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# Journal of Applied Analysis

Editor-in-Chief: Liczberski, Piotr / Ciesielski, Krzysztof

Managing Editor: Gajek, Leslaw

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Volume 24, Issue 1

# Deferred weighted 𝒜-statistical convergence based upon the (p,q)-Lagrange polynomials and its applications to approximation theorems

H. M. Srivastava
• Corresponding author
• Department of Mathematics and Statistics, University of Victoria, Victoria, BC, V8W 3R4, Canada; and Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan, P. R. China
• Email
• Other articles by this author:
/ Bidu Bhusan Jena
/ Susanta Kumar Paikray
/ U. K. Misra
• Department of Mathematics, National Institute of Science and Technology, Palur Hills, Golanthara 761008, Odisha, India
• Email
• Other articles by this author:
Published Online: 2018-05-03 | DOI: https://doi.org/10.1515/jaa-2018-0001

## Abstract

Recently, the notion of positive linear operators by means of basic (or q-) Lagrange polynomials and $\mathcal{𝒜}$-statistical convergence was introduced and studied in [M. Mursaleen, A. Khan, H. M. Srivastava and K. S. Nisar, Operators constructed by means of q-Lagrange polynomials and A-statistical approximation, Appl. Math. Comput. 219 2013, 12, 6911–6918]. In our present investigation, we introduce a certain deferred weighted $\mathcal{𝒜}$-statistical convergence in order to establish some Korovkin-type approximation theorems associated with the functions 1, t and ${t}^{2}$ defined on a Banach space $C\left[0,1\right]$ for a sequence of (presumably new) positive linear operators based upon $\left(p,q\right)$-Lagrange polynomials. Furthermore, we investigate the deferred weighted $\mathcal{𝒜}$-statistical rates for the same set of functions with the help of the modulus of continuity and the elements of the Lipschitz class. We also consider a number of interesting special cases and illustrative examples in support of our definitions and of the results which are presented in this paper.

MSC 2010: 40A05; 41A36; 40G15

## 1 Introduction, definitions and motivation

In the study of sequence spaces, classical convergence has got innumerable applications where the convergence of a sequence requires that almost all elements are to satisfy the convergence condition, that is, all the elements of the sequence need to be in an arbitrarily small neighborhood of the limit. However, such restriction is relaxed in statistical convergence, where the validity of the convergence condition is achieved only for a majority of elements. The notion of statistical convergence was introduced and studied by Fast [12] and Steinhaus [36]. Recently, statistical convergence has been a dynamic research area, basically due to the fact that it is more general than classical convergence, and such theory is discussed in the study in the areas of (for instance) Fourier Analysis, Number Theory and Approximation Theory. Gradually the idea has been extended to different summability methods. For more details of statistical convergence as well as statistical summability, see the recent works [6, 14, 16, 19, 29, 31, 32, 33, 35].

An application of statistical summability gave rise to the theory of statistical approximation, which has been an active area of research in the last decade. For more details, the statistical approximation properties have been investigated for q-analogues of several operators. For example, in [2] a q-analogue of Butzer and Hahn operators, in [3, 24] a q-analogue of Stancu–Beta operators, in [11] a q-Bleimann–Butzer–Hahn operator, in [15] a q-Baskakov–Kantorovich operator, in [18] a q-analogue of the Meyer–König and Zeller operator, in [34] a q-Szász–Mirakjan–Kantorovich-type operator, in [28] a q-Szász–Mirakjan operator, and in [30] a q-analogue of the Bernstein–Kantorovich operator were defined and their statistical approximation properties were investigated.

Subsequently, the statistical approximation properties have also been investigated for $\left(p,q\right)$-analogues of several operators. For example, Mursaleen et al. [21] first applied the concept of the $\left(p,q\right)$-calculus in approximation theory as an alternative to the q-calculus. Later, based upon $\left(p,q\right)$-integers, a $\left(p,q\right)$-analogue of the Bernstein–Stancu operators [22], a Kantorovich variant of the $\left(p,q\right)$-Szász–Mirakjan operators [20], the $\left(p,q\right)$-Lorentz polynomials on a compact disk [26], a $\left(p,q\right)$-analogue of the Bernstein–Schurer operators [27], and so on, have also been introduced.

Let $ℕ$ be the set of natural numbers and let $K\subseteq ℕ$. Also let

and suppose that $|{K}_{n}|$ is the cardinality of ${K}_{n}$. Then the natural density of K is defined by

provided that the limit exists.

