Weighted (E_λ, q)(C_λ, 1) Statistical Convergence and Some Results Related to This Type of Convergence

Abstract

In this paper, we defined weighted $(E_{\lambda },q)(C_{\lambda },1)$ statistical convergence. We also proved some properties of this type of statistical convergence by applying $(E_{\lambda },q)(C_{\lambda },1)$ summability method. Moreover, we used $(E_{\lambda },q)(C_{\lambda },1)$ summability theorem to prove Korovkin’s type approximation theorem for functions on general and symmetric intervals. We also investigated some of the results of the rate of weighted $(E_{\lambda },q)(C_{\lambda },1)$ statistical convergence and studied some sequences spaces defined by Orlicz functions.

1. Introduction

The concept of statistical convergence of a sequence of real numbers was presented in 1951 by Fast [1] and Steinhaus [2], and has been further studied by many researchers.

We have referred to some of the papers that have different results related to statistical convergence, namely [3,4,5,6,7,8,9,10,11,12,13,14]. These results have contributed to developing the field of functional analysis and approximation theory. The theory of approximation of functions, formulated by Weierstrass and improved by Korovkin-type approximation, has been a field of interest for over a century [15]. Korovkin-type theorem, which plays an important role in approximation theory, has had its generalizations in weighted space expanded to a more wide space of sequence by using summation process and convergence methods. Generally speaking, this theorem demonstrates a variety of test subsets of function that guarantee the convergence (or the approximation). The Korovkin-type theorems have been studied by several researchers in different ways, as seen in [4,5,16,17,18,19].

Next, we present the basic concepts and definitions needed in our work. Let H be a subset of the set $\mathbb{N}$, the set of all positive integers and to denote $H_n=\{k\le n:k\in H\}$. Then, the natural density (also known as asymptotic density) of H is represented by ${\displaystyle d\left(H\right)=\underset{n\to \infty }{lim}{\displaystyle \frac{|H_n|}{n}}}$ if the limit exists, here $|H_n|$ denotes the cardinality of the enclosed set $H_n$. Therefore, in any sequence $x=(x_k;k=0,1,2,\dots )$ is said to be statistically convergent to a definite number L, if for each $\epsilon >0,$ we have ${\displaystyle \underset{n\to \infty }{lim}{\displaystyle \frac{\left|\right\{k\le n:|x_k-L|\ge \epsilon \left\}\right|}{n}}=0.}$

In this case, we denote this by $\mathrm{stat-lim} x=L.$

Note that for every convergence, the sequences are statistically convergent. However, in general, the converse is not true.

We now recall the definition of weighted $(E,q)(C,1)$ statistical convergence, which is needed in our present study as follows:

The series ${\displaystyle \sum _{n=0}^{\infty }{x}_n}$ is said to be summable to a definite number S if $E_n^q\to S$ as $n\to \infty ,$ here $E_n^q={\displaystyle \frac{1}{{(1+q)}^n}}{\sum }_{k=0}^n\left(\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{S}_k.$ In this case, we write $S_n\to S(E,q)$ as $n\to \infty$.

In general, the concept of $(C,1)$ method can also be defined as ${\displaystyle C_k^1={\displaystyle \frac{1}{(k+1)}}\sum _{l=0}^k{S}_l\to S}$ and if $C_k^1\to S$ as $k\to \infty$, we can say that this summability method is convergent. For this case, the series ${\displaystyle \sum _{k=0}^{\infty }{x}_k}$ is $(C,1)$—summable to the definite number S, which can be written by $S_k\to S(C,1)$ as $k\to \infty$ [16,20].

From the above conditions, we obtain the method $(E,q)(C,1)$ as a product of $(E,q)$ method and $(C,1)$ method, which can be defined by:

$$t_n^{(E,q)(C,1)}={\displaystyle \frac{1}{{(1+q)}^n}}\sum _{k=0}^n\left(\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{C}_k^1={\displaystyle \frac{1}{{(1+q)}^n}}\sum _{k=0}^n\left(\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v=0}^k{S}_v.$$

If $t_n^{(E,q)(C,1)}\to S$ as $n\to \infty ,$ then we say that a series ${\displaystyle \sum _{n=0}^{\infty }{x}_n}$ or sequence $\left\{S_n\right\}$ is summable to S by $(E,q)(C,1)$ method and it can be represented by $S_n\to S(E,q)(C,1)$.

In functional analysis, an Orlicz space is a type of function space which was discovered by Władysław Orlicz in 1932. In 1971, Lindentrauss and Tzafriri [21] studied Orlicz spaces of measurable functions and used his idea to define the sequence space as follows:

$${\displaystyle l_M=\left\{x=\left(x_i\right):\sum _{i=1}^{\infty }M\left({\displaystyle \frac{|x_i|}{\varrho }}\right)<\infty \mathrm{for}\mathrm{some}\varrho >0\right\}.}$$

This is called an Orlicz sequence space.

The space $l_M$ is a Banach space with a norm and can be expressed as follows:

$${\displaystyle \left|\right|x\left|\right|=inf\left\{\varrho >0:\sum _{i=1}^{\infty }M\left({\displaystyle \frac{|x_i|}{\varrho }}\right)\le 1\right\}.}$$

The space $l_M$ is closely related to the space $l_p$, which is an Orlicz sequence space with,

$$M\left(x\right)=x^p,1\le p<\infty .$$

In the past years, many researchers have published articles that deal with the relationship between an Orlicz function and convergence in general. In 2000, Bhardwaj, V.K. and Singh, N. studied some sequence spaces defined by Orlicz functions [22]. Later, Mursaleen et al. studied a new convergent sequence space defined by Orlicz functions. They also established the relationship between strong $\sigma$-convergence and uniform $\sigma$-statistical convergence [23]. In 2004, E. Savas and. R. Savas introduced a new concept of $\lambda$-strong convergence in relation to an Orlicz function [24]. Recently, a new class of sequences that have been defined with respect to Orlicz functions has been of interest for researchers. More details can be found in reference [3,25,26,27].

Motivated essentially by the above mentioned works, the objectives of this research are: (1) to define a new weighted $(E_{\lambda },q)(C_{\lambda },1)$ statistical convergence and applying the $(E_{\lambda },q)(C_{\lambda },1)$ summability method to prove some properties of this type of statistical convergence; (2) to use $(E_{\lambda },q)(C_{\lambda },1)$ summability theorem to prove Korovkin’s type approximation theorem; (3) to investigate some results of the rate of weighted $(E_{\lambda },q)(C_{\lambda },1)$ statistical convergence and (4) to study some sequence spaces defined by Orlicz functions.

2. Weighted $({\mathit{E}}_{\mathit{\lambda }},\mathit{q})({\mathit{C}}_{\mathit{\lambda }},\mathbf{1})$ Statistical Convergence

The concept of the weighted $(C,1)(E,1)$ was given by Tuncer Acar and S.A. Mohiuddine, one can read [28] for more details. This method was generalized by Aljimi. E and Sirimark. P in 2021, see [29] for more information. Thus, in this section, we introduce the weighted $(E_{\lambda },q)(C_{\lambda },1)$ statistical convergence, which is a generalization of the weighted $(E,1)(C,1)$, see [6] for more details.

Let $\lambda =\left({\lambda }_n\right)$ be a non-decreasing sequence of positive numbers which tends to infinity as $n\to \infty$, that is, let

$${\lambda }_{n+1}\le {\lambda }_n+1,{\lambda }_1=1.$$

Next, let us write the new $(E_{\lambda },q)(C_{\lambda },1)$ summability method

$${\displaystyle t_n^{(E_{\lambda },q)(C_{\lambda },1)}={\displaystyle \frac{1}{{(1+q)}^n}}\sum _{k\in I_n}\left(\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}{S}_v,}$$

here $I_n=[n-{\lambda }_n+1,n]$.

The series $\sum x_k$ is summable to S by weighted $(E_{\lambda },q)(C_{\lambda },1)$ method if

$${\displaystyle t_n^{(E_{\lambda },q)(C_{\lambda },1)}={\displaystyle \frac{1}{{(1+q)}^n}}\sum _{k\in I_n}\left(\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}{S}_v\to Sasn\to \infty }$$

and it is denoted by $S_n\to S(E_{\lambda },q)(C_{\lambda },1)$.

Remark 1

([16]). If we replace ${\lambda }_n$ with n and q with number 1 in $(E_{\lambda },q)(C_{\lambda },1)$, then we obtain the $(E,1)(C,1)$ summability method.

Theorem 1.

The summability method $(E_{\lambda },q)(C_{\lambda },1)$ is a regular method.

Proof. 

To begin with, the definition of $(E_{\lambda },q)(C_{\lambda },1)$ is as follows:

$${\displaystyle t_n^{(E_{\lambda },q)(C_{\lambda },1)}={\displaystyle \frac{1}{{(1+q)}^n}}\sum _{k\in I_n}\left(\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}{x}_v.}$$

We also know that the Cesaro $(C,1)$ summability method is a regular method. From the regularity of Cesaro method, we have ${\displaystyle \sum _{k\in I_n}{x}_k\to S}$ and ${\displaystyle (C,1)={\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}{x}_v\to S}$. Therefore, the summability method $(E_{\lambda },q)(C_{\lambda },1)$ can be rewritten as:

$$\begin{array}{cc}\hfill t_n^{(E_{\lambda },q)(C_{\lambda },1)}& ={\displaystyle \frac{1}{{(1+q)}^n}}\sum _{k\in I_n}\left(\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}{x}_v\hfill \\ & ={\displaystyle \frac{1}{{(1+q)}^n}}\sum _{k\in I_n}\left(\begin{array}{c}n\\ k\end{array}\right)q^{n-k}·S\hfill \\ \hfill & ={\displaystyle \frac{1}{{(1+q)}^n}}{(1+q)}^n·S=S.\hfill \end{array}$$

Since ${\displaystyle \sum _{k\in I_n}{x}_k\to S}$ and $t_n^{(E_{\lambda },q)(C_{\lambda },1)}\to S.$ From the above conditions, we have shown that the summability method $(E_{\lambda },q)(C_{\lambda },1)$ is regular.  □

Next, we introduce the key definitions in this work. The details are as follows:

Definition 1.

