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On the αβ-statistical convergence of the modified discrete operators
Advances in Difference Equations volume 2018, Article number: 252 (2018)
Abstract
The concept of αβ-statistical convergence was introduced by Aktuğlu (J. Comput. Appl. Math. 259:174–181, 2014). In this paper, we apply αβ-statistical convergence to investigate modified discrete operator approximation properties.
1 Introduction and preliminaries
The notion of statistical convergence was introduced by Fast [2] and Steinhaus [3] independently in the same year 1951 as follows.
Let K \(\subset\mathbb{N} \) and \(K_{n}=\{k\leq n:k\in K\}\). Then the natural density of K is defined by \(\delta(K)=\lim_{n}\frac {|K_{n}|}{n} \) if the limit exists, where \(|K_{n}|\) denotes the cardinality of \(K_{n}\).
A sequence \(x=(x_{k})\) is said to be statistically convergent to L if for every \(\varepsilon> 0\), \(\delta\{k \in\mathbb{N} :|x_{k}-L|\geq \varepsilon\}=0\) or \(\lim_{n} \frac{{ \vert { \{ {k \le n: \vert {{x_{k}} - L} \vert \ge\varepsilon} \}} \vert }}{n} = 0\). We write \(\mbox{st-}\!\lim x_{k} =L\).
Statistical convergence is a generalization of concept of ordinary convergence. So, every convergent sequence is statistically convergent, but not conversely. For example, let
Then, \(\mbox{st-}\!\lim{x_{k}} = 0\), but \((x_{k})\) is not convergent.
Approximation theory has important applications in the theory of polynomial approximation, various areas of functional analysis, and numerical solutions of differential and integral equations. In the recent years, with the help of the concept of statistical convergence, various statistical approximation results have been proved.
Gadjiev and Orhan [4] studied a Korovkin-type approximation theorem by using the notion of statistical convergence for the first time in 2002. Later, generalizations and applications of this concept have been investigated by various authors [5–11].
Aktuğlu [1] introduced αβ-statistical convergence as follows. Let \(\alpha(n)\) and \(\beta(n)\) be two sequences of positive numbers satisfying the following conditions:
- \({P_{1}}\)::
-
α and β are both nondecreasing,
- \({P_{2}}\)::
-
\(\beta(n) \ge\alpha(n)\),
- \({P_{3}}\)::
-
\(\beta(n) - \alpha(n) \to\infty\) as \(n \to\infty\).
Let Λ denote the set of pairs \(( {\alpha,\beta} )\) satisfying \({P_{1}}\), \({P_{2}}\), \({P_{3}}\).
For a pair \(( {\alpha,\beta} ) \in\Lambda\), \(0 < \gamma \le1\), and \(K \subset\mathbb{N}\), we define
where \(P_{n}^{\alpha,\beta}\) is the closed interval \([ {\alpha (n), \beta(n)} ]\), and \(\vert S \vert \) represents the cardinality of S.
Definition 1.1
([1])
A sequence x is said to be αβ-statistically convergent of order γ to L, denoted by \(\mathrm{st}_{\alpha\beta}^{\gamma}\mbox{-}\! \lim_{n \to\infty} {x_{n}} = L\) if for every \(\varepsilon > 0\),
For \(\gamma = 1\), we say that x is αβ-statistically convergent to L, and this is denoted by \(\mathrm{st}_{\alpha\beta}\mbox{-}\!\lim_{n \to\infty} {x_{n}} = L\).
Let X be a compact subset of \(\mathbb{R}\), and let \(0<\gamma\leq1\); then we can consider the following definition for a sequence of functions \(f_{r}:X\rightarrow\mathbb{R}\).
