.
-
(v)
If we take \(\gamma =1\), \(\alpha (r)=k_{r-1}+1\), \(\beta (r)=k_{r}\) and \(s_{k}=1\) for all \(k\) then, weighted \(\alpha \beta \)-summable is reduced to lacunary summable sequences, and \([\overline{N}_{\alpha \beta },s]\)-summable sequences are reduced to \(N_{\theta }\) summbale sequences introduced in [6].
Definition 2.2
A sequence \(x=(x_{k})\) is said to be weighted \(\alpha \beta \)-statistically convergent of order \(\gamma \) to \(\ell \) or \(S^{\gamma }_{\alpha \beta }\)-convergent, if for every \(\epsilon > 0\)
and denote \(st^{\gamma }_{\alpha \beta }-\lim x=\ell \) or \(x_{k}\longrightarrow \ell [\overline{S}^{\gamma }_{\alpha \beta }]\), where \(\overline{S}^{\gamma }_{\alpha \beta }\) denotes the set of all weighted \(\alpha \beta \)-statistically convergent sequences of order \(\gamma \) .
Theorem 2.1
Let \(0<\gamma \le \delta \le 1\). Then, we have \([\overline{N}^{\gamma }_{\alpha \beta },s]\subseteq [\overline{N}^{\delta }_{\alpha \beta },s]\) and the inclusion is strict for some \(\gamma ,\delta \) such that \(\gamma <\delta \).
Proof
Let \(x=(x_{k})\in [\overline{N}^{\gamma }_{\alpha \beta },s]\) and \(\gamma ,\delta \) be given such that \(0<\gamma \le \delta \le 1\). Then, we obtain that
which gives \([\overline{N}^{\gamma }_{\alpha \beta },s]\subseteq [\overline{N}^{\delta }_{\alpha \beta },s]\). Now, we show that this inclusion is strict. Let us consider the sequence \(t = (t_{k})\) defined by,
If we choose \(s_{k}=1\) for all \(k\in \mathbb {N}\), it is clear that
Since \(\frac{1}{(\beta (n)-\alpha (n)+1)^{\gamma -1/2}} \longrightarrow 0\) as \(n\longrightarrow \infty \) for \(1/2<\beta \le 1\), we have \(t=(t_{k})\in [\overline{N}^{\gamma }_{\alpha \beta },s] \). On the other hand, we get
and \(\frac{\sqrt{\beta (n)-\alpha (n)+1}-1}{(\beta (n)-\alpha (n)+1)^{\delta }} \longrightarrow \infty \) as \(n\longrightarrow \infty \) for \(0<\delta < 1/2\) then, we have \(t=(t_{k})\notin [\overline{N}^{\delta }_{\alpha \beta },s] \). This completes the proof. \(\square \)
Theorem 2.2
Let \((\alpha ,\beta )\in \Lambda \). Then, we have following statements:
-
(a)
If a sequence \(x=(x_{k})\) is strongly weighted \((\alpha \beta )\)-summable of order \(\gamma \) to limit \(\ell \), then it is weighted \(\alpha \beta \)-statistically convergent of order \(\gamma \) to \(\ell \), that is \( [\overline{N}^{\gamma }_{\alpha \beta },s]\subseteq \overline{S}^{\gamma }_{\alpha \beta }\) and this inclusion is strict.
-
(b)
If \(x=(x_{k})\) bounded and weighted \(\alpha \beta \)-statistically convergent of order \(\gamma \) to \(\ell \) then \(x_{k}\longrightarrow [\overline{N}^{\gamma }_{\alpha \beta },s]\).