We say that a sequence $x=\left(x_k\right)$ is weighted $(E_{\lambda },q)(C_{\lambda },1)$ summable to the definite number L if

$${\displaystyle \underset{n}{lim}{t}_n^{(E_{\lambda },q)(C_{\lambda },1)}=\underset{n}{lim}{\displaystyle \frac{1}{{(1+q)}^n}}\sum _{k\in I_n}\left(\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}{x}_v=L.}$$

Definition 2.

A sequence $x=\left(x_k\right)$ is said to be strongly $(E_{\lambda },q)(C_{\lambda },1)$ summable to the definite number L if

$${\displaystyle \underset{n}{lim}{\displaystyle \frac{1}{{(1+q)}^n}}\sum _{k\in I_n}\left(\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|x_v-L|=0.}$$

We shall write $x_k\to L\left[(E_{\lambda },q)(C_{\lambda },1)\right]$ for this case and the set of all strongly $(E_{\lambda },q)(C_{\lambda },1)$-summable sequences is designated by $\left[(E_{\lambda },q)(C_{\lambda },1)\right]$.

Definition 3.

A sequence $x=\left(x_k\right)$ is said to be weighted statistically summable $(E_{\lambda },q)(C_{\lambda },1)$ to the definite number L if

$${\displaystyle st-\underset{n}{lim}{t}_n^{(E_{\lambda },q)(C_{\lambda },1)}=L.}$$

In this case, we write that $L={\left(EC\right)}_{\lambda }\left(st\right)-\lim x.$

We now obtain the weighted Euler–Cesáro $\lambda$-statistical convergence by using the notion of $(E_{\lambda },q)(C_{\lambda },1)$-summability. Let $K\subseteq \mathbb{N}$. The number ${\delta }_{{\left(EC\right)}_{\lambda }}\left(K\right)$ is said to be weighted Euler–Cesáro $\lambda$-density, of K which is expressed as follows:

$${\displaystyle {\delta }_{{\left(EC\right)}_{\lambda }}\left(K\right)=\underset{n}{lim}{\displaystyle \frac{1}{{(1+q)}^n}}\left|\{k\le {(1+q)}^n:k\in K\}\right|.}$$

Remark 2.

If we select ${\lambda }_n$ as equal to n, the weighted Euler–Cesáro λ-density is reduced to generalized weighted Euler–Cesáro density.

Definition 4.

We say that the sequence $x=\left(x_k\right)$ is weighted Euler–Cesáro λ-statistically convergent (or $S_{{\left(EC\right)}_{\lambda }}$-convergent) to the definite number L if for each $\epsilon >0$; i.e.,

$${\displaystyle \underset{n}{lim}{\displaystyle \frac{1}{{(1+q)}^n}}\left|\right\{{\{k\le (1+q)}^n:(\begin{array}{c}n\\ k\end{array})q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|x_v-L|\ge \epsilon \left\}\right|=0.}$$

In this case, we can write $S_{{\left(EC\right)}_{\lambda }}\left(st\right)-limx=L$ and the set of all weighted Euler–Cesáro λ-statistically convergent sequences is designated by $S_{{\left(EC\right)}_{\lambda }}.$

Definition 5.

We say that the sequence $x=\left(x_k\right)$ is weighted ${\left[(E_{\lambda },q)(C_{\lambda },1)\right]}_r^{\lambda }$-summable $(0<r<\infty )$ to the definite number L, if

$${\displaystyle \underset{n}{lim}{\displaystyle \frac{1}{{(1+q)}^n}}\left|\right\{k\le {(1+q)}^n:(\begin{array}{c}n\\ k\end{array})q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}{|x_v-L|}^r\ge \epsilon \left\}\right|=0.}$$

In this case, we can write $x_k\to L{\left[(E_{\lambda },q)(C_{\lambda },1)\right]}_r^{\lambda }.$ We therefore say that $L{\left[(E_{\lambda },q)(C_{\lambda },1)\right]}_r^{\lambda }$ limit of the sequence $x=\left(x_k\right).$

These lead us to the following results:

Theorem 2.

Let ${\displaystyle (\begin{array}{c}n\\ k\end{array})q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|x_v-L|\le M}$, for all $n,k\in \mathbb{N}$. If a sequence $x=\left(x_k\right)$ is $S_{{\left(EC\right)}_{\lambda }}$-statistically convergent to the definite number L, then it is also weighted as statistically $(E_{\lambda },q)(C_{\lambda },1)$-summable to L, but not conversely.

Proof. 

We accept that $x=\left(x_k\right)$ is $S_{{\left(EC\right)}_{\lambda }}$-statistically convergent to L, and it implies that

$${\displaystyle \underset{n}{lim}{\displaystyle \frac{1}{{(1+q)}^n}}\left|\right\{k\le {(1+q)}^n:(\begin{array}{c}n\\ k\end{array})q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|x_v-L|\ge \epsilon \}|=0.}$$

Firstly, we denote that

$${\displaystyle K_n=\{k\le {(1+q)}^n:(\begin{array}{c}n\\ k\end{array})q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|x_v-L|\ge \epsilon \},}$$

and

$${\displaystyle K_n^C=\{k\le {(1+q)}^n:(\begin{array}{c}n\\ k\end{array})q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|x_v-L|<\epsilon \}.}$$

Therefore,

$$\begin{array}{cc}\hfill {\displaystyle |(E_{\lambda },q)(C_{\lambda },1)-L|}& =|{\displaystyle \frac{1}{{(1+q)}^n}}{\displaystyle \sum _{k\in I_n}}(\begin{array}{c}n\\ k\end{array})q^{n-k}{\displaystyle \frac{1}{k+1}}{\displaystyle \sum _{v\in I_k}}(x_v-L)|\hfill \\ \hfill & {\displaystyle \le |{\displaystyle \frac{1}{{(1+q)}^n}}\sum _{\begin{array}{c}k\in I_n\\ k\in K_n\end{array}}(\begin{array}{c}n\\ k\end{array})q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}(x_v-L)|}\hfill \\ \hfill & {\displaystyle +|{\displaystyle \frac{1}{{(1+q)}^n}}\sum _{\begin{array}{c}k\in I_n\\ k\in K_n^C\end{array}}(\begin{array}{c}n\\ k\end{array})q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}(x_v-L)|}\hfill \\ \hfill & {\displaystyle \le {\displaystyle \frac{1}{{(1+q)}^n}}\sum _{\begin{array}{c}k\in I_n\\ k\in K_1\end{array}}(\begin{array}{c}n\\ k\end{array})q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|x_v-L|}\hfill \\ \hfill & {\displaystyle +{\displaystyle \frac{1}{{(1+q)}^n}}\sum _{\begin{array}{c}k\in I_n\\ k\in K_n^C\end{array}}(\begin{array}{c}n\\ k\end{array})q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|x_v-L|}\hfill \\ \hfill & {\displaystyle \le {\displaystyle \frac{1}{{(1+q)}^n}}·M·\left|K_1\right|+{\displaystyle \frac{1}{{(1+q)}^n}}\sum _{\begin{array}{c}k\in I_n\\ k\in K_n^C\end{array}}\epsilon \to 0+1·\epsilon }\hfill \\ \hfill & =\epsilon \hfill \end{array}$$

as $n\to \infty$, which means that $t_n^{(E_{\lambda },q)(C_{\lambda },1)}\to L$, i.e., sequence $x=\left(x_k\right)$ is $(E_{\lambda },q)(C_{\lambda },1)$-summable to L. As a result, $x=\left(x_n\right)$ is statistically weighted $(E_{\lambda },q)(C_{\lambda },1)$-summable to L.  □

Next, we present that the converse of Theorem 2 is not true by using the example below:

Example 1.

Consider ${\lambda }_n=n$ for all $n\in \mathbb{N}$. Additionally, we can define the sequence $x=\left(x_k\right)$ as follows:

$$x_k=\left\{\begin{array}{cc}\sqrt{k}\hfill & if,k=s^2\hfill \\ 0\hfill & if,k\ne s.\hfill \end{array}\right.$$

From the above condition, we have

$${\displaystyle {\displaystyle \frac{1}{{(1+q)}^n}}|\left\{k\le {(1+q)}^n:\sum _{k=0}^n(\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v=0}^k|x_v-L|\ge \epsilon \}|\le {\displaystyle \frac{\sqrt{{(1+q)}^n}}{{(1+q)}^n}}\to 0\mathrm{as}n\to \infty }$$

On the other hand,

$$\begin{array}{cc}\hfill {\displaystyle {\displaystyle \frac{1}{{(1+q)}^n}}\sum _{k=0}^n(\begin{array}{c}n\\ k\end{array})q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v=0}^k|x_v-0|}& ={\displaystyle \frac{1}{{(1+q)}^n}}\sum _{k=0}^n(\begin{array}{c}n\\ k\end{array})q^{n-k}{\displaystyle \frac{1}{k+1}}{\displaystyle \sum _{v=0}^k}v\hfill \\ \hfill & {\displaystyle ={\displaystyle \frac{1}{{(1+q)}^n}}\sum _{k=0}^n(\begin{array}{c}n\\ k\end{array}){\displaystyle \frac{1}{k+1}}·{\displaystyle \frac{k(k+1)}{(1+q)}}}\hfill \\ \hfill & {\displaystyle ={\displaystyle \frac{1}{{(1+q)}^{n+1}}}\sum _{k=0}^n(\begin{array}{c}n\\ k\end{array})k}\hfill \\ \hfill & {\displaystyle ={\displaystyle \frac{1}{{(1+q)}^{n+1}}}·{(1+q)}^{n-1}·n}\hfill \\ \hfill & {\displaystyle ={\displaystyle \frac{n}{{(1+q)}^2}}\to \infty }\hfill \end{array}$$

as $n\to \infty$. Therefore, the inclusion is strict.

