1 INTRODUCTION

The classical Korovkin type approximation theory provides a simple criteria whether a sequence of a positive linear operators on \(C[0,1]\), the space of all real continuous functions defined on the closed interval \([0,1]\), converges to the identity operator or not [25]. This theory has been studied by many mathematicians with various motivations such as extending the interval to whole real line, relaxing the continuity of the functions, relaxing the positivity of the operators or considering some special sequences of positive linear operators (see e.g., [1, 2]). Another motivation is to introduce the summability theory whenever a sequence of positive linear operator is not convergent in the ordinary sense. Actually, the main aim of the summability theory is to make a non-convergent sequence or series to converge in a more general sense. Therefore, summability theory has many applications in probability limit theorems, approximation theory with positive linear operators, and differential equations, whenever the ordinary limit does not exist (see, [5, 21, 22, 26]). Gadjiev and Orhan [21] proved a Korovkin type theorem by considering statistical convergence instead of ordinary convergence. Following that study many authors have given several approximation results via summability theory (see, e.g., [4, 8, 12, 13, 30, 33, 34, 42]).

Let \(x=\left(x_{j}\right)\) be a real sequence and let \(A=(a_{nj})\) be a summability matrix. If the sequence \(\left\{\left(Ax\right)_{n}\right\}\) is convergent to a real number \(L\), then we say that the sequence \(x\) is \(A\) -summable to the real number \(L\), where the series\((Ax)_{n}:=\sum_{j=0}^{\infty}a_{nj}x_{j}\) is convergent for any \(n\in\mathbb{N}_{0}\) and \(\mathbb{N}_{0}=\{0,1,...\}\). A summability matrix \(A\) is said to be regular if \(\lim(Ax)_{n}=L\) whenever \(\lim x=L\) (see, [9]). In fact, any summability method is said to be regular if it preserves ordinary limit.

Let \(A=(a_{nj})\) be a non-negative regular summability matrix and let \(E\subset\mathbb{N}\). Then the number

$$\delta_{A}(E):=\lim\sum_{j\in E}a_{nj}$$

is said to be the \(A\) -density of \(E\) whenever the limit exists (see, [10, 18, 24]). Regularity of the summability matrix \(A\) ensures that \(0\leq\delta_{A}(E)\leq 1\) whenever \(\delta_{A}(E)\) exists. If we consider \(A=C\), the Cesàro matrix, then \(\delta(E):=\delta_{C}(E)\) is called the (natural or asymptotic) density of \(E\) (see, [17]), where \(C=(c_{nj})\) is the summability matrix defined by

$$c_{nj}=\begin{cases}1/(n+1),\quad\text{if }\quad j\leq n,\\ 0,\quad\text{otherwise.}\end{cases}$$

A real sequence \(x=\left(x_{j}\right)\) is said to be \(A\) -statistically convergent (see, [11, 19]) to a real number \(L\) if for any \(\varepsilon>0\),

$$\delta_{A}(\left\{j\in\mathbb{N}_{0}:\left|x_{j}-L\right|\geq\varepsilon\right\})=0.$$

In this case we write \(st_{A}-\lim x=L\). If we consider the Cesàro matrix, then \(C\)-statistical convergence is called statistical convergence [16, 31, 36]. In general, \(A\)-statistical convergence is regular and there exists some sequences which are \(A\)-statistically convergent but not ordinary convergent.

Now, we recall the concept of power series (summability) method [9]. Let \(\left(p_{j}\right)\) be a real sequence with \(p_{0}>0\) and \(p_{1},p_{2},...\geq 0,\) and such that the corresponding power series \(p(t):={\displaystyle\sum\limits_{j=0}^{\infty}}p_{j}t^{j}\) has radius of convergence \(R\) with \(0<R\leq\infty.\) If

$$\lim_{0<t\rightarrow R^{-}}\frac{1}{p(t)}\sum_{j=0}^{\infty}x_{j}p_{j}t^{j}=L,$$

then we say that \(x=\left(x_{j}\right)\) is convergent in the sense of power series method (\(P_{p}\) convergent). This summability method is a general version of Abel and Borel summability methods. Korovkin type theorems related to these methods can be found in [4, 6, 7, 37, 38, 41, 42].