Proof
-
(a)
Let \(\epsilon >0\) and \( x_{k}\longrightarrow \ell [\overline{N}^{\gamma }_{\alpha \beta },s]\). Then, we get
$$\begin{aligned} \frac{1}{S_{n}^{\gamma }}\sum _{k\in P^{\alpha ,\beta }_{n}}|x_{k}-\ell |\!=\! \frac{1}{S_{n}^{\gamma }}\sum _{\begin{array}{c} k\in P^{\alpha ,\beta }_{n} \\ s_{k}|x_{k}\!-\!\ell |\ge \epsilon \end{array}}s_{k}|x_{k}\!-\!\ell |\!+\!\frac{1}{S_{n}^{\gamma }}\sum _{\begin{array}{c} k\in P^{\alpha ,\beta }_{n} \\ s_{k}|x_{k}-\ell |< \epsilon \end{array}}s_{k}|x_{k}-\ell |\ge \frac{\epsilon \left| K^{\alpha \beta }_{s_{n}}(\epsilon )\right| }{S_{n}^{\gamma }}. \end{aligned}$$This implies that \(\lim _{n \rightarrow \infty }\frac{\left| K^{\alpha \beta }_{s_{n}}(\epsilon )\right| }{S_{n}^{\gamma }}=0\) which means \(\delta ^{\alpha ,\beta }(K^{\alpha \beta }_{s_{n}}(\epsilon ),\gamma )=0\), where \(K^{\alpha \beta }_{s_{n}}(\epsilon )=\left\{ k\le S_{n}:s_{k}|x_{k}-\ell |\ge \epsilon \right\} \). Therefore, \(x=(x_{k})\) is weighted \(\alpha \beta \)-statistically convergent of order \(\gamma \) to \(\ell \). To prove \([\overline{N}^{\gamma }_{\alpha \beta },s]\subseteq S^{\gamma }_{\alpha \beta }\) in \((a)\) is strict, let the sequence \(x=(x_{k})\) be defined by
$$\begin{aligned} x_{k}=\left\{ \begin{array}{ll} k\text {,} \quad 1\le k\le [(\beta (n)-\alpha (n)+1)^{\gamma /2}]\text {,} \\ 0\text {,} \quad \text {otherwise}. \end{array} \right. \end{aligned}$$(2.1)Then \(x\) is not bounded and for every \(\epsilon >0\). Let \(s_{k}=1\) for all \(k\). Then we have
$$\begin{aligned}&\frac{1}{(\beta (n)-\alpha (n)+1)^{\gamma }}\left| \left\{ k\le \beta (n)-\alpha (n)+1:|x_{k}-0|\ge \epsilon \right\} \right| \\&\quad =\frac{[(\beta (n)-\alpha (n)+1)^{\gamma /2}]}{(\beta (n)-\alpha (n)+1)^{\gamma }}\longrightarrow 0\,\, \mathrm as \,\,n\longrightarrow \infty . \end{aligned}$$That is, \(x_{k}\longrightarrow 0[S^{\gamma }_{\alpha \beta }]\). But
$$\begin{aligned}&\frac{1}{(\beta (n)-\alpha (n)+1)^{\gamma }}\sum _{k\in P^{\alpha ,\beta }_{n}}|x_{k}-0|\\&\quad =\frac{[(\beta (n)-\alpha (n)+1)^{\gamma /2}]\left( [(\beta (n)-\alpha (n)+1)^{\gamma /2}]+1\right) }{2(\beta (n)-\alpha (n)+1)^{\gamma }}\longrightarrow \frac{1}{2}, \end{aligned}$$i.e., \(x_{k}\nrightarrow 0[\overline{N}^{\gamma }_{\alpha \beta },s]\).
-
(b)
Assume that \(x=(x_{k})\) is bounded and weighted \(\alpha \beta \)-statistically convergent of order \(\gamma \) to \(\ell \). Then for \(\epsilon >0 \), we have \(\delta ^{\alpha ,\beta }(K^{\alpha \beta }_{\epsilon },\gamma )=0\). Since \(x=(x_{k})\) is bounded, there exists \(M>0\) such that \(s_{k}|x_{k}-\ell |\le M\) for all \(k \in \mathbb {N}\).
$$\begin{aligned} \frac{1}{S_{n}^{\gamma }}\sum _{k\in P^{\alpha ,\beta }_{n}}s_{k}|x_{k}-\ell |&= \frac{1}{S_{n}^{\gamma }}\sum _{\begin{array}{c} k\in P^{\alpha ,\beta }_{n} \\ s_{k}|x_{k}-\ell |\ge \varepsilon \end{array}}s_{k}|x_{k}-\ell |+\frac{1}{S_{n}^{\gamma }}\sum _{\begin{array}{c} k\in P^{\alpha ,\beta }_{n} \\ s_{k}|x_{k}-\ell |< \varepsilon \end{array}}|x_{k}-\ell |\\&\le \frac{M}{S_{n}^{\gamma }}\left| \left\{ k\in P^{\alpha ,\beta }_{n}:s_{k}|x_{k}-\ell |\ge \epsilon \right\} \right| +\varepsilon , \end{aligned}$$this implies that \(x_{k}\longrightarrow [\overline{N}^{\gamma }_{\alpha \beta },s]\).
\(\square \)
3 Application to Korovkin type approximation
In this section, we get an analogue of classical Korovkin Theorem by using the concept of \(\alpha \beta \)-statistical convergence. Also we estimate, in rates of \(\alpha \beta \)-statistical convergence. Recently, such types of approximation theorems are proved, [1, 3, 4, 11, 13, 14].
Let \(C[a,b]\) be the linear space of all real-valued continuous functions \(f\) on \([a,b]\) and let \(L\) be a linear operator which maps \(C[a,b]\) into itself. We say \(L\) is positive operator, if for every non-negative \(f\in C[a,b]\), we have \(L(f,x)\ge 0\) for \(x\in [a,b]\). It is well-known that \(C[a,b]\) is a Banach space with the norm given by
The classical Korovkin approximation theorem states as follows (see [7, 9])
where \(e_{i}=x^{i}\) and \(f\in C[a,b]\).