We next determine the conditions of the $S_{{\left(EC\right)}_{\lambda }}$-summability, which imply the ${\left[(E_{\lambda },q)(C_{\lambda },1)\right]}_r^{\lambda }$-statistical convergence and vice versa. The details are as follows:

Proposition 1.

Let the sequence $x=\left(x_n\right)$ be weighted ${\left[(E_{\lambda },q)(C_{\lambda },1)\right]}_r^{\lambda }$-summable and convergent to the definite number L. The sequence $x=\left(x_n\right)$ is said to be $S_{{\left(EC\right)}_{\lambda }}$-statistically convergent to the definite number L if the following two conditions are satisfied:

1. 

$0<r<1$ and $0\le |x_k-L|<1$

2. 

$1\le r<\infty$ and $0\le |x_k-L|<1$.

Proof. 

Assume that the sequence $x=\left(x_n\right)$ is weighted ${\left[(E_{\lambda },q)(C_{\lambda },1)\right]}_r^{\lambda }$-summable and convergent to the definite number L. Therefore, under above conditions (1) and (2), we obtain:

$${\displaystyle (\begin{array}{c}n\\ k\end{array})q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|x_v{-L|}^r\ge (\begin{array}{c}n\\ k\end{array})q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|x_v-L|}$$

and

$${\displaystyle (\begin{array}{c}n\\ k\end{array})q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|x_v{-L|}^r\le )\begin{array}{c}n\\ k\end{array})q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|x_v-L|}$$

respectively, thus we find that:

$$\begin{array}{cc}\hfill {\displaystyle {\displaystyle \frac{1}{{(1+q)}^n}}\left|K_n\right|}& ={\displaystyle \frac{1}{\epsilon ·{(1+q)}^n}}{\displaystyle \sum _{\begin{array}{c}k\in I_n\\ k\in K_n\end{array}}}\epsilon \hfill \\ \hfill & {\displaystyle \le {\displaystyle \frac{1}{\epsilon ·{(1+q)}^n}}\sum _{\begin{array}{c}k\in I_n\\ k\in K_n\end{array}}(\begin{array}{c}n\\ k\end{array})q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|x_v-L|}\hfill \\ \hfill & {\displaystyle \le {\displaystyle \frac{1}{\epsilon ·{(1+q)}^n}}\sum _{k\in I_n}(\begin{array}{c}n\\ k\end{array})q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|x_v-L|}\hfill \\ \hfill & {\displaystyle \le {\displaystyle \frac{1}{\epsilon ·{(1+q)}^n}}\sum _{k\in I_n}(\begin{array}{c}n\\ k\end{array})q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}{|x_v-L|}^r\to 0,}\hfill \end{array}$$

when the number of n tends to infinity as $n\to \infty .$ Therefore, the sequence $x=\left(x_n\right)$ is $S_{{\left(EC\right)}_{\lambda }}$-statistically convergent to the definite number L.  □

Proposition 2.

Let the sequence $x=\left(x_n\right)$ be $S_{{\left(EC\right)}_{\lambda }}$-statistically convergent to the definite number L and assume that:

$$\left(\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|x_v-L{|}^r\le M(k\in \mathbb{N}).$$

If the following two conditions are satisfied:

1. 

$0<r<1$ and $0\le M<\infty$

2. 

$1\le r<\infty$ and $0\le M<1$,

then the sequence $x=\left(x_n\right)$ is weighted ${\left[(E_{\lambda },q)(C_{\lambda },1)\right]}_r^{\lambda }$-summable and convergent to the definite number L.

Proof. 

Assume that the sequence $x=\left(x_n\right)$ is $S_{{\left(EC\right)}_{\lambda }}$-statistically convergent to the definite number L. For $\epsilon >0$ we have:

$${\displaystyle S_{\left(EC\right)}\left(\{k\in \mathbb{N}:(\begin{array}{c}n\\ k\end{array})q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|x_v-L|>\epsilon \right\})=0.}$$

Since

$${\displaystyle \left(\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|x_v-L{|}^r\le M(k\in \mathbb{N})}$$

we have:

$$\begin{array}{cc}\hfill {\displaystyle {\displaystyle \frac{1}{{(1+q)}^n}}\sum _{k\in I_n}(\begin{array}{c}n\\ k\end{array})q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}{|x_v-L|}^r}& ={\displaystyle \frac{1}{{(1+q)}^n}}{\displaystyle \sum _{\begin{array}{c}k\in I_n\\ k\in K_n^C\end{array}}}(\begin{array}{c}n\\ k\end{array})q^{n-k}{\displaystyle \frac{1}{k+1}}{\displaystyle \sum _{v\in I_k}}{|x_v-L|}^r\hfill \\ \hfill & {\displaystyle +{\displaystyle \frac{1}{{(1+q)}^n}}\sum _{\begin{array}{c}k\in I_n\\ k\in K_n\end{array}}(\begin{array}{c}n\\ k\end{array})q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}{|x_v-L|}^r}\hfill \\ \hfill & {\displaystyle =S_1+S_2}\hfill \end{array}$$

where

$${\displaystyle S_1\left(n\right)={\displaystyle \frac{1}{{(1+q)}^n}}\sum _{\begin{array}{c}k\in I_n\\ k\in K_n^C\end{array}}(\begin{array}{c}n\\ k\end{array})q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}{|x_v-L|}^r}$$

$${\displaystyle S_2\left(n\right)={\displaystyle \frac{1}{{(1+q)}^n}}\sum _{\begin{array}{c}k\in I_n\\ k\in K_n\end{array}}(\begin{array}{c}n\\ k\end{array})q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}{|x_v-L|}^r}$$

Now, if the number k is element of set $K_n^C$, then,

$$\begin{array}{cc}\hfill {\displaystyle S_1\left(n\right)}& ={\displaystyle \frac{1}{{(1+q)}^n}}{\displaystyle \sum _{\begin{array}{c}k\in I_n\\ k\in K_n^C\end{array}}}(\begin{array}{c}n\\ k\end{array})q^{n-k}{\displaystyle \frac{1}{k+1}}{\displaystyle \sum _{v\in I_k}}{|x_v-L|}^r\hfill \\ \hfill & {\displaystyle \le {\displaystyle \frac{1}{{(1+q)}^n}}\sum _{\begin{array}{c}k\in I_n\\ k\in K_n^C\end{array}}(\begin{array}{c}n\\ k\end{array})q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|x_v-L|}\hfill \\ \hfill & {\displaystyle =\epsilon ·{\displaystyle \frac{1}{{(1+q)}^n}}\left|K_n^C\right|}\hfill \\ \hfill & {\displaystyle =\epsilon }\hfill \end{array}$$

If the number k is an element of set $K_n$, we thus have

$$\begin{array}{cc}\hfill {\displaystyle S_2\left(n\right)}& ={\displaystyle \frac{1}{{(1+q)}^n}}{\displaystyle \sum _{\begin{array}{c}k\in I_n\\ k\in K_n\end{array}}}(\begin{array}{c}n\\ k\end{array})q^{n-k}{\displaystyle \frac{1}{k+1}}{\displaystyle \sum _{v\in I_k}}{|x_v-L|}^r\hfill \\ \hfill & {\displaystyle \le {\displaystyle \frac{1}{{(1+q)}^n}}\sum _{\begin{array}{c}k\in I_n\\ k\in K_n\end{array}}(\begin{array}{c}n\\ k\end{array})q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|x_v-L|}\hfill \\ \hfill & {\displaystyle \le (\underset{k}{sup}\sum _{\begin{array}{c}k\in I_n\\ k\in K_n\end{array}}(\begin{array}{c}n\\ k\end{array})q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|x_v-L|)·\left({\displaystyle \frac{K_n}{{(1+q)}^n}}\right)}\hfill \\ \hfill & {\displaystyle \le M·{\displaystyle \frac{K_n}{{(1+q)}^n}}\to 0}\hfill \end{array}$$

as $n\to 0$. Since

$${\displaystyle S_{\left(EC\right)}\left(\{k\in \mathbb{N}:(\begin{array}{c}n\\ k\end{array})q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|x_v-L|>\epsilon \right\})=0.}$$

Consequently,

$${\displaystyle x_k\to L{\left[(E_{\lambda },q)(C_{\lambda },1)\right]}_r^{\lambda }.}$$

□

Theorem 3.

A sequence $x=\left(x_k\right)$ is weighted as statistically summable $(E_{\lambda },q)(C_{\lambda },1)$ to the definite number L if and only if there exists a set of $K=\{k_1<k_2<\cdots <k_n<\cdots \}\subseteq \mathbb{N}$ such that $\delta \left(K\right)$ is equal to one and $\left(x_{k_n}\right)$ is weighted $(E_{\lambda },q)(C_{\lambda },1)$-summable to the definite number L.

Proof. 