The following theorem characterizes the regularity of a power series method.

Theorem 1 [9]. A power series method \(P_{p}\) is regular if and only if for any \(j\in\mathbb{N}_{0}\)

$$\lim_{0<t\rightarrow R^{-}}\frac{p_{j}t^{j}}{p(t)}=0.$$

Ünver and Orhan [43] introduced the concept of power series statistical convergence which is stronger than the ordinary convergence. Power series statistical convergence is defined with the help of the concept of density with respect to the power series methods.

Definition 1 [43]. Let \(P_{p}\) be a regular power series method and let \(E\subset\mathbb{N}_{0}.\) If

$$\delta_{P_{p}}\left(E\right):=\lim_{0<t\rightarrow R^{-}}\frac{1}{p(t)}\sum_{j\in E}^{\infty}p_{j}t^{j}$$

exists then \(\delta_{P_{p}}\left(E\right)\) is called the \(P_{p}\) -density of \(E\) . It is easy to see that \(0\leq\delta_{P_{p}}\left(E\right)\leq 1\) whenever \(\delta_{P_{p}}\left(E\right)\) exists.

Definition 2 [43]. Let \(x=(x_{j})\) be a sequence and let \(P_{p}\) be a regular power series method. Then \(x\) is said to be \(P_{p}\)-statistically convergent to \(L\) if for any \(\varepsilon>0\)

$$\lim_{0<t\rightarrow R^{-}}\frac{1}{p(t)}\sum_{\left|x_{j}-L\right|\geq\varepsilon}p_{j}t^{j}=0,$$

i.e., \(\delta_{P_{p}}\left(\{j\in\mathbb{N}_{0}:\left|x_{j}-L\right|\geq\varepsilon\}\right)=0\) . In this case we write \(st_{P_{p}}-\lim x=L.\)

The main goals of this paper are to give the rates of \(P_{p}\)-statistical convergence and to prove some approximation properties of \(q\)-Meyer–König and Zeller operators and Durrmeyer variant of the \(q\)-Meyer König and Zeller operators by considering \(P_{p}\)-statistical convergence. Both sequences of operators are constructed with the use of \(q\)- calculus.

Now, let us recall some notations from \(q\)-calculus [29]: for any fixed real number \(q>0\), the \(q\)-integer \(\left[j\right]\) is defined by

$$\left[j\right]:=\left[j\right]_{q}=\sum_{k=1}^{j}q^{k-1}=\begin{cases}\frac{1-q^{j}}{1-q},\quad q\neq 1,\\ j,\quad q=1,\end{cases}$$

where \(j\) is a positive integer and \(\left[0\right]=0,\) the \(q\)-factorial \(\left[j\right]!\) of \([j]\) is given with

$$\left[j\right]!:=\begin{cases}\prod\limits_{k=1}^{j}[k],\quad j=1,2,...,\\ 1,\quad j=0.\end{cases}$$

For integers \(j\geq r\geq 0,\) the \(q\)-binomial coefficient is defined by

$$\genfrac{[}{]}{0.0pt}{}{j}{r}_{q}=\frac{\left[j\right]_{q}!}{\left[r\right]_{q}!\left[j-r\right]_{q}!}$$

and \(q\)-shifted factorial is defined by

$$\left(t;q\right)_{j}:=\begin{cases}1,\quad j=0,\\ \prod\limits_{j=0}^{j-1}\left(1-tq^{j}\right),\quad j=1,2,...\end{cases}$$

Thomae [39] introduced the \(q\)-integral of function \(f\) defined on the interval \(\left[0,a\right]\) as follows:

$${\displaystyle\int\limits_{0}^{a}}f\left(t\right)d_{q}t:=a\left(1-q\right)\sum\limits_{j=0}^{\infty}f\left(aq^{j}\right)q^{j},\quad 0<q<1.$$

