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

$${x_{k}} = \textstyle\begin{cases} 1,& k = {m^{2}}, \\ 0,& k \ne{m^{2}}, \end{cases}\displaystyle \quad m = 1,2,3,\ldots. $$

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

$${\delta^{\alpha,\beta}}(K,\gamma) = \lim_{n \to\infty} \frac {{ \vert {K \cap P_{n}^{\alpha,\beta}} \vert }}{{{{ ( {\beta(n) - \alpha(n) + 1} )}^{\gamma}}}}, $$

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 \(\varepsilon > 0\),

$${\delta^{\alpha,\beta}}\bigl(\bigl\{ k: \vert {{x_{k}} - L} \vert \ge \varepsilon\bigr\} ,\gamma\bigr) = \lim_{n \to\infty} \frac{{ \vert { \{ {k \in P_{n}^{\alpha,\beta}: \vert {{x_{k}} - L} \vert \ge\varepsilon } \}} \vert }}{{{{ ( {\beta(n) - \alpha(n) + 1} )}^{\gamma}}}} = 0. $$

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 \(\varepsilon>0\),

$${\delta^{\alpha,\beta}}\bigl(\bigl\{ k: \bigl\Vert f_{k}(x)-f(x) \bigr\Vert _{C(X)} \ge \varepsilon\bigr\} ,\gamma\bigr) = \lim _{n \to\infty} \frac{{ \vert { \{ {k \in P_{n}^{\alpha,\beta}: \Vert f_{k}(x)-f(x) \Vert _{C(X)} \ge\varepsilon} \}} \vert }}{{{{ ( {\beta(n) - \alpha(n) + 1} )}^{\gamma}}}} = 0. $$

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

$$L_{n}(e_{v},x)\rightrightarrows f \bigl(\alpha \beta^{\gamma}\textit{-stat}\bigr),\quad v=0,1,2. $$

Then for all \(f\in C(X)\),

$$L_{n}(f,x)\rightrightarrows f \bigl(\alpha\beta^{\gamma} \textit{-stat}\bigr). $$

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

$$ {L_{n}}(f;x) = \sum_{k = 0}^{\infty}{{\phi_{n,k}}(x)f({x_{n,k}})},\quad x\geq 0, f\in F, $$
(1)

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 \(n\in\mathbb{N}\):

  1. (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.

  2. (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)
  3. (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:

$$\begin{aligned}& \zeta_{n,0}(x)=1, \qquad \zeta_{n,1}(x)=0, \end{aligned}$$
(4)
$$\begin{aligned}& \zeta_{n,r+1}(x)=\psi(n,x) \bigl(\zeta'_{n,r}(x)+r \zeta_{n,r+1}(x)\bigr), \quad r\in \mathbb{N}, \end{aligned}$$
(5)
$$\begin{aligned}& \zeta_{n,2}(x)=\psi(n,x). \end{aligned}$$
(6)

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

$$\begin{aligned}& L_{n}(e_{0};x)=\sum_{k = 0}^{\infty}\phi_{n,k}=e_{0}, \\& L_{n}(e_{1};x)=\sum_{k = 0}^{\infty}\phi_{n,k}x_{n,k}=e_{1}, \end{aligned}$$

we can obtain that

$$\begin{aligned}& \mathrm{st}_{\alpha\beta}^{\gamma}\mbox{-}\! \lim_{n \to\infty} \bigl\Vert {L_{n}}(e_{0};x)-e_{0} \bigr\Vert =0, \\& \mathrm{st}_{\alpha\beta}^{\gamma}\mbox{-}\! \lim_{n \to\infty} \bigl\Vert {L_{n}}(e_{1};x)-e_{1} \bigr\Vert =0. \end{aligned}$$

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

$$ \begin{aligned} &x_{n,k}=\frac{k}{a_{n}}\leq k,\quad k\in\mathbb{N}, \mbox{with } \lim_{n \to\infty} \frac{1}{{{a_{n}}}} = 0, \\ &\psi(n,x)=\sum_{i = 1}^{l} { \frac{{{\psi_{i}}(x)}}{{a_{n}^{i}}}},\quad x\geq0. \end{aligned} $$
(7)

Under these assumptions, the requirement of Theorem 2.1 is fulfilled. Starting from (1), under the additional assumptions (7), Agratini defined

$$ L_{n,\delta}(f;x)=\sum_{k = 0}^{[a_{n}(x+\delta(n))]} \phi_{n,k}f\biggl(\frac {k}{a_{n}}\biggr), \quad x\geq0, f\in F, $$
(8)

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

$$ \zeta_{n,2m}(x)\leq\frac{C(m,K)}{a_{n}},\quad x\in K, $$
(9)

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

$$ t^{2m}\leq2^{2m-1}\bigl(x^{2m}+(t-x)^{2m} \bigr),\quad t\geq0, x\geq0, m\in\mathbb{N}. $$
(10)