Assume that there exists a set of $K\subset \mathbb{N}$ such that $\delta \left(K\right)$ is equal to one and $\left(x_{k_n}\right)$ is $(E_{\lambda },q)(C_{\lambda },1)$-summable to the definite number L. Consequently, there is a positive integer $n_0$, such that for every $n>n_0$, we have:

$${\displaystyle |t_n^{(E_{\lambda },q)(C_{\lambda },1)}-L|<\epsilon .}$$

To put $K_t\left(\epsilon \right)=\left\{n\in \mathbb{N}:|t_{k_n}^{(E_{\lambda },q)(C_{\lambda },1)}-L|\ge \epsilon \right\}$ and $K^{\prime }=\{k_{n_0+1},k_{n_0+2},\cdots \}.$

Therefore, $\delta \left(K^{\prime }\right)$ is equal to one and $K_t\left(\epsilon \right)\subset \mathbb{N}-K^{\prime },$ which implies that $\delta \left(K_t\left(\epsilon \right)\right)$ is equal to zero.

From the above conditions, we can conclude that $x=\left(x_k\right)$ is statistically summable $(E_{\lambda },q)(C_{\lambda },1)$ to the definite number L.

Conversely, let $x=\left(x_k\right)$ be statistically summable $(E_{\lambda },q)(C_{\lambda },1)$ to the definite number L. r is positive integer if we put $K_t\left(r\right):=\left\{j\in \mathbb{N}:|t_{k_j}^{(E_{\lambda },q)(C_{\lambda },1)}-L|\ge {\displaystyle \frac{1}{r}}\right\}$ and $M_t\left(r\right):=\left\{j\in \mathbb{N}:|t_{k_j}^{(E_{\lambda },q)(C_{\lambda },1)}-L|<{\displaystyle \frac{1}{r}}\right\}.$ Then, $\delta \left(K_t\left(r\right)\right)=0,$

$$M_t\left(1\right)\supset M_t\left(2\right)\supset \cdots \supset M_t\left(i\right)\supset M_t(i+1)\supset ,$$

(1)

and

$$\delta \left(M_t\left(r\right)\right)=1.$$

(2)

Next, we need to show that $j\in M_t\left(r\right),\left(x_{k_j}\right)$ is $(E_{\lambda },q)(C_{\lambda },1)$ is summable to the definite number L.

Assume that $\left(x_{k_j}\right)$ is not $(E_{\lambda },q)(C_{\lambda },1)$-summable to the definite number L. Therefore, there is $\epsilon >0$ such that $|t_{k_j}^{(E_{\lambda },q)(C_{\lambda },1)}-L|\ge \epsilon$ for infinitely many terms. Let $M_t\left(\epsilon \right):=\left\{j\in \mathbb{N}:|t_{k_j}^{(E_{\lambda },q)(C_{\lambda },1)}-L|<\epsilon \right\}$ and $\epsilon >{\displaystyle \frac{1}{r}},(r=1,2,3,\dots ).$ From (1), we obtain $M_t\left(r\right)\subset M_t\left(\epsilon \right).$ Therefore, $\delta \left(M_t\left(r\right)\right)$ is equal to zero, which contradicts (2) and hence $\left(x_{k_j}\right)$ is $(E_{\lambda },q)(C_{\lambda },1)$ summable to the definite number L.  □

3. Application to Korovkin Type Theorem

Throughout this section, we will use the $C[a,b]$ and $F[a,b]$ to notate function spaces, where $C[a,b]$ denotes the space of all bounded and continuous functions defined in $[a,b]$ and $F[a,b]$ denotes the linear space of all real-valued functions defined in $[a,b]$. It is well-known that $C[a,b]$ is a Banach space with a norm and can be defined as follows:

$${\displaystyle {\left|\right|f\left|\right|}_{\infty }=\underset{x\in [a,b]}{sup}\left|f\left(x\right)\right|,f\in C[a,b].}$$

Definition 6.

Let $B=\left(B_n\right)$ be a sequence of positive linear operators from $[a,b]$ into $[a,b]$. The map B is positive if it satisfies the following condition:

$$B(f;x)\ge 0forallf\left(x\right)\ge 0,x\in [a,b].$$

The classical Korovkin first theorem is presented as follows:

Theorem 4.

Assume that $\left(B_n\right)$ is a sequence of positive linear operators from $C[0,1]$ into $F[0,1]$. Then, for all $f\in C[0,1]$, we have

$${\displaystyle \underset{n\to \infty }{lim}\left|\right|B_n(f,x)-f\left(x\right){\left|\right|}_{\infty }=0,}$$

if and only if

$${\displaystyle \underset{n\to \infty }{lim}\left|\right|B_n(f_i,x)-f_i\left(x\right){\left|\right|}_{\infty }=0(1\le i\le 3),}$$

here, i is the natural number, $f_0\left(x\right)$ is equal to 1, $f_1\left(x\right)$ is equal to x, and $f_2\left(x\right)$ is equal to $x^2$.

In the present paper, we extend the same test functions $1,e^{-x}$ and $e^{-2x}$ with the results seen in [30]. Let $C\left(I\right)$ denote the Banach space with a uniform norm ${\left|\right|,\left|\right|}_{\infty }$ of all real two-dimensional continuous functions on $I=[0,\infty )$, yield a finite of ${\displaystyle \underset{n\to \infty }{lim}f\left(x\right)}$. Assume that $B_n:C\left(I\right)\to C\left(I\right)$, we therefore write that $B_n(f;x)$ for $B_n(f\left(s\right);x)$ in a more convenient way.

Theorem 5.

Let $\left(B_k\right)$ be a sequence of positive linear operators from $C\left(I\right)$ into $C\left(I\right)$. Then,

$$E_{\lambda }{C}_{\lambda }\left(st\right)-\underset{n}{lim}\left|\right|B_k(f;x)-f\left(x\right){\left|\right|}_{\infty }=0,$$

(3)

if and only if

$$E_{\lambda }{C}_{\lambda }\left(st\right)-\underset{n}{lim}\left|\right|B_k(f;x)-1{\left|\right|}_{\infty }=0,$$

(4)

$$E_{\lambda }{C}_{\lambda }\left(st\right)-\underset{n}{lim}\left|\right|B_k(f;x)-e^{-x}{\left|\right|}_{\infty }=0,$$

(5)

and

$$E_{\lambda }{C}_{\lambda }\left(st\right)-\underset{n}{lim}\left|\right|B_k(f;x)-e^{-2x}{\left|\right|}_{\infty }=0,$$

(6)

for all $f\in C\left(I\right).$

Proof. 

Assume that condition (3) is true. Since the functions $1,e^{-x}$, and $e^{-2x}$ belong to $C\left(I\right)$, then the conditions (4)–(6) follow immediately from (3). To begin with, we prove the converse of the theorem by supposing that conditions (4)–(6) are satisfied, and then we can prove that the condition (3) is true.

Let $f\in C\left(I\right)$. Then, there exists a constant $M>0$, such that $\left|f\right(x\left)\right|\le M$ for $x\in I$. Hence,

$$\left|f\right(s)-f(x\left)\right|\le 2M,-\infty <s,x<\infty .$$

(7)

For a given $\epsilon >0$, there is a $\delta =\delta \left(\epsilon \right)>0$ such that:

$$\left|f\right(t)-f(x\left)\right|<\epsilon ,$$

(8)

whenever $|e^{-t}-e^{-x}|<\delta$ for all $x\in I$.

Using (7) and (8), putting $\psi =\psi (s,t)={(e^{-t}-e^{-x})}^2$, we obtain

$${\displaystyle |f\left(t\right)-f\left(x\right)<\epsilon +{\displaystyle \frac{2M}{{\delta }^2}}\psi .}$$

By these, we mean:

$${\displaystyle -\epsilon -{\displaystyle \frac{2M}{{\delta }^2}}\psi <f\left(t\right)-f\left(x\right)<\epsilon +{\displaystyle \frac{2M}{{\delta }^2}}\psi .}$$

We now include operator $B_k(1,x)$ to this inequality. Since $B_k(f;x)$ is monotone and linear, therefore,

$$B_k(1,x)(-\epsilon -{\displaystyle \frac{2M}{{\delta }^2}}\psi )<B_k(1,x)(f\left(t\right)-f\left(x\right))<B_k(1,x)(\epsilon +{\displaystyle \frac{2M}{{\delta }^2}}\psi ).$$

(9)

Note that: x is fixed and so $f\left(x\right)$ is a constant number.