Finally, the \(q\)-beta function [39] is defined by

$$B_{q}\left(m,j\right)={\displaystyle\int\limits_{0}^{1}}t^{m-1}\left(qt;q\right)_{j-1}d_{q}t.$$

In [3], it can be found some different type operators which were constructed via \(q\)-calculus. In [40], \(q\)-Meyer–König and Zeller operators were defined by

$$M_{j}^{q}\left(f;x\right)=\begin{cases}\prod\limits_{m=0}^{j}\left(1-q^{m}x\right)\sum\limits_{k=0}^{\infty}f\left(\frac{\left[k\right]}{\left[j+k\right]}\right)\left[\begin{matrix}j+k\\ k\end{matrix}\right]x^{k},\quad x\in\left[0,1\right),\\ f\left(1\right),\quad x=1\end{cases}$$
(1.1)

for \(f\in C\left[0,1\right]\), \(x\in\left[0,1\right]\), \(j\in\mathbb{N}\) and \(q\in\left(0,1\right]\). In [13], a different modification of \(q\)-Meyer–König and Zeller operators were defined. In [23], Durrmeyer variant of the \(q\)-Meyer–König and Zeller operators were introduced for \(f\in C\left[0,1\right]\), \(x\in\left[0,1\right]\), \(j\in\mathbb{N}\) and \(\alpha>0,q\in\left(0,1\right]\) as follows:

$$D_{j}^{q}\left(f;x\right)=\begin{cases}{\displaystyle\sum\limits_{k=1}^{\infty}}M_{j}^{q}\left(x\right){\displaystyle\int\limits_{0}^{1}}\tfrac{1}{B_{q}\left(j,k\right)}t^{k-1}\left(qt;q\right)_{j-1}f\left(t\right)d_{q}t+m_{j,0,q}\left(x\right)f\left(0\right),\quad x\in\left[0,1\right)\\ f\left(1\right),\quad x=1,\end{cases}$$
(1.2)

where

$$m_{j,k,q}\left(x\right)=\left(x;q\right)_{j+1}\left[\begin{matrix}j+k\\ k\end{matrix}\right]x^{k}.$$

Statistical convergence of the operators (1.1) and (1.2) and different modifications of these operators were examined in [13, 14, 27, 32]. Using the Abel convergence some properties of the operators (1.1) and (1.2) were studied in [35]. In that work, the authors compare the conditions of Abel convergence results with the below classical conditions (1.3) which are necessary for ordinary convergence.

It is well known that if the classical conditions

$$\lim\limits_{j\rightarrow\infty}q_{j}=1\text{ and }\lim\limits_{j\rightarrow\infty}\frac{1}{\left[j\right]}=0$$
(1.3)

hold, then for each \(f\in C[0,1]\) the sequences \((M_{j}^{q}f)\) and \((D_{j}^{q}f)\) converge uniformly to \(f\) over \([0,1]\) (see [23, 40]). We use the norm of the Banach space \(B[0,1]\) defined for any \(f\in C[0,1]\) by \(||f||:=\sup_{0\leq x\leq 1}|f(x)|,\) where \(B[0,1]\) is the space of all bounded real functions defined over \([0,1]\).

We need the following known lemmas in our proofs:

Lemma 1 [40]. Let \(j\geq 3\) be a positive integer. Then the following hold for the operators (1.1):

$$M_{j}^{q}\left(e_{0};x\right)=1,$$
(1.4)
$$M_{j}^{q}\left(e_{1};x\right)=x,$$
(1.5)
$$x^{2}\leq M_{j}^{q}\left(e_{2};x\right)\leq\frac{x}{\left[j-1\right]}+x^{2},$$
(1.6)

where \(e_{i}(x)=x^{i}\) for \(i=0,1,2\).