A given sequence $\left({x}_{n}\right)$ is said to be statistically convergent to $\mathrm{\ell }$ if, for each $ϵ>0$, the set

has natural density zero (see [12, 36]). This means that, for each $ϵ>0$, we have

In this case, we write

$\mathrm{stat}\underset{n\to \mathrm{\infty }}{lim}{x}_{n}=\mathrm{\ell }.$

We present below an example to illustrate that every convergent sequence is statistically convergent but the converse is not true.

#### Example 1.1.

Let us consider the sequence $x=\left({x}_{n}\right)$ by

Then it is easy to see that the sequence $\left({x}_{n}\right)$ is divergent in the ordinary sense, while 0 is the statistical limit of $\left({x}_{n}\right)$ since $d\left(K\right)=0$, where

$K=\left\{{m}^{2}:m\in ℕ\right\}.$

In the year 2009, Karakaya and Chishti [17] introduced the fundamental concept of weighted statistical convergence, and their definition was later modified by Mursaleen, Karakaya, Ertürk and Gürsoy [23].

Suppose that $\left({s}_{k}\right)$ is a sequence of non-negative numbers with n-th partial sum ${S}_{n}$ such that

${S}_{n}=\sum _{k=0}^{n}{s}_{k}\mathit{ }\left({s}_{0}>0,n\to \mathrm{\infty }\right).$

Then, by setting

${t}_{n}=\frac{1}{{S}_{n}}\sum _{k=0}^{n}{s}_{k}{x}_{k}\mathit{ }\left(n\in {ℕ}_{0}:=ℕ\cup \left\{0\right\}\right),$

the given sequence $\left({x}_{n}\right)$ is said to be weighted statistically convergent (or ${\mathrm{stat}}_{\overline{N}}$-convergent) to a number $\mathrm{\ell }$ if, for each $ϵ>0$, the set

has zero weighted density (see [23]). This means that, for each $ϵ>0$, we have

In this case, we write

${\mathrm{stat}}_{\overline{N}}lim{x}_{n}=\mathrm{\ell }.$

In the year 2013, Belen and Mohiuddine [4] established a new technique for weighted statistical convergence in terms of the de la Vallée Poussin mean, and it was subsequently investigated further by Braha, Srivastava and Mohiuddine [7] as the ${\mathrm{\Lambda }}_{n}$-weighted statistical convergence. Very recently, a certain class of weighted statistical convergence and associated Korovkin-type approximation theorems involving trigonometric functions were established by Srivastava, Jena, Paikray and Misra [33].

Let X and Y be two sequence spaces and let $\mathcal{𝒜}=\left({a}_{n,k}\right)$ be an infinite matrix. If, for each ${x}_{k}\in X$, the series given by

${\mathcal{𝒜}}_{n}x=\sum _{k=1}^{\mathrm{\infty }}{a}_{n,k}{x}_{k}$

converges for each $n\in ℕ$ and the sequence $\left({\mathcal{𝒜}}_{n}x\right)$ belongs to Y, then we say that the matrix $\mathcal{𝒜}$ maps X into Y. By the symbol $\left(X,Y\right)$ we denote the set of all matrices which map X into Y.

We say that $\mathcal{𝒜}$ is regular if

$\underset{n\to \mathrm{\infty }}{lim}{\mathcal{𝒜}}_{n}x=\mathrm{\ell }\mathit{ }\text{whenever}\mathit{ }\underset{k\to \mathrm{\infty }}{lim}{x}_{k}=\mathrm{\ell }.$

The well-known Silverman–Toeplitz theorem (see, for details, [5]) asserts that $\mathcal{𝒜}=\left({a}_{n,k}\right)$ is regular if and only if the following conditions hold:

• (i)

${sup}_{n\to \mathrm{\infty }}{\sum }_{k=1}^{\mathrm{\infty }}|{a}_{n,k}|<\mathrm{\infty }$.

• (ii)

${lim}_{n\to \mathrm{\infty }}{a}_{n,k}=0$ for each k.

• (iii)

${lim}_{n\to \mathrm{\infty }}{\sum }_{k=1}^{\mathrm{\infty }}{a}_{n,k}=1$.