Consequently, we have

$$-\epsilon B_k(1,x)-{\displaystyle \frac{2M}{{\delta }^2}}{B}_k(\psi ,x)<B_k(f,x)-f\left(x\right)B_k(1,x)<\epsilon B_k(1,x)+{\displaystyle \frac{2M}{{\delta }^2}}{B}_k(\psi ,x).$$

(10)

However,

$$\begin{array}{cc}\hfill {\displaystyle B_k(f,x)-f\left(x\right)}& =B_k(f,x)-f\left(x\right)B_k(1,x)+f\left(x\right)B_k(1,x)-f\left(x\right)\hfill \\ \hfill & {\displaystyle =[B_k(f,x)-f\left(x\right)B_k(1,x)]+f\left(x\right)[B_k(1,x)-1.}\hfill \end{array}$$

Using (9) and (10), we obtain:

$$B_k(f,x)-f\left(x\right)<\epsilon B_k(1,x)+{\displaystyle \frac{2M}{{\delta }^2}}{B}_k(\psi ,x)+f\left(x\right)(B_k(1,x)-1).$$

(11)

Let us estimate $B_k(\psi ,x)$ as follows:

$$\begin{array}{cc}\hfill {\displaystyle B_k(\psi ,x)}& =B_k({(e^{-x}-e^{-t})}^2,x)\hfill \\ \hfill & {\displaystyle =B_k(e^{-2t}-2e^{-t}{e}^{-x}+e^{-2x},x)}\hfill \\ \hfill & {\displaystyle =B_k(e^{-2t},x)+2e^{-x}{B}_k(e^{-t},x)+e^{-2x}{B}_k(1,x)}\hfill \\ \hfill & {\displaystyle =[B_k(e^{-2t},x)-x^2]+2e^{-x}[B_k(e^{-t},x)-x]+e^{-2x}[B_k(1,x)-1].}\hfill \end{array}$$

Substituting the value from the above equation into (11), we obtain

$$\begin{array}{cc}\hfill {\displaystyle B_k(f,x)-f\left(x\right)}& <\epsilon B_k(1,x)\hfill \\ \hfill & {\displaystyle +{\displaystyle \frac{2M}{{\delta }^2}}\{[B_k(e^{-2t},x)-e^{-2t}]+2e^{-x}[B_k(e^{-t},x)-e^{-x}]+e^{-2x}[B_k(1,x)-1]\}}\hfill \\ \hfill & {\displaystyle +f\left(x\right)(B_k(1,x)-1)}\hfill \\ \hfill & {\displaystyle =\epsilon (B_k(1,x)-1)+\epsilon }\hfill \\ \hfill & {\displaystyle +{\displaystyle \frac{2M}{{\delta }^2}}\{[B_k(e^{-2t},x)-e^{-2x}]+2e^{-x}[B_k(e^{-t},x)-e^{-x}]+e^{-2x}[B_k(1,x)-1]\}}\hfill \\ \hfill & {\displaystyle +f\left(x\right)(B_k(1,x)-1).}\hfill \end{array}$$

Thus,

$$\begin{array}{cc}\hfill {\displaystyle |B_k(f,x)-f\left(x\right)|}& =\epsilon +\left(\epsilon +{\displaystyle \frac{2M}{{\delta }^2}}+M\right)|B_k(1,x)-1|+{\displaystyle \frac{2M}{{\delta }^2}}|B_k(e^{-2t},x)-e^{-2x}|\hfill \\ \hfill & {\displaystyle +{\displaystyle \frac{4M}{{\delta }^2}}|B_k(e^{-t},x)-e^{-x}|.}\hfill \end{array}$$

Then, taking the supremum over $x\in I$, we obtain

$${\displaystyle \left|\right|B_k(f;x)-f\left(x\right){\left|\right|}_{C\left(I\right)}\le \epsilon +K\left(\sum _{i=0}^2\left|\right|B_k(f_i;x)-f_i\left(x\right){\left|\right|}_{C\left(I\right)}\right),}$$

here $K=\mathit{max}\left\{\epsilon +{\displaystyle \frac{2M}{{\delta }^2}}+M,{\displaystyle \frac{2M}{{\delta }^2}},{\displaystyle \frac{4M}{{\delta }^2}}\right\}.$

Replace $B_n(f;x)$ with

$$t_n^{(E_{\lambda },q)(C_{\lambda },1)}={\displaystyle \frac{1}{{(1+q)}^n}}\sum _{k\in I_n}(\begin{array}{c}n\\ k\end{array})q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}{B}_v(·,x),$$

(12)

on both sides of the results. Given $r>0$, choose a $\eta >0$ such that $\eta <r$ and define the following sets:

$${\displaystyle D=\left\{k\le n:\left|\right|t_n^{(E_{\lambda },q)(C_{\lambda }.1)}(f;x)-f\left(x\right)|{|}_{C\left(I\right)}\ge r\right\}}$$

and

$${\displaystyle D_i=\left\{k\le n:\left|\right|t_n^{(E_{\lambda },q)(C_{\lambda },1)}(f_i;x)-f_i\left(x\right)|{|}_{C\left(I\right)}\ge {\displaystyle \frac{r-\eta }{3K}}\right\},i=0,1,2.}$$

Therefore, $D\subset {\cup }_{i=0}^2{D}_i$ and their densities satisfy the relation below:

$${\displaystyle \delta \left(D\right)\le \sum _{i=0}^2\delta \left(D_i\right).}$$

Finally, from the relations (3)–(5) as well as the above estimates, we obtain:

$${\displaystyle E_{\lambda }{C}_{\lambda }\left(st\right)-\underset{k}{lim}\left|\right|T_k(f;x)-f\left(x\right){\left|\right|}_{\infty }=0,\mathrm{for}\mathrm{all}f\in C\left(I\right)}$$

which completes the proof for this theorem.  □

4. Some Results of Rate of Weighted $({\mathit{E}}_{\mathit{\lambda }},\mathit{q})({\mathit{C}}_{\mathit{\lambda }},\mathbf{1})$—Statistical Convergence

In this section, we consider the rate of weighted $(E_{\lambda },q)(C_{\lambda },1)$-statistical convergence. Rate of statistical convergence was studied by Syed Mohiuddine et al. [17]. The relationship between summability theory and statistical convergence was done by Schoenberg [31]. The statistical convergence as a summability method has been investigated by many authors, including Fridy [32], Freedman et al. [33], and Kolk [34,35].

Next, we give some results about the rate of weighted $(E_{\lambda },q)(C_{\lambda },1)$-statistical convergence.

Definition 7.

Let $\left(a_n\right)$ be a positive non-increasing sequence. We say that the sequence $x=\left(x_n\right)$ is weighted $(E_{\lambda },q)(C_{\lambda },1)$-statistically convergent to the number L with the rate $o\left(a_n\right)$ if for every $\epsilon >0$, we have

$${\displaystyle \underset{n\to \infty }{lim}{\displaystyle \frac{1}{n{a}_n}}\left|\left\{k\le n:|{\displaystyle \frac{1}{{(1+q)}^n}}\sum _{k\in I_n}\left(\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}{x}_v-L|\ge \epsilon \right\}\right|.}$$

In this case, we write

$${\displaystyle x_n-L={\left(EC\right)}_{\lambda }\left(st\right)-o\left(a_n\right).}$$

Now, we have the following results:

Lemma 1.

Let $\left(a_n\right)$ and $\left(b_n\right)$ be positive non-increasing sequences. Let $x=\left(x_n\right)$ and $y=\left(y_n\right)$ be sequences such that $x_n-L_1={\left(EC\right)}_{\lambda }\left(st\right)-o\left(a_n\right)$ and $x_n-L_2={\left(EC\right)}_{\lambda }\left(st\right)-o\left(a_n\right)$. Then,

1. 

$(x_n-L_1)+(y_n-L_2)={\left(EC\right)}_{\lambda }\left(st\right)-o\left(c_n\right)$

2. 

$(x_n-L_1)-(y_n-L_2)={\left(EC\right)}_{\lambda }\left(st\right)-o\left(c_n\right)$

3. 

$\alpha (x_n-L)={\left(EC\right)}_{\lambda }\left(st\right)-o\left(a_n\right)$, for any scalar α

4. 

$(x_n-L_1)(y_n-L_2)={\left(EC\right)}_{\lambda }\left(st\right)-o\left(d_n\right)$

here $c_n=max\{a_n,b_n\}$ and $d_n=a_n{b}_n.$

Proof. 

We first prove $(x_n-L_1)+(y_n-L_2)={\left(EC\right)}_{\lambda }\left(st\right)-o\left(c_n\right).$

For $\epsilon >0$, let

$$M_1{=\{k\le n:|\left(EC\right)}_{\lambda }(x_n+y_n)-(L_1+L_2)|\ge \epsilon \},$$

$$M_2{=\{k\le n:|\left(EC\right)}_{\lambda }\left(x_n\right)-L_1|\ge {\displaystyle \frac{\epsilon }{2}}\},$$

and

$$M_3{=\{k\le n:|\left(EC\right)}_{\lambda }\left(y_n\right)-L_2|\ge {\displaystyle \frac{\epsilon }{2}}\}.$$

We can observe the relation $M_1\subset M_2\cup M_3$. Furthermore, since $c_n=max\{a_n,b_n\}$ we obtain this inequality

$${\displaystyle {\displaystyle \frac{|M_1|}{n·c_n}}\le {\displaystyle \frac{|M_2|}{n·c_n}}+{\displaystyle \frac{|M_3|}{n·c_n}}.}$$

We act with a limit on the last inequality and using the assumption of lemma, we achieve

$${\displaystyle {\displaystyle \frac{|M_1|}{n·c_n}}\to 0,}$$

when $n\to \infty .$ Therefore, we have proved that the first statement of the lemma is true. Statements 2, 3 and 4 are proved similarly to Statement 1.  □

The concept of modulus of continuity is important in proving some statements, so for that reason, we recall this notion. For the function $f\left(x\right)\in C[a,b]$, the modulus of continuity is defined as follows:

$${\displaystyle \omega (f,\delta )=\underset{|x-y|\le \delta }{sup}|f\left(x\right)-f\left(y\right)|.}$$

It is known that, for any value of $|x-y|$, the inequality below is correct:

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

(13)

then we have a result as follows:

Theorem 6.

Let $\left(B_n\right)$ be a sequence of positive linear operators from $C[a,b]$ into $C[a,b]$. Suppose that

1. 

$\parallel B_n{(1;x)-1\parallel }_{\infty }={\left(EC\right)}_{\lambda }\left(st\right)-o\left(a_n\right),$

2. 

$\omega (f,{\lambda }_n)={\left(EC\right)}_{\lambda }\left(st\right)-o\left(b_n\right),$ where ${\lambda }_n=\sqrt{B_n(\mu ;x)}$ and $\mu =\mu (t,x)={(e^{-t}-e^{-x})}^2.$

Then, for all $f\in C[a,b]$, we have

$${\displaystyle \parallel B_n{(f;x)-f\left(x\right)\parallel }_{\infty }={\left(EC\right)}_{\lambda }\left(st\right)-o\left(c_n\right)}$$

where $c_n=max\{a_n,b_n\}.$

Proof. 