Lemma 2 [23]. Let \(j\geq 3\) be a positive integer. Then the following hold for the operators (1.2):

$$D_{j}^{q}\left(e_{0};x\right)=1,$$
(1.7)
$$D_{j}^{q}\left(e_{1};x\right)=x,$$
(1.8)
$$D_{j}^{q}\left(e_{2};x\right)=x^{2}+\frac{\left[2\right]x\left(1-x\right)\left(1-q^{j}x\right)}{\left[j-1\right]}-E_{j,q}\left(x\right),$$
(1.9)

where

$$0\leq E_{j,q}\left(x\right)\leq\frac{x\left[2\right]\left[3\right]q^{j-1}}{\left[j-1\right]\left[j-2\right]}\left(1-x\right)\left(1-qx\right)\left(1-q^{j}x\right).$$

For the operators (1.2) the following lemma was given in [35].

Lemma 3. Let \(j\geq 3\) be a positive integer. Then we have

$$D_{j}^{q}\left(e_{2};x\right)-x^{2}\geq\frac{\left[2\right]x\left(1-x\right)\left(1-q^{j}x\right)}{\left[j-1\right]}-\frac{x\left[2\right]\left[3\right]q^{j-1}}{\left[j-1\right]\left[j-2\right]}\left(1-x\right)\left(1-qx\right)\left(1-q^{j}x\right)\geq 0.$$

Note that in Lemma 1 and Lemma 2 the results were obtained for \(j\geq 3.\) Like ordinary convergence and statistical convergence, an exclusion of a finite number of terms does not affect the \(P_{p}\)-statistical convergence of a sequence since the \(P_{p}\)-density of a finite set is equal to zero.

The remainder of this paper is organized as follows: In Section 2, we study the rates of \(P_{p}\)-statistical convergence by means of modulus of continuity and class of Lipschitz functions. Furthermore, we give some examples to prove the results obtained in this paper are stronger than some previous ones. In Section 3 and Section 4, we study some Korovkin type approximation properties of \(q\)-Meyer–König and Zeller operators and Durrmeyer variant of \(q\)-Meyer–König and Zeller operators via \(P_{p}\)-statistical convergence, respectively.

2 RATES OF THE \(P_{p}\)-STATISTICAL CONVERGENCE

In this section, we compute the rates of the \(P_{p}\)-statistical convergence of the sequences of positive linear operators by means of the modulus of continuity and the elements of the Lipschitz class. Some results related to the rate of the convergence may be found in [15, 20, 28].

Let \((L_{j})\) be a sequence of positive linear operators from \(C[0,1]\) into \(B[0,1]\). The modulus of continuity of \(\omega\left(f,\delta\right)\) is defined by

$$\omega\left(f,\delta\right)=\sup_{\begin{subarray}{c}\left|x-y\right|\leq\delta\\ x,y\in\left[0,1\right]\end{subarray}}\left|f\left(x\right)-f\left(y\right)\right|.$$

It is well known that, for any \(f\) \(\in C\left[0,1\right],\)

$$\lim_{\delta\rightarrow 0^{+}}\omega\left(f,\delta\right)=0$$
(2.1)

and for any \(\delta>0\)

$$\left|f\left(x\right)-f\left(y\right)\right|\leq\omega\left(f,\delta\right)\left(\frac{\left|x-y\right|}{\delta}+1\right)$$
(2.2)

and for all \(c>0\)

$$\omega(f,c\delta)\leq(1+\left\lfloor c\right\rfloor)\omega(f,\delta),$$

where \(\left\lfloor c\right\rfloor\) is the greatest integer less than or equal to \(c.\) Now we are ready to give the following theorem.