Definition 1.2
A sequence of functions \(f_{r}\) is said to be αβ-statistically uniformly convergent to f on X of order γ and denoted by \(f_{k}\rightrightarrows f (\alpha\beta^{\gamma}\mbox{-stat})\) if for every \(\varepsilon>0\),
Theorem 1.3
([1])
Let \((\alpha,\beta)\in\Lambda\), \(0<\gamma\leq1\), and let \(L_{n}:C(X)\rightarrow C(X)\) be a sequence of positive linear operators satisfying
Then for all \(f\in C(X)\),
Throughout this paper, K represents a compact subinterval of \(\mathbb {R^{+}}\), and \(e_{j}\) stands for \(e_{j}(t)=t^{j}\), \(j\in\mathbb {N}_{0}=\{0\}\cup\mathbb{N}\).
Agratini [12] investigated a general class of positive approximation processes of discrete type expressed by series and modified them into finite sums. Agratini [12] defined the operator
where F stands for the domain of \(L_{n}\) containing the set of all continuous functions on \(\mathbb{R^{+}}\) for which the series in (1) is convergent, by using the following three requirements:
For each \(n\in\mathbb{N}\):
-
(i)
For every \(k\in\mathbb{N}_{0}\), there exists a sequence of \(\gamma _{k}\) such that \(x_{n,k}=O(n^{-\gamma_{k}})\) (\(n\rightarrow\infty\)) a net on \(\mathbb{R^{+}}\), \(\Delta_{n}=(x_{n,k})_{k\geq0}\) is fixed.
-
(ii)
A sequence \((\phi_{n,k})_{k\geq0}\) is given, where \(\phi_{n,k}\in C'(\mathbb{R^{+}})\) and \(C'(\mathbb{R^{+}})\) is the space of all real-valued functions continuously differentiable in \(\mathbb{R^{+}}\). This sequence satisfies the following conditions:
$$ \phi_{n,k}\geq0, k\in\mathbb{N}_{0},\quad \sum _{k = 0}^{\infty}{{\phi_{n,k}}(x)} = {e_{0}},\qquad \sum_{k = 0}^{\infty}{{ \phi_{n,k}}(x)} {x_{n,k}} = {e_{1}}. $$(2) -
(iii)
There exists a positive function \(\psi\in\mathbb{\mathbb {R^{\mathbb{N}\times\mathbb{R}^{+}}}}\), \(\psi\in C(\mathbb{R^{+}})\), with the property
$$ \psi(n,x)\phi'_{n,k}(x)=(x_{n,k}-x) \phi_{n,k}(x),\quad k \in\mathbb{N_{0}}, x\geq0. $$(3)
Agratini [12] indicated the following technical result.
Lemma 1.4
Let \({L_{n}}(f;x) = \sum_{k = 0}^{\infty}{{\phi _{n,k}}(x)f({x_{n,k}})}\), \(x\geq0\), \(f\in{F}\), and let \(\zeta _{n,r}\) be the rth central moment of \(L_{n}\). For every \(x\in\mathbb {R^{+}}\), we have the following identities:
In this paper, we present αβ-statistical convergence approximation properties of the operator investigated by Agratini [12].
2 Main results
Theorem 2.1
Let \(L_{n}(f;x)=\sum_{k = 0}^{\infty}\phi_{n,k}f(x_{n,k})\). If \(\textit{st}_{\alpha\beta}^{\gamma}\textit{-}\! \lim_{n \to\infty}\psi(n,x) = 0\) uniformly on K, then for every \(f\in F\), we have \(\textit{st}_{\alpha\beta}^{\gamma}\textit{-}\! \lim_{n \to\infty} \|{L_{n}}(f;x)-f(x)\|=0\).