Let $f\in C[a,b]$, and $x\in [a,b].$ Using inequations (11) and (13) we have:

$$\begin{array}{c}{\displaystyle |B_n(f;x)-f\left(x\right)|\le B_n\left(\right|f\left(x\right)-f\left(y\right)|;x)+\left|f\left(x\right)\right|·|B_n(1;x)-1|}\hfill \\ {\displaystyle \le B_n({\displaystyle \frac{|e^{-t}-e^{-x}|}{\delta }}+1;x)·\omega (f,\delta )+|f\left(x\right)|·|B_n(1;x)-1|}\hfill \\ {\displaystyle \le B_n({\displaystyle \frac{{(e^{-t}-e^{-x})}^2}{{\delta }^2}}+1;x)·\omega (f,\delta )+|f\left(x\right)|·|B_n(1;x)-1|}\hfill \\ {\displaystyle \le (B_n(1;x)+{\displaystyle \frac{B_n(\mu ;x)}{{\delta }^2}}+1;x)·\omega (f,\delta )+|f\left(x\right)|·|B_n(1;x)-1|}\hfill \\ {\displaystyle \le (B_n(1;x)-1)·\omega (f,\delta )+\left|f\left(x\right)\right|·|B_n(1;x)-1|+\omega (f,\delta )}\hfill \\ {\displaystyle +{\displaystyle \frac{\omega (f,\delta )B_n(\mu ;x)}{{\delta }^2}}.}\hfill \end{array}$$

If we put $\delta ={\lambda }_n=\sqrt{B_n(\mu ;x)}$, we obtain

$$\begin{array}{cc}\hfill {\displaystyle \parallel B_n{(f;x)-f\left(x\right)\parallel }_{\infty }}& \le {\parallel f\parallel }_{\infty }·\parallel B_n{(1;x)-1\parallel }_{\infty }+2·\omega (f,{\lambda }_n)+\omega (f,{\lambda }_n)·{\parallel B_n(1;x)-1\parallel }_{\infty }\hfill \\ \hfill & {\displaystyle \le \mathcal{H}\{\parallel B_n(1;x)-1{\parallel }_{\infty }+\omega (f,{\lambda }_n)+\omega (f,{\lambda }_n)·\parallel B_n(1;x)-1{\parallel }_{\infty }\}}\hfill \end{array}$$

where $\mathcal{H}=max\{\parallel f{\parallel }_{\infty },2\}.$ Therefore,

$$\begin{array}{c}{\displaystyle \parallel B_n(f;x)(\begin{array}{c}n\\ k\end{array})q^{n-k}\frac{1}{k+1}\sum _{v\in I_k}{B}_v(.,x)-f(x){\parallel }_{\infty }}\hfill \\ {\displaystyle \le \mathcal{H}\left\{\parallel B_n(1;x)(\begin{array}{c}n\\ k\end{array}\right)q^{n-k}\frac{1}{k+1}\sum _{v\in I_k}{B}_v(.,x)-1{\parallel }_{\infty }+\omega (f,{\lambda }_n)}\hfill \\ {\displaystyle +\omega (f,{\lambda }_n)·\left(\begin{array}{c}n\\ k\end{array}\right)q^{n-k}\frac{1}{k+1}\sum _{v\in I_k}{B}_v(.,x)·\parallel B_n(1;x)(\begin{array}{c}n\\ k\end{array}\left)q^{n-k}\frac{1}{k+1}\sum _{v\in I_k}{B}_v(.,x)-1{\parallel }_{\infty }\right\}.}\hfill \end{array}$$

Now using Definition 7 and Conditions 1 and 2 of Theorem 6, we obtain the desired result. This completes the proof of the theorem.  □

5. Weighted $({\mathit{E}}_{\mathit{\lambda }},\mathit{q})({\mathit{C}}_{\mathit{\lambda }},\mathbf{1})$-Statistical Convergence and Orlicz Function

We have seen throughout that the relationship between statistical convergence and summabilities methods plays a major role in this article. As seen in the introduction of the paper, Orlicz functions have been investigated by different authors. E. Savas and R. Savash studied some results for the sequence spaces $[V,M,p],{[V,M,p]}_0,{[V,M,p]}_{\infty }$ (see [24]). Naim Braha, Hari Mohan Srivastava and Mikail At (see [4]) defined the sequences spaces ${[N_{p,q}{C}_n^1,M,p]}_0$, ${[N_{p,q}{C}_n^1,M,p]}_1$, ${[N_{p,q}{C}_n^1,M,p]}_{\infty }$. We emphasize that the above works have inspired us to investigate some topological results of sequence spaces ${[(E_{\lambda },q)(C_{\lambda },1),M,p]}_0$, ${[(E_{\lambda },q)(C_{\lambda },1),M,p]}_1$, ${[(E_{\lambda },q)(C_{\lambda },1),M,p]}_{\infty }.$

We recall the definition of sequence space $l_M$, which is given as follows:

$${\displaystyle l_M=\left\{x=\left(x_i\right):\sum _{i=1}^{\infty }M\left({\displaystyle \frac{|x_i|}{\varrho }}\right)<\infty \mathrm{for}\mathrm{some}\varrho >0\right\}.}$$

also, this is a Banach space with a norm, which can be expressed as:

$${\displaystyle \parallel x\parallel =inf\left\{\varrho >0:\sum _{i=1}^{\infty }M\left({\displaystyle \frac{|x_i|}{\varrho }}\right)\le 1\right\}.}$$

Remark 3.

If we put $M\left(x\right)=x^p,$ for $1\le p<\infty ,$ then the space $l_M$ complies with the space $l_p.$

Definition 8.

An Orlicz function is a type of function $M:[0,\infty )\to [0,\infty ),$ that is non-decreasing, continuous, and convex with $M\left(0\right)=0,$ $M\left(x\right)>0$ for $x>0$ and $M\left(x\right)\to \infty$ as $x\to \infty .$

If $M(x+y)\le M\left(x\right)+M\left(y\right)$ is substituted for the convexity of an Orlicz function M, then the function M is referred to as a modulus function.

For all values of $u,{▵}_2$-condition is said to satisfy an Orlicz function M, if there exists a constant $K>0$, e.g., $M\left(2u\right)\le KM\left(u\right),$$u\ge 0.$

There are different sequence spaces, which have been introduced by many researchers. Vinod and Niranja introduced a lacunary sequence, which is a strong convergence with respect to an Orlicz function [25]. In 2010, Osama et al. studied $\lambda$-statistically convergent sequences and used them to prove some analogues of the classical Korovkin theorem [16]. In 2012, the concept of statistical summability $(C,1)$ was studied by applying the sequence of classical Baskakov operator to construct examples for supporting the results [8].

In this work, we will consider the sequence space of:

${[(E_{\lambda },q)(C_{\lambda },1),M,p]}_0$, ${[(E_{\lambda },q)(C_{\lambda },1),M,p]}_1,$${[(E_{\lambda },q)(C_{\lambda },1),M,p]}_{\infty }.$ The details are in the following definitions:

Definition 9.

Let $p=\left(p_k\right)$ be a sequence of strictly positive real numbers, and M be an Orlicz function, the sequence spaces ${[(E_{\lambda },q)(C_{\lambda },1),M,p]}_0$ can be defined as follows:

$${\displaystyle {[(E_{\lambda },q)(C_{\lambda },1),M,p]}_0=\left\{x\in \omega :\underset{n}{lim}{\displaystyle \frac{1}{{(1+q)}^n}}\sum _{k\in I_n}{\left[M\left({\displaystyle \frac{\left(\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{\displaystyle \frac{1}{k+1}}{\displaystyle \sum _{v\in I_k}}|x_v|}{\rho }}\right)\right]}^{p_k}=0\right\}}$$

for some $\rho >0.$

Definition 10.

Let $p=\left(p_k\right)$ be a sequence of strictly positive real numbers, and M be an Orlicz function, the sequence spaces ${[(E_{\lambda },q)(C_{\lambda },1),M,p]}_1$ can be defined as follows:

$${[(E_{\lambda },q)(C_{\lambda },1),M,p]}_1=\left\{x\in \omega :\underset{n}{lim}{\displaystyle \frac{1}{{(1+q)}^n}}\sum _{k\in I_n}{\left[M\left({\displaystyle \frac{{\displaystyle \left(\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|x_v-L|}}{\rho }}\right)\right]}^{p_k}=0\right\}$$

for some $\rho >0.$ If $x\in {[(E,q)(C,1),M,p]}_1$, we say that the sequence $x=\left(x_k\right)$ is weighted strongly Euler $(E,q)(C,1)$-summable to the definite number L with respect to the Orlicz function M. We write $x_k\to L\left([(E_{\lambda },q)(C_{\lambda },1),M,p]\right)$ in this case.

Definition 11.

Let $p=\left(p_k\right)$ be a sequence of strictly positive real numbers, and M be an Orlicz function, the sequence spaces ${[(E_{\lambda },q)(C_{\lambda },1),M,p]}_{\infty }$ can be defined as follows:

$${\displaystyle {[(E_{\lambda },q)(C_{\lambda },1),M,p]}_{\infty }=\left\{x\in \omega :\underset{n}{sup}{\displaystyle \frac{1}{{(1+q)}^n}}\sum _{k\in I_n}{\left[M\left({\displaystyle \frac{{\displaystyle \left(\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|x_v|}}{\rho }}\right)\right]}^{p_k}<\infty \right\}}$$

for some $\rho >0$.