Theorem 2. Let \((L_{j})\) be a sequence of positive linear operators from \(C[0,1]\) into \(B[0,1]\) . If

(i) \(st_{P_{p}}\)-\(\lim\left|\left|L_{j}e_{0}-1\right|\right|=0,\)

(ii) \(st_{P_{p}}\)-\(\lim\omega\left(f,\varphi_{j}\left(t\right)\right)=0\) for any \(t\)

then, for any \(f\in C\left[0,1\right]\) we have \(st_{P_{p}}\text{-}\!\lim\left|\left|L_{j}f-f\right|\right|=0,\) where

$$\varphi_{j}\left(t\right):=\left(\sup_{0\leq x\leq 1}L_{j}\left(\left(t-x\right)^{2};x\right)\right)^{\frac{1}{2}}.$$
(2.3)

Proof. Using (2.2), for any \(f\in C\left[0,1\right]\), any \(x,t\in[a,b]\) and any \(\delta>0\) we can write

$$\left|L_{j}\left(f;x\right)-f\left(x\right)\right|\leq L_{j}(\left|f\left(t\right)-f\left(x\right)\right|;x)+\left|f(x)\right|\left|L_{j}(e_{0})-1)\right|\leq L_{j}\left(\left(1+\left\lfloor\dfrac{\left|t-x\right|}{\delta}\right\rfloor\right)\omega\left(f,\delta\right);x\right)$$
$$+\left|f(x)\right|\left|L_{j}(e_{0})-1)\right|{}\leq\omega\left(f,\delta\right)L_{j}\left(e_{0}\right)+\frac{1}{\delta^{2}}\omega\left(f,\delta\right)L_{j}\left(\left(t-x\right)^{2};x\right)+\left|f(x)\right|\left|L_{j}(e_{0})-1)\right|.$$

Thus, we obtain

$$\left|L_{j}\left(f;x\right)-f\left(x\right)\right|\leq\omega\left(f;\delta\right)L_{j}\left(e_{0}\right)+\frac{1}{\delta^{2}}\omega\left(f;\delta\right)L_{j}\left(\left(t-x\right)^{2};x\right)+\left|f(x)\right|\left|L_{j}(e_{0})-1)\right|.$$

Now, if we take \(\delta=\left\{\sup_{0\leq x\leq 1}L_{j}\left(\left(t-x\right)^{2};x\right)\right\}^{\frac{1}{2}},\) then we get \(0\leq\left|\left|L_{j}f-f\right|\right|\leq 2\omega\left(f,\varphi_{j}\left(t\right)\right)+\omega\left(f,\varphi_{j}\left(t\right)\right)\left|L_{j}(e_{0})-1)\right|+C\left|L_{j}(e_{0})-1)\right|,\) where \(C=\left|\left|f\right|\right|\). From \((ii)\), we have \(st_{P_{p}}\text{-}\!\lim\left|\left|L_{j}f\!-\!f\right|\right|=0\) which completes the proof. \(\Box\)

Now, we give the rate of \(P_{p}\)-statistical convergence with the help of Lipschitz class \(Lip_{M}\left(\alpha\right)\) where \(0<\alpha\leq 1,\) \(M>0.\) Recall that a function belongs to \(Lip_{M}\left(\alpha\right)\) if for any \(t,x\in\left[0,1\right]\)

$$\left|f\left(t\right)-f\left(x\right)\right|\leq M\left|t-x\right|^{\alpha}.$$

We have the following theorem.

Theorem 3. Let \((L_{j})\) be a sequence of positive linear operators from \(C[0,1]\) into \(B[0,1]\) . If

(i) \(st_{P_{p}}\)-\(\lim\left|\left|L_{j}e_{0}-1\right|\right|=0,\)

(ii) \(st_{P_{p}}\)-\(\lim\varphi_{j}\left(t\right)=0\) for any \(t\) then, for any \(f\in Lip_{M}\left(\alpha\right)\) such that \(0<\alpha\leq 1,\) \(M\in\mathbb{R}^{+},\) we have \(st_{P_{p}}\text{-}\lim\left|\left|L_{j}f-f\right|\right|=0.\)