Theorem 3.1
Let \((L_{k})\) be a sequence of positive linear operator from \(C[a,b]\) in to \(C[a,b]\). Then for all \(f\in C[a,b]\)
if and only if
Proof
Because of \(e_{i}\in C[a,b]\) for \((i=0,1,2)\), conditions (3.2)–(3.4) follow immediately from (3.1). Let the conditions (3.2)–(3.4) hold and \(f\in C[a,b]\). By the continuity of \(f\) at \(x\), it follows that for given \(\varepsilon > 0\) there exists \(\delta \) such that for all \(t\)
Since \(f\) is bounded, we get
Hence
By using (3.5) and (3.6), we have
This implies that
By using the positivity and linearity of \((L_{k})\), we get
where \(x\) is fixed and so \( f(x)\) is constant number. Therefore,
On the other hand
By inequality (3.7) and (3.8), we obtain
Now, we compute second moment
By (3.9), we have
Because of \(\varepsilon \) is arbitrary, we obtain
where \(R=\max \left( \varepsilon +M+\frac{2Mb^{2}}{\delta ^{2}},\frac{4Mb}{\delta ^{2}}\right) \).
For \(\varepsilon ^{\prime }>0\), we can write
Then, \(\mathcal {C}\subset \mathcal {C}_{1}\cup \mathcal {C}_{2}\cup \mathcal {C}_{3}\), so we have \(\delta ^{\alpha ,\beta }(\mathcal {C},\gamma )\le \delta ^{\alpha ,\beta }(\mathcal {C}_{1},\gamma )+\delta ^{\alpha ,\beta }(\mathcal {C}_{2},\gamma )+\delta ^{\alpha ,\beta }(\mathcal {C}_{3},\gamma )\). Thus, by conditions (3.2)–(3.4), we obtain
which completes the proof. \(\square \)
We remark that our Theorem 3.1 is stronger than that of classical Korovkin approximation theorem. For this claim, we consider the following example:
Example 1
Considering the sequence of Bernstein operators
We define the sequence of linear operators as \(T_{n}:C[0,1]\longrightarrow C[0,1]\) with \(T_{n}(f,x)=(1+x_{n})B_{n}(f,x)\), where \(x=(x_{n})\) is defined in (2.1). Now let \(s_{k}=1\) for all \(k\). Note that the sequence \(x=(x_{n})\) is weighted \(\alpha \beta \)-statistically convergent but not convergent. Then, \(B_{n}(1,x)=1\), \(B_{n}(t,x)=x\) and \(B_{n}(t^{2},x)=x^{2}+\frac{x-x^{2}}{n}\) and sequence \((T_{n})\) satisfies the conditions (3.2)–(3.4). Therefore, we get
On the other hand, we have \(T_{n}(f,0)=(1+x_{n})f(0)\), since \(B_{n}(f,0)=f(0)\), thus we obtain
One can see that \((T_{n})\) does not satisfy the classical Korovkin theorem, since \(x=(x_{n})\) is not convergent.
4 Rate of weighted \(\alpha \beta \)-statistically convergent of order \(\gamma \)
In this section, we estimate the rate of weighted \(\alpha \beta \)-statistically convergent of order \(\gamma \) of a sequence of positive linear operator which is defined from \(C[a,b]\) into \(C[a,b]\) .
Definition 4.1
Let \((u_{n})\) be a positive non-increasing sequence. We say that the sequence \(x=(x_{k})\) is \(\alpha \beta \)-statistically convergent of order \(\gamma \) to \(\ell \) with the rate \(o(u_{n})\) if for every, \(\epsilon >0\)
At this stage, we can write \(x_{k}-\ell =\overline{S}^{\gamma }_{\alpha \beta }-o(u_{n})\).
Before proceeding further, let us give basic definition and notation on the concept of the modulus of continuity.
The modulus of continuity of \(f\), \(\omega (f,\delta )\) is defined by
It is well-known that for a function \(f \in C[a,b]\),
for any \(\delta >0\)
Theorem 4.1
Let \((L_{k})\) be sequence of positive linear operator from \(C[a,b]\) into \(C[a,b]\). Assume that
Then for all \(f\in C[a,b]\), we get
where \(z_{n}= \max \{u_{n},v_{n}\}\).
Proof
Let \(f\in C[a,b]\) and \(x\in [a,b]\). From (3.8) and (4.1), we can write
By choosing \(\sqrt{\psi _{k}}=\delta \), we get
where \(H=\max \{2,\parallel f\parallel _{C[a,b]}\}\). By Definition 4.1 and conditions (i)–(ii), we get the desired the result. \(\square \)
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We thank the referees for their careful reading of the original manuscript and for the valuable comments.
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Communicated by S. K. Jain.
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Karakaya, V., Karaisa, A. Korovkin type approximation theorems for weighted \(\alpha \beta \)-statistical convergence . Bull. Math. Sci. 5, 159–169 (2015). https://doi.org/10.1007/s13373-015-0065-y
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DOI: https://doi.org/10.1007/s13373-015-0065-y
Keywords
- Korovkin type theorems
- Weighted \(\alpha \beta \)-summability
- Rate of the weighted \(\alpha \beta \)-statistical convergent
- Positive linear operator