From the Definitions 9–11 if $M\left(x\right)$ is equal to x, then instead of ${[(E_{\lambda },q)(C_{\lambda },1),M,p]}_0$, ${[(E_{\lambda },q)(C_{\lambda },1),M,p]}_1$ and ${[(E_{\lambda },q)(C_{\lambda },1),M,p]}_{\infty },$ respectively, we will write ${[(E_{\lambda },q)(C_{\lambda },1),p]}_0$, ${[(E_{\lambda },q)(C_{\lambda },1),p]}_1$ and ${[(E_{\lambda },q)(C_{\lambda },1),p]}_{\infty },$ In a special case, when $M\left(x\right)$ is equal to x and $p_k$ is equal to $p_0$ for all $k\in \mathbb{N},$ instead of writing ${[(E_{\lambda },q)(C_{\lambda },1),M,p]}_0$, ${[(E_{\lambda },q)(C_{\lambda },1),M,p]}_1$ and ${[(E_{\lambda },q)(C_{\lambda },1),M,p]}_{\infty },$ we write ${\left[(E_{\lambda },q)(C_{\lambda },1)\right]}_0$, ${\left[(E_{\lambda },q)(C_{\lambda },1)\right]}_1$ and ${\left[(E_{\lambda },q)(C_{\lambda },1)\right]}_{\infty }.$

Lemma 2

([22]). Let M be an Orlicz function, which satisfies the ${▵}_2$-condition, and let $0<\delta <1.$ Then, for each $x\ge \delta ,$

$$M\left(x\right)<Kx{\delta }^{-1}M\left(2\right)$$

for some constant $K>0$.

Theorem 7.

Let M be Orlicz function and $p=\left(p_k\right)$ be a sequence of strictly positive real numbers, the sequence spaces ${[(E_{\lambda },q)(C_{\lambda },1),M,p]}_0,$${[(E_{\lambda },q)(C_{\lambda },1),M,p]}_1,$${[(E_{\lambda },q)(C_{\lambda },1),M,p]}_{\infty }$ are linear spaces over the set of complex numbers.

Proof. 

We begin by proving that the sequence space ${[(E_{\lambda },q)(C_{\lambda },1),M,p]}_0,$ is a linear space over the complex numbers.

To show that this space is linear, we must find positive ${\rho }_3$ such that

$${\displaystyle \underset{n}{lim}{\displaystyle \frac{1}{{(1+q)}^n}}\sum _{k\in I_n}{\left[M\left({\displaystyle \frac{{\displaystyle \left(\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|\alpha x_v+\beta y_v|}}{{\rho }_3}}\right)\right]}^{p_k}=0.}$$

Let $x,y\in {[(E_{\lambda },q)(C_{\lambda },1),M,p]}_0$ and $\alpha ,\beta \in \mathbb{C},$ then there exist ${\rho }_1>0$ and ${\rho }_2>0$ such that

$${\displaystyle \underset{n}{lim}{\displaystyle \frac{1}{{(1+q)}^n}}\sum _{k\in I_n}{\left[M\left({\displaystyle \frac{{\displaystyle \left(\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|x_v|}}{{\rho }_1}}\right)\right]}^{p_k}=0}$$

and

$${\displaystyle \underset{n}{lim}{\displaystyle \frac{1}{{(1+q)}^n}}\sum _{k\in I_n}{\left[M\left({\displaystyle \frac{{\displaystyle \left(\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|y_v|}}{{\rho }_2}}\right)\right]}^{p_k}=0.}$$

Now, we define ${\rho }_3=max\left(2\right|\alpha |{\rho }_1,2\left|\beta \right|{\rho }_2).$ Since M is non-decreasing and convex,

$$\begin{array}{c}{\displaystyle {\displaystyle \frac{1}{{(1+q)}^n}}\sum _{k\in I_n}{\left[M\left({\displaystyle \frac{{\displaystyle \left(\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|\alpha x_v+\beta y_v|}}{{\rho }_3}}\right)\right]}^{p_k}}\hfill \\ {\displaystyle \le {\displaystyle \frac{1}{{(1+q)}^n}}\sum _{k\in I_n}{\left[M\left({\displaystyle \frac{{\displaystyle \left(\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|\alpha x_v|}}{{\rho }_3}}+{\displaystyle \frac{{\displaystyle \left(\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|\beta y_v|}}{{\rho }_3}}\right)\right]}^{p_k}}\hfill \\ {\displaystyle ={\displaystyle \frac{1}{{(1+q)}^n}}\sum _{k\in I_n}{\left[M\left({\displaystyle \frac{{\displaystyle \alpha \left(\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|x_v|}}{2\alpha {\rho }_1}}\right)+M\left({\displaystyle \frac{{\displaystyle \beta \left(\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|y_v|}}{2\beta {\rho }_2}}\right)\right]}^{p_k}}\hfill \\ {\displaystyle \le {\displaystyle \frac{1}{{(1+q)}^n}}\sum _{k\in I_n}{\displaystyle \frac{1}{2^{p_k}}}{\left[M\left({\displaystyle \frac{{\displaystyle \left(\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|x_v|}}{{\rho }_1}}\right)+M\left({\displaystyle \frac{{\displaystyle \left(\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|y_v|}}{{\rho }_2}}\right)\right]}^{p_k}}\hfill \\ {\displaystyle \le {\displaystyle \frac{1}{{(1+q)}^n}}\sum _{k\in I_n}{\left[M\left({\displaystyle \frac{{\displaystyle \left(\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|x_v|}}{{\rho }_1}}\right)+M\left({\displaystyle \frac{{\displaystyle \left(\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|y_v|}}{{\rho }_2}}\right)\right]}^{p_k}}\hfill \\ {\displaystyle \le K·{\displaystyle \frac{1}{{(1+q)}^n}}\sum _{k\in I_n}{\left[M\left({\displaystyle \frac{{\displaystyle \left(\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|x_v|}}{{\rho }_1}}\right)\right]}^{p_k}}\hfill \\ {\displaystyle +K·{\displaystyle \frac{1}{{(1+q)}^n}}\sum _{k\in I_n}{\left[M\left({\displaystyle \frac{{\displaystyle \left(\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|y_v|}}{{\rho }_2}}\right)\right]}^{p_k}}\hfill \end{array}$$

if we act with a limit to the above relations when n tends to infinity, we have:

$$\begin{array}{c}{\displaystyle K·{\displaystyle \frac{1}{{(1+q)}^n}}\sum _{k\in I_n}{\left[M\left({\displaystyle \frac{{\displaystyle \left(\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|x_v|}}{{\rho }_1}}\right)\right]}^{p_k}}\hfill \\ {\displaystyle +K·{\displaystyle \frac{1}{{(1+q)}^n}}\sum _{k\in I_n}{\left[M\left({\displaystyle \frac{{\displaystyle \left(\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|y_v|}}{{\rho }_2}}\right)\right]}^{p_k}\to 0}\hfill \end{array}$$

which means

$${\displaystyle {\displaystyle \frac{1}{{(1+q)}^n}}\sum _{k\in I_n}{\left[M\left({\displaystyle \frac{{\displaystyle \left(\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|\alpha x_v+\beta y_v|}}{{\rho }_3}}\right)\right]}^{p_k}\to 0}$$

where $K=max(1,2^{H-1}),H=sup{p}_k.$ So, we show that $\alpha x+\beta y\in {[(E_{\lambda },q)(C_{\lambda },1),M,p]}_0.$

Thus, this completes the proof of theorem.

Similarly, we can prove that the space ${[(E_{\lambda },q)(C_{\lambda },1),M,p]}_1,$${[(E_{\lambda },q)(C_{\lambda },1),M,p]}_{\infty }$ are linear spaces over the set of complex numbers.  □

The following results are proved directly, hence we will appropriate them without proof.

Theorem 8.

Let M be an Orlicz function, then

$${\displaystyle {[(E_{\lambda },q)(C_{\lambda },1),M,p]}_0\subset {[(E_{\lambda },q)(C_{\lambda },1),M,p]}_1\subset {[(E_{\lambda },q)(C_{\lambda },1),M,p]}_{\infty }.}$$

Theorem 9.

Let $M_1$ and $M_2$ be two Orlicz functions. Then,

$${[(E_{\lambda },q)(C_{\lambda },1),M_1,p]}_z\cap {[(E_{\lambda },q)(C_{\lambda },1),M_2,p]}_z\subseteq {[(E_{\lambda },q)(C_{\lambda },1),M_1+M_2,p]}_z,$$

for $Z=l_{\infty },c$ and $c_0.$

Theorem 10.

Let $0<p_k<q_k$ and $\left({\displaystyle \frac{q_k}{p_k}}\right)$ be bounded. Then,

$${\displaystyle {[(E_{\lambda },q)(C_{\lambda },1),M,q]}_1\subset [E_{\lambda },q)(C_{\lambda },1){,M,p]}_1.}$$

We now prove the following result.

Theorem 11.

Let M be any Orlicz function, let $p=\left(p_k\right)$ be a bounded sequence of strictly positive real numbers, and the space ${[(E_{\lambda },q)(C_{\lambda },1),M,p]}_0$ is a paranormed space (not necessarily total paranormed) with

$${\displaystyle g\left(x\right)=inf\left\{{\rho }^{p_n{/}_H}\in \omega :{\left({\displaystyle \frac{1}{{(1+q)}^n}}\sum _{k\in I_n}{\left[M\left({\displaystyle \frac{\left(\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{\displaystyle \frac{1}{k+1}}{\displaystyle \sum _{v\in I_k}}|x_v|}{\rho }}\right)\right]}^{p_k}\right)}^{{\displaystyle \frac{1}{H}}}\le 1,n=1,2,\dots \right\}}$$

where $H=max(1,sup{p}_k)$.