Proof. Since \(f\in Lip_{M}\left(\alpha\right),\) we can write from linearity

$$\left|L_{j}\left(f;x\right)-f\left(x\right)\right|\leq L_{j}\left(\left|f\left(t\right)-f\left(x\right)\right|;x\right)+\left|f(x)\right|\left|L_{j}(e_{0})-1)\right|$$
$$\leq ML_{j}\left(\left|t-x\right|^{\alpha};x\right)+\left|f(x)\right|\left|L_{j}(e_{0})-1)\right|.$$

By applying the Hölder inequality, we get

$$\left|\left|L_{j}f-f\right|\right|\leq M\left(\varphi_{j}\left(t\right)\right)^{\alpha}\left|\left(L_{j}(e_{0})\right)^{\frac{2-\alpha}{\alpha}}-1\right|+M\left(\varphi_{j}\left(t\right)\right)^{\alpha}+C\left|L_{j}(e_{0})-1\right|,$$

from (i) and (ii) we have \(st_{P_{p}}-\lim\left|\left|L_{j}f-f\right|\right|=0,\) which ends the proof. \(\Box\)

3 POWER SERIES STATISTICAL CONVERGENCE OF THE SEQUENCE \((M_{j}^{q}\))

In this section, using power series statistical convergence, we obtain Korovkin type approximation of the operators \(\left(M_{j}^{q}\right)\) defined with (1.1). Throughout this section, we deal with the sequence \(\left(q_{j}\right)\) such that \(0<q_{j}\leq 1\) , \(q_{0}=0\) and we assume that \(M_{0}^{q}f=0\) for any \(f\in C[0,1]\).

Now, we recall the following Korovkin type \(P_{p}\)-statistical approximation theorem which is given in [43].

Theorem 4. Let \(P_{p}\) be regular power series method and let \(\left(L_{j}\right)\) be a sequence of linear positive operators on \(C\left[0,1\right]\) such that for \(i=0,1,2\)

$$st_{P_{p}}\text{-}\lim\left|\left|L_{j}e_{i}-e_{i}\right|\right|=0,$$
(3.1)

then for any \(f\in C\left[0,1\right]\) we have \(st_{P_{p}}\text{-}\lim\left|\left|\left(L_{j}f-f\right)\right|\right|=0.\)

We are ready to prove the following Korovkin type \(P_{p}\)-statistical approximation theorem:

Theorem 5. Assume that \(P_{p}\) is a regular power series method. If \(st_{P_{p}}\) - \(\lim q_{j}=1\) and \(st_{P_{p}}\) - \(\lim\dfrac{1}{\left[j-1\right]}=0\) , then for each \(f\in C\left[0,1\right]\) we have \(st_{P_{p}}\text{-}\lim\left|\left|M_{j}^{q}f-f\right|\right|=0.\)

Proof. From Theorem 4, it is enough to demonstrate that (3.1) holds for \((M_{j}^{q})\). Now, considering Lemma 1, we get for \(i=0,1\) that \(st_{P_{p}}\text{-}\lim\left|\left|M_{j}^{q}e_{i}-e_{i}\right|\right|=0.\) Moreover, using (1.6), we have for \(j\geq 3\) that

$$0\leq M_{j}^{q}\left(e_{2};x\right)-x^{2}\leq\frac{x}{\left[j-1\right]}.$$

Here, let us define the following sets for any \(\varepsilon>0,\)

$$N:=\left\{j\in\mathbb{N}:\left|M_{j}^{q}\left(e_{2};x\right)-x^{2}\right|\geq\varepsilon\right\},\quad N_{1}:=\left\{j\in\mathbb{N}:\frac{1}{\left[j-1\right]}\geq\varepsilon\right\}.$$

It is obvious that \(N\subset N_{1}\) which implies with the the hypothesis that

$$0\leq\delta_{P_{p}}\left(\left\{j\in\mathbb{N}:\left|\left|M_{j}^{q}e_{2}-e_{2}\right|\right|_{C_{B}}\geq\varepsilon\right\}\right)\leq\delta_{P_{p}}\left\{j\in\mathbb{N}:\frac{1}{\left[j-1\right]}\geq\varepsilon\right\}=0.$$