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

$$\biggl\vert {f \biggl( {\frac{k}{{{a_{n}}}}} \biggr)} \biggr\vert \le{g_{m}}(x) + {2^{2m - 1}}B{ \biggl( {\frac{k}{{{a_{n}}}} - x} \biggr)^{2m}},\quad k\in\mathbb {\mathbb{N}}_{0}, x\geq0. $$

Since \(x, \delta(n)\), and \(a_{n}\) are positive, if \(k\geq[a_{n}(x+\delta (n))]+1\), then

$$\frac{k}{{{a_{n}}}} \ge x. $$

So we can write

$$\bigl\{ k\in\mathbb{N}_{0}: k\geq\bigl[a_{n}\bigl(x+ \delta(n)\bigr)\bigr]+1\bigr\} \subset\biggl\{ k\in\mathbb {N}_{0}: \biggl\vert {\frac{k}{{{a_{n}}}} - x} \biggr\vert >\delta(n)\biggr\} :=I_{n,x,\delta}. $$

Let \({R_{n}}: = {L_{n}} - {L_{n,\delta}}\). Thus it follows that

$$\begin{aligned} \bigl\vert {{R_{n}}(f;x)} \bigr\vert =& \Biggl\vert {\sum _{k = [{a_{n}}(x + \delta (n))] + 1}^{\infty}{{\phi_{n,k}}f \biggl( { \frac{k}{{{a_{n}}}}} \biggr)} } \Biggr\vert \\ \le&\sum_{k = [{a_{n}}(x + \delta(n))] + 1}^{\infty}{{\phi_{n,k}} \biggl[ {{g_{m}}(x) + {2^{2m - 1}}B{{ \biggl( {\frac{k}{{{a_{n}}}} - x} \biggr)}^{2m}}} \biggr]} \\ \le&\sum_{k \in{I_{n,x,\delta}}}^{\infty}{{ \phi_{n,k}}(x){g_{m}}(x) + {2^{2m - 1}}B\sum _{k \in{I_{n,x,\delta}}}^{\infty}{{\phi _{n,k}}(x){{ \biggl( { \frac{k}{{{a_{n}}}} - x} \biggr)}^{2m}}} } \\ \le&{g_{m}}(x)\frac{1}{{{\delta^{2m}} ( n )}}\sum_{k = 0}^{\infty}{{\phi_{n,k}}(x){{ \biggl( {\frac{k}{{{a_{n}}}} - x} \biggr)}^{2m}} + } {2^{2m - 1}}B\sum_{k = 0}^{\infty}{{\phi_{n,k}}(x){{ \biggl( {\frac{k}{{{a_{n}}}} - x} \biggr)}^{2m}}} \\ =& {g_{m}}(x)\frac{1}{{{\delta^{2m}} ( n )}}{\zeta_{n,2m}}(x) + {2^{2m - 1}}B{\zeta_{n,2m}}(x). \end{aligned}$$

Using \({\zeta_{n,2m}}(x) \le\frac{{C ( {m,K} )}}{{a_{n}^{m}}}\), we get

$$ \bigl\vert {{R_{n}}(f;x)} \bigr\vert \le \biggl( {{g_{m}}(x)\frac{1}{{{\delta ^{2m}} ( n )}} + {2^{2m - 1}}B} \biggr) \frac{{C(m,K)}}{{a_{n}^{m}}}. $$
(11)

Taking the norm on K, we have

$$\bigl\Vert {{R_{n}}(f;x)} \bigr\Vert _{C(K)} \le \Vert {{g_{m}}} \Vert C(m,K){ \biggl[ {\frac{1}{{\sqrt{{a_{n}}} \delta(n)}}} \biggr]^{2m}} + {2^{2m - 1}}B\frac{{C(m,K)}}{{a_{n}^{m}}}. $$

For a given \(\varepsilon>0\), define the sets

$$\begin{aligned}& A:= \bigl\{ {k \le P_{n}^{\alpha,\beta}: } \bigl\Vert R_{k}(f,x) \bigr\Vert \geq\varepsilon \bigr\} , \\& {A_{1}}: = \biggl\{ {k \le P_{n}^{\alpha,\beta}: {{ \bigl( {\sqrt{{a_{k}}} \delta(k)} \bigr)}^{ - 2m}} \ge \frac{\varepsilon}{{2 \Vert {{g_{m}}} \Vert C(m,K)}}} \biggr\} , \end{aligned}$$

and

$${A_{2}}: = \biggl\{ {k \le P_{n}^{\alpha,\beta}: {a_{k}}^{ - m} \ge\frac {\varepsilon}{{{2^{2m}}BC(m,K)}}} \biggr\} . $$

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

$${\delta^{\alpha,\beta}}\bigl(\bigl\{ k: \vert {{x_{k}} - L} \vert \ge \varepsilon\bigr\} ,\gamma\bigr) = \lim_{n \to\infty} \frac{{ \vert { \{ {k\leq n: \vert {{x_{k}} - L} \vert \ge\varepsilon} \}} \vert }}{{n}}. $$

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.