Proof. 

It is quite clear that we have two statements; $g\left(x\right)=g(-x)$ and $g(x+y)\le g\left(x\right)+g\left(y\right).$ From $M\left(0\right)$, which is equal to zero, we obtain

$${\displaystyle inf\left\{{\rho }^{r_n{/}_H}\right\}\mathrm{for}x=0.}$$

In contrast, if we assume that $g\left(x\right)$ is equal to zero, then x is equal to zero. Finally, using the same technique as in Bhardwaj and Singh, it is clear that scalar multiplication is continuous, see [2]. Therefore, this proves the Theorem 11.  □

Theorem 12.

For any Orlicz function M, let ${\displaystyle 0<h=\underset{k}{inf}{p}_k\le p_k\le \underset{k}{sup}{p}_k=H}$. Then, the following statement is true:

$${\displaystyle {[(E_{\lambda },q)(C_{\lambda },1),M,p]}_1\subset S_{{\left(EC\right)}_{\lambda }}.}$$

Proof. 

Firstly, we assume that $x\in {[(E_{\lambda },q)(C_{\lambda },1),M,p]}_1.$ Then, there exists the value $\rho >0$ such that:

$${\displaystyle {\displaystyle \frac{1}{{(1+q)}^n}}\sum _{k\in I_n}{\left[M\left({\displaystyle \frac{{\displaystyle \left(\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|x_v-L|}}{\rho }}\right)\right]}^{p_k}\to 0,\mathrm{as}n\to \infty .}$$

Let us denote that ${\sum }_1$ is the sum over $k\le {(1+q)}^n$ with ${\displaystyle \left(\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|x_v-L|\ge \epsilon }$ and the sum over $k\le {(1+q)}^n$ with ${\displaystyle \left(\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|x_v-L|<\epsilon }$ is also symbolized by ${\sum }_2.$

Hence, for any given $\epsilon >0$, we have

$$\begin{array}{l}{\displaystyle {\displaystyle \frac{1}{{(1+q)}^n}}\sum _{k\in I_n}{\left[M\left({\displaystyle \frac{{\displaystyle \left(\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|x_v-L|}}{\rho }}\right)\right]}^{p_k}}\\ \hspace{1em}={\displaystyle \frac{1}{{(1+q)}^n}}{\sum }_1{\left[M\left({\displaystyle \frac{{\displaystyle \left(\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|x_v-L|}}{\rho }}\right)\right]}^{p_k}\\ \hspace{1em}+{\displaystyle \frac{1}{{(1+q)}^n}}{\sum }_2{\left[M\left({\displaystyle \frac{{\displaystyle \left(\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|x_v-L|}}{\rho }}\right)\right]}^{p_k}\\ \hspace{1em}\ge {\displaystyle \frac{1}{{(1+q)}^n}}{\sum }_1{\left[M\left({\displaystyle \frac{{\displaystyle \left(\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|x_v-L|}}{\rho }}\right)\right]}^{p_k}\\ \hspace{1em}\ge {\displaystyle \frac{1}{{(1+q)}^n}}{\sum }_1{\left[M\left({\displaystyle \frac{\epsilon }{\rho }}\right)\right]}^{p_k}\\ \hspace{1em}={\displaystyle \frac{1}{{(1+q)}^n}}{\sum }_1{\left[M\left({\epsilon }_1\right)\right]}^{p_k}\\ \hspace{1em}{\displaystyle \ge {\displaystyle \frac{1}{{(1+q)}^n}}\sum _{k\le {(1+q)}^n}min\{{[M\left({\epsilon }_1\right)]}^h,{[M({\epsilon }_1]}^H\}}\\ \hspace{1em}{\displaystyle \ge {\displaystyle \frac{1}{{(1+q)}^n}}|\left\{k\le {(1+q)}^n:(\begin{array}{c}n\\ k\end{array})\right.q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|x_v-L|\ge \epsilon \left\}\right|·min\{{[M({\epsilon }_1)]}^h,{[M({\epsilon }_1]}^H\}.}\end{array}$$

Therefore, $x=\left(x_k\right)\in S_{{\left(EC\right)}_{\lambda }}$. This completes the proof of Theorem 12.  □

Theorem 13.

In the definition of Orlicz function, if a bounded function M does not satisfy the condition $M\left(x\right)\to \infty (x\to \infty ),$ then $S_{{\left(EC\right)}_{\lambda }}\subset [(E_{\lambda },q)(C_{\lambda },1),M].$

Proof. 

Let us assume that $M\left(y\right)\le K$ for some positive constant K and all $y\ge 0$ and let $\epsilon >0$ and select $\delta >0$ such that $M\left(t\right)<\epsilon$ for $0\le t\le \delta .$ Then, for (11) proof, which is written in the ∑-notation, we have:

$$\begin{array}{c}{\displaystyle {\displaystyle \frac{1}{{(1+q)}^n}}\sum _{k\le {(1+q)}^n}\left[M\left((\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|x_v-L|\right)]}\hfill \\ {\displaystyle ={\displaystyle \frac{1}{{(1+q)}^n}}{\sum }_1\left[M\left((\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|x_v-L|\right)]}\hfill \\ {\displaystyle +{\displaystyle \frac{1}{{(1+q)}^n}}{\sum }_2\left[M\left((\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|x_v-L|\right)]}\hfill \\ {\displaystyle \le {\displaystyle \frac{K}{{(1+q)}^n}}|\left\{k\le {(1+q)}^n:(\begin{array}{c}n\\ k\end{array})\right.q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|x_v-L|\ge \epsilon \}|+{\displaystyle \frac{M\left(\delta \right)}{{(1+q)}^n}}·{(1+q)}^n}\hfill \\ {\displaystyle \le {\displaystyle \frac{K}{{(1+q)}^n}}|\left\{k\le {(1+q)}^n:(\begin{array}{c}n\\ k\end{array})\right.q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|x_v-L|\ge \epsilon \}|+\epsilon .}\hfill \end{array}$$

Therefore, $x_n\to L[(E_{\lambda },q)(C_{\lambda },1),M].$ The proof of Theorem 13 is thus completed.  □

Theorem 14.

Let M be an Orlicz function which satisfies the condition of ${▵}_2$, and it is asserted as ${\left[(E_{\lambda },q)(C_{\lambda },1)\right]}_1\subset {[(E_{\lambda },q)(C_{\lambda },1),M]}_1.$

Proof. 

Let $x\in [\omega ,\eta ],$ so that

$${\displaystyle A_n\equiv {\displaystyle \frac{K}{{(1+q)}^n}}\sum _{k\in I_n}\left[(\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|x_v-L|]\to 0,(n\to \infty )}$$

for some definite number L. We assume that $\epsilon >0$ and select $\delta$ with $0<\delta <1$ such that $M\left(t\right)<\epsilon (0\le t\le \delta ).$ Next, we consider the ∑-notation, which is used in the proof of (11), and we can write:

$$\begin{array}{c}{\displaystyle {\displaystyle \frac{1}{{(1+q)}^n}}\sum _{k\in I_n}\left[M\left((\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|x_v-L|\right)]}\hfill \\ {\displaystyle ={\displaystyle \frac{1}{{(1+q)}^n}}{\sum }_1\left[M\left((\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|x_v-L|\right)]}\hfill \\ {\displaystyle +{\displaystyle \frac{1}{{(1+q)}^n}}{\sum }_2\left[M\left((\begin{array}{c}n\\ k\end{array}\right)q^{n-k}{\displaystyle \frac{1}{k+1}}\sum _{v\in I_k}|x_v-L|\right)]}\hfill \\ <{\displaystyle \frac{\epsilon {(1+q)}^n}{{(1+q)}^n}}+{\displaystyle \frac{KM\left(2\right)A_n}{\delta }},\hfill \end{array}$$

for some constant K. Hence, in Lemma 2, by letting n tends to infinity, it is clearly seen that $x\in {\left[(E_{\lambda },q)(C_{\lambda },1)\right]}_1,$ for $\rho =1.$ This completes the proof of the theorem.  □

6. Conclusions

In this research, we have generalized Euler–Cesaro summability method, which we symbolically write as $(E_{\lambda },q)(C_{\lambda },1)$. Moreover, we have given the definition of: weighted $(E_{\lambda },q)(C_{\lambda },1)$ summablity, weighted $(E_{\lambda },q)(C_{\lambda },1)$ strong summablity, weighted $(E_{\lambda },q)(C_{\lambda },1)$ statistical convergence, $S_{{\left(EC\right)}_{\lambda }}$ convergence, weighted ${\left[(E_{\lambda },q)(C_{\lambda },1)\right]}_r^{\lambda }$ summability. We have also defined sequence spaces by Orlicz functions as

${[(E_{\lambda },q)(C_{\lambda },1),M,p]}_0,{[(E_{\lambda },q)(C_{\lambda },1),M,p]}_1$, and ${[(E_{\lambda },q)(C_{\lambda },1),M,p]}_{\infty }$. Using this definition, we have proved some topological and algebraic results as well as applied the Korovkin type theorem.

Future research based on this summability method may point in a new direction:

• Positive linear operators, also known as modified Baskakov operators, satisfy a Voronovskaya type theorem involving the $(E_{\lambda },q)(C_{\lambda },1)$ statistical convergence.
• Rate of weighted $(E_{\lambda },q)(C_{\lambda },1)$ statistical convergence for generalized of some opertaors like Blending-Type Bernstein–Kantorovich operators.