Hence, we obtain

$$st_{P_{p}}\text{-}\lim\left|\left|\left(M_{j}^{q}e_{2}-e_{2}\right)\right|\right|=0.$$

So, the proof is completed. \(\Box\)

The following remark shows that the conditions of Theorem 5 are weaker than the classical conditions:

Remark 1. It is not difficult to see that the classical conditions (1.3) entail that the sequence \(\left(\dfrac{1}{\left[j-1\right]}\right)_{j=3}^{\infty}\) is \(P_{p}\) -statistically convergent to zero. Conversely, we assume that \(P_{p}\) is the power series method with \(\left(p_{j}\right)\)

$$p_{j}:=\begin{cases}0,\quad j=2k\\ 1,\quad j=2k+1\end{cases}$$

and we consider the sequence \((q_{j})\)

$$q_{j}:=\begin{cases}0,\quad j=2k\\ 1,\quad j=2k+1\end{cases}$$

for some non-negative integer \(k\) . Note that \((q_{j})\) does not satisfy the classical conditions. Besides, we have for any \(j\geq 3\) that

$$\frac{1}{\left[j-1\right]}=\begin{cases}1,\quad j=2k\\ \dfrac{1}{j-1},\quad j=2k+1.\end{cases}$$

Thus, the sequence \(\left(\dfrac{1}{\left[j-1\right]}\right)_{j=3}^{\infty}\) is \(P_{p}\) -statistically convergent to zero.

The rate of \(P_{p}\)-statistical convergence by means of modulus of continuity for the operators \(\left(M_{j}^{q}\right)\) is given in the following

Theorem 6. If \(0<q_{j}\leq 1\) , \(st_{P_{p}}\) - \(\lim q_{j}=1\) and \(st_{P_{p}}\) - \(\lim\frac{1}{\left[j-1\right]}=0\) , then for any \(f\) \(\in C\left[0,1\right]\) we have \(\left|\left|M_{j}^{q}\left(f\right)-f\right|\right|\leq 2\omega\left(f,\alpha_{j}\right),\) where \(\alpha_{j}=\frac{1}{\left[j-1\right]}.\)

4 POWER SERIES STATISTICAL CONVERGENCE OF THE OPERATORS \(\left(D_{j}^{q}\right)\)

In this section, we study the Korovkin type approximation of the operators \(\left(D_{j}^{q}\right)\) defined with (1.2) by considering the \(P_{p}\)-statistical convergence. Throughout this section, we deal with the sequence \(\left(q_{j}\right)\) such that \(0<q_{j}<1\) , \(q_{0}=0\) and we assume that \(D_{0}^{q}f=0\) for any \(f\in C[0,1]\).

Theorem 7. If the sequence \(st_{P_{p}}\) - \(\lim q_{j}=1\) and \(st_{P_{p}}\) - \(\lim\dfrac{1}{\left[j-1\right]}=0\) then for each \(f\in C\left[0,1\right]\) we have

$$st_{P_{p}}\text{-}\lim\left|\left|D_{j}^{q}f-f\right|\right|=0.$$

Proof. From Theorem 4, it sufficies to show that (3.1) holds for \((D_{j}^{q}).\) Using (1.7) and (1.8), we obtain for \(i=0,1\) that \(st_{P_{p}}-\lim\left|\left|D_{j}^{q}e_{i}-e_{i}\right|\right|=0.\) On the other hand from (1.9) and Lemma 3 we have for any \(j\geq 3\) that

$$\frac{\left[2\right]x\left(1-x\right)\left(1-q^{j}x\right)}{\left[j-1\right]}-\frac{x\left[2\right]\left[3\right]q^{j-1}}{\left[j-1\right]\left[j-2\right]}\left(1-x\right)\left(1-qx\right)\left(1-q^{j}x\right)$$
$${}\leq D_{j}^{q}\left(e_{2};x\right)-x^{2}\leq\frac{\left[2\right]x\left(1-x\right)\left(1-q^{j}x\right)}{\left[j-1\right]},$$

which implies

$$0\leq\left|\left|D_{j}^{q}e_{2}-e_{2}\right|\right|\leq\sup_{0\leq x\leq 1}\left(\frac{\left[2\right]x\left(1-x\right)\left(1-q^{j}x\right)}{\left[j-1\right]}\right)\leq\frac{\left[2\right]}{\left[j-1\right]}.$$