Proof
Due to
we can obtain that
We know from [12] that \(\zeta _{n,r}(x)=L_{n}((e_{1}-xe_{0})^{r};x)\), \(r\in\mathbb{N}_{0}\). If we choose \(r=2\), then we can write \(\psi(n,x)=\zeta _{n,2}(x)=L_{n}((e_{1}-xe_{0})^{2};x)\). Since \(L_{n}\) is a linear operator, we can easily see that \(\psi(n,x)=L_{n}(e_{2};x)-x^{2}\). So, \(\|\psi(n,x)\|_{C(K)}=\|L_{n}(e_{2};x)-x^{2}\|_{C(K)}\). Since \(\mathrm{st}_{\alpha\beta}^{\gamma}\mbox{-}\! \lim_{n \to\infty}\psi(n,x) = 0\) uniformly on K, we have \(\|\psi(n,x)\|_{C(K)}=\|L_{n}(e_{2};x)-x^{2}\| _{C(K)}\). Then from Theorem 1.3 we obtain that \(\mathrm{st}_{\alpha\beta }^{\gamma}\mbox{-}\! \lim_{n \to\infty} \|{L_{n}}(f;x)-f(x)\|_{C(K)}=0\). □
We give some information to investigate αβ-statistical approximation properties of modified discrete operators defined by Agratini [12]. If we specialize the net \(\Delta_{n}\) and the function ψ, then we consider that a positive sequence \((a_{n})_{n\geq1}\) and the function \(\psi_{i}\in C(\mathbb{R}^{+})\), \(i=1,2,\ldots,l\), exist such that, for every \(n\in\mathbb{N}\), we have
Under these assumptions, the requirement of Theorem 2.1 is fulfilled. Starting from (1), under the additional assumptions (7), Agratini defined
where \(\delta=(\delta(n))_{n\geq1}\) is a sequence of positive numbers. The study of these operators was developed in polynomial weighted spaces connected to the weights \(\omega_{m}\), \(m\in\mathbb{N}_{0}\), \(\omega_{m}(x)=\frac{1}{1+x^{2m}}\), \(x\geq0\). For every \(m\in\mathbb {N}_{0}\), the spaces \(E_{m}:=\{f\in\mathbb {C}({R^{+}}):\|f\|_{m}:=\sup_{x\geq0}\omega _{m}(x) \vert f(x) \vert <\infty\}\) are endowed with the norm \(\|\cdot\|_{m}\).
Lemma 2.2
([12])
Let \(L_{n}\), \(n\in\mathbb{N}\), be defined by (1), and let assumptions (7) be fufilled. If \(\psi_{i}\in C^{2m-2}(\mathbb{R^{+}})\), \(i=1,2,3,\ldots,l\), then the central moment of \((2m)\)th order satisfies
where \(C(m,K)\) is a constant depending only on m and the compact set \(K\subset\mathbb{R}^{+}\).
Theorem 2.3
Let \(L_{n,\delta}(f;x)=\sum_{k = 0}^{[a_{n}(x+\delta (n))]}\phi_{n,k}f(\frac{k}{a_{n}})\) be defined by [13] If \(\psi _{i}\in C^{2m-2}(\mathbb{R}^{+})\), \(i=1,2,\ldots,l\), and \(\textit{st}_{\alpha\beta }^{\gamma}\textit{-}\! \lim_{n \to\infty}\sqrt{a_{n}}\delta(n) = 0\), then \(\textit{st}_{\alpha\beta}^{\gamma}\textit{-}\! \lim_{n \to\infty} \|{L_{n,\delta }}(f;x)-f(x)\|_{C(K)}=0\), for every \(f\in E_{m}\cap F\).