Now, let us define the following sets for any \(\varepsilon>0,\)

$$M:=\left\{j\in\mathbb{N}:\left|M_{j}^{q}\left(e_{2};x\right)-x^{2}\right|\geq\varepsilon\right\}\quad M_{1}:=\left\{j\in\mathbb{N}:\frac{2}{\left[j-1\right]}\geq\varepsilon\right\}.$$

It is obvious that \(M\subset M_{1}\) which implies with the hypothesis that

$$0\leq\delta_{P_{p}}\left(\left\{j\in\mathbb{N}:\left|\left|M_{j}^{q}e_{2}-e_{2}\right|\right|_{C_{B}}\geq\varepsilon\right\}\right)\leq\delta_{P_{p}}\left\{j\in\mathbb{N}:\frac{\left[2\right]}{\left[j-1\right]}\geq\varepsilon\right\}=0.$$

Therefore, we obtain\(st_{P_{p}}\text{-}\lim\left|\left|D_{j}^{q}e_{2}-e_{2}\right|\right|=0.\) Hence, the proof ends. \(\Box\)

Following remark shows that the condition of Theorem 7 is weaker than the classical conditions (1.3):

Remark 2. Note that if the classical conditions (1.3) hold then condition of Theorem 7 holds. In fact, if \(\lim\limits_{j\rightarrow\infty}q_{j}=1\) and \(\lim\limits_{j\rightarrow\infty}\frac{1}{\left[j\right]}=0\) then we have

$$\lim_{j\rightarrow\infty}\frac{1}{\left[j-1\right]}=\lim_{j\rightarrow\infty}\frac{1}{\left[j\right]-q_{j}^{j-1}}=0.$$

Therefore, it is \(P_{p}\) -statistically convergent to zero. Conversely, we assume that \(P_{p}\) is the power series method with \(\left(p_{j}\right)\)

$$p_{j}:=\begin{cases}0,\quad j=k^{2},\\ 1,\quad{otherwise}\end{cases}$$

and we consider the sequence \((q_{j})\) given by

$$q_{j}:=\begin{cases}0,\quad j=k^{2},\\ 1-\frac{1}{j},\quad{otherwise}\end{cases}$$

for some non-negative integer \(k\) . We see that \((q_{j})\) does not satisfy the conditions of classical Korovkin theorem. On the other hand, we have for any \(j\geq 2\) that

$$\frac{1-q_{j}}{1-q_{j}^{j-1}}=\begin{cases}1,\quad j=k^{2},\\ \frac{\frac{1}{j}}{1-\left(1-\frac{1}{j}\right)^{j-1}},\quad{otherwise}.\end{cases}$$

Thus, the sequence \(\left(\frac{1}{\left[j-1\right]}\right)_{j=2}^{\infty}\) is \(P_{p}\) -statistically convergent to zero.

The following theorem which gives the rate of the \(P_{p}\)-statistical convergence by means of modulus of continuity for \((D_{j}^{q})\) can be proved easily.

Theorem 8. If \(0<q_{j}\leq 1\) , \(st_{P_{p}}\) - \(\lim q_{j}=1\) and \(st_{P_{p}}\) - \(\lim\dfrac{1}{\left[j-1\right]}=0\) , then for any \(f\) \(\in C\left[0,1\right]\) we have \(\left|\left|D_{j}^{q}\left(f\right)-f\right|\right|\leq 2\omega\left(f,\beta_{j}\right),\) where \(\beta_{j}=\frac{\left[2\right]}{\left[j-1\right]}.\)