Proof
To prove Theorem 2.3, we need the elementary inequality
On the other hand, for \(f\in E_{m}\), there exist constants \(A,B\in \mathbb{R}^{+}\) and \(m\in\mathbb{N}\) such that \(|f|\leq A+Bt^{2m}\). Thus, using (1), we get \(|f(t)|\leq A+B(2^{2m-1}(x^{2m}+(t-x)^{2m}))=A+B2^{2m-1}x^{2m}+B2^{2m-1}(t-x)^{2m} =g_{m}(x)+2^{2m-1}B(t-x)^{2m}\), where \(g_{m}:=A+B2^{2m-1}e^{2m}\). Then
Since \(x, \delta(n)\), and \(a_{n}\) are positive, if \(k\geq[a_{n}(x+\delta (n))]+1\), then
So we can write
Let \({R_{n}}: = {L_{n}} - {L_{n,\delta}}\). Thus it follows that
Using \({\zeta_{n,2m}}(x) \le\frac{{C ( {m,K} )}}{{a_{n}^{m}}}\), we get
Taking the norm on K, we have
For a given \(\varepsilon>0\), define the sets
and
Then from (8) we clearly have \(A \subset{A_{1}} \cup{A_{2}}\) and \({\delta ^{\alpha,\beta}}(A;\gamma) \le{\delta^{\alpha,\beta }}({A_{1}};\gamma) + {\delta^{\alpha,\beta}}({A_{2}};\gamma)\). Since \(\mathrm{st}_{\alpha\beta}^{\gamma}\mbox{-}\! \lim_{n \to\infty} \sqrt{{a_{n}}} \delta(n) = \infty\) and \(\mathrm{st}_{\alpha\beta}^{\gamma}\mbox{-}\! \lim_{n \to \infty} {a_{n}}^{ - 1} = 0\), the proof is complete. □
If we take \(\alpha(n)=1\), \(\beta(n)=n\), and \(\gamma=1\), then
Therefore, if we take \(\alpha(n)=1\), \(\beta(n)=n\), and \(\gamma=1\), then αβ-statistical convergence reduces to statistical convergence. Thus, Theorems 2.1 and 2.3 reduce to Theorems 1 and 2 of [14], respectively.
References
Aktuğlu, H.: Korovkin type approximation theorems proved via αβ-statistical convergence. J. Comput. Appl. Math. 259, 174–181 (2014)
Fast, H.: Sur la convergence statistique. Colloq. Math. 2, 241–244 (1951)
Steinhaus, H.: Sur la convergence ordinaire et la convergence asymptotique. Colloq. Math. 2, 73–74 (1951)
Gadjiev, A.D., Orhan, C.: Some approximation theorems via statistical convergence. Rocky Mt. J. Math. 32, 129–138 (2002)
Mohiuddine, S.A., Alotaibi, A., Hazarika, B.: Weighted A-statistical convergence for sequences of positive linear operator. Sci. World J. (2014). https://doi.org/10.1155/2014/437863
Karakaya, V., Karaisa, A.: Korovkin type approximation theorems for weighted αβ-statistical convergence. Bull. Math. Sci. 5, 159–169 (2015)
Mohiuddine, S.A.: An application of almost convergence in approximation theorems. Appl. Math. Lett. 23, 1382–1387 (2010)
Duman, O., Orhan, C.: Statistical approximation by positive linear operators. Stud. Math. 161, 187–197 (2004)
Dirik, F., Demirci, K.: Korovkin type approximation theorem for functions of two variables in statistical sense. Turk. J. Math. 34, 73–83 (2010)
Kiriṣci, M., Karaisa, A.: Fibonacci statistical convergence and Korovkin type approximation theorems. J. Inequal. Appl. 2017, 229 (2017)
Kadak, U.: Weighted statistical convergence based on generalized difference operator involving \((p, q)\)-gamma function and its applications to approximation theorems. J. Math. Anal. Appl. 448, 1633–1650 (2017)
Agratini, O.: On the convergence of truncated class of operators. Bull. Inst. Math. Acad. Sin. 3, 213–223 (2003)
Çakar, Ö., Gadjiev, A.D.: On uniform approximation by Bleimann, Butzer and Hahn on all positive semiaxis. Trans. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. 19, 21–26 (1999)
Canatan, R.: A note on the statistical approximation properties of the modified discrete operators. Open J. Discrete Math. 2, 114–117 (2012)
Acknowledgements
The first author would like thank TUBITAK (The Scientific and Technological Research Council of Turkey) for financial support during her doctorate studies.
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Abay Tok, M., Kara, E.E. & Altundaǧ, S. On the αβ-statistical convergence of the modified discrete operators. Adv Differ Equ 2018, 252 (2018). https://doi.org/10.1186/s13662-018-1661-4
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DOI: https://doi.org/10.1186/s13662-018-1661-4
MSC
- 40A05
- 41A25
- 41A36
Keywords
- Korovkin-type theorems
- αβ-statistical convergence
- Modified discrete operator