Freedman and Sember [13] extended the definition of statistical convergence with the help of the non-negative regular matrix $\mathcal{𝒜}=\left({a}_{n,k}\right)$, which they called the $\mathcal{𝒜}$-statistical convergence. For any non-negative regular matrix $\mathcal{𝒜}$, we say that a sequence $\left({x}_{n}\right)$ is $\mathcal{𝒜}$-statistically convergent (or ${\mathrm{stat}}_{\mathcal{𝒜}}$-convergent) to a number $\mathrm{\ell }$ if, for each $ϵ>0$, the set

has zero $\mathcal{𝒜}$-density. This means that, for each $ϵ>0$, we have

$\underset{n\to \mathrm{\infty }}{lim}\sum _{k:|{x}_{k}-\mathrm{\ell }|\geqq ϵ}{a}_{n,k}=0.$

In this case, we write

${\mathrm{stat}}_{\mathcal{𝒜}}lim{x}_{n}=\mathrm{\ell }.$

In the year 2016, Mohiuddine [19] introduced the notion of weighted $\mathcal{𝒜}$-summability by using a weighted regular matrix. He also introduced the definitions of statistically weighted $\mathcal{𝒜}$-summability and weighted $\mathcal{𝒜}$-statistical convergence. In particular, he proved a Korovkin-type approximation theorem by means of statistically weighted $\mathcal{𝒜}$-summable sequences of real or complex numbers. Subsequently, Kadak, Braha and Srivastava [16] have investigated the notion of statistical weighted $\mathcal{ℬ}$-summability by using a weighted regular matrix to establish some approximation theorems.

Motivated essentially by the above-mentioned works, here we establish the concept of deferred weighted $\mathcal{𝒜}$-statistical convergence.

Let $\left({a}_{n}\right)$ and $\left({b}_{n}\right)$ be sequences of non-negative integers satisfying the following conditions:

• (i)

${a}_{n}<{b}_{n}$ ($n\in ℕ$).

• (ii)

${lim}_{n\to \mathrm{\infty }}{b}_{n}=\mathrm{\infty }$.

The above conditions (i) and (ii) are known as the regularity conditions for the deferred weighted mean (see Agnew [1]).

We next suppose that $\left({s}_{n}\right)$ is the sequence of non-negative real numbers such that

${S}_{n}=\sum _{m={a}_{n}+1}^{{b}_{n}}{s}_{m}.$

Now, in order to simply define the deferred weighted mean ${\sigma }_{n}$, we first set

${\sigma }_{n}=\frac{1}{{S}_{n}}\sum _{m={a}_{n}+1}^{{b}_{n}}{s}_{m}{x}_{m}.$

We then say that a sequence $\left({x}_{n}\right)$ is deferred weighted summable (or ${c}^{D\left(\overline{N}\right)}$-summable) to $\mathrm{\ell }$ if

$\underset{n\to \mathrm{\infty }}{lim}{\sigma }_{n}=\mathrm{\ell }.$

In this case, we write

${c}^{D\left(\overline{N}\right)}-lim{x}_{n}=\mathrm{\ell },$

and let ${c}^{D\left(\overline{N}\right)}$ denote the space of all deferred weighted summable sequences.

Next, we present below the following definitions.

#### Definition 1.2.

A sequence $\left({x}_{n}\right)$ is said to be deferred weighted $\mathcal{𝒜}$-summable (or $\left[D{\left(\overline{N}\right)}_{\mathcal{𝒜}},{p}_{n}\right]$-summable) to a number $\mathrm{\ell }$ if the $\mathcal{𝒜}$-transform of $\left({x}_{n}\right)$ is deferred weighted summable to the same number $\mathrm{\ell }$, that is,

$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{{S}_{n}}\sum _{m={a}_{n}+1}^{{b}_{n}}\sum _{k=1}^{\mathrm{\infty }}{s}_{m}{a}_{m,k}{x}_{k}=\mathrm{\ell }.$

In this case, we write

$\left[D{\left(\overline{N}\right)}_{\mathcal{𝒜}},{p}_{n}\right]-{x}_{n}=\mathrm{\ell }.$

Here we denote the set of all deferred weighted $\mathcal{𝒜}$-summable sequences by $\left[D{\left(\overline{N}\right)}_{\mathcal{𝒜}},{p}_{n}\right)\right]$.

Definition 1.2 is a generalization of several known definitions as discussed in Remark 2.1 below.

#### Remark 1.3.

If ${a}_{n}=0$ and ${b}_{n}=n$ for all n, then the deferred weighted $\mathcal{𝒜}$-summability reduces to the weighted $\mathcal{𝒜}$-summability (see [19]). Also, if ${a}_{n}=0$ and ${b}_{n}=n$ for all n, and $\mathcal{𝒜}=I$ (I being the identity matrix), then the deferred weighted $\mathcal{𝒜}$-summability reduces simply to the weighted summability (see [23]). Moreover, if ${a}_{n}=0$ and ${b}_{n}=n$ for all n, $\mathcal{𝒜}=\left(\mathcal{𝒞},1\right)$ and ${s}_{n}=1$, then the deferred weighted $\mathcal{𝒜}$-summability reduces to the statistical convergence (see [12]).

For simplicity in notation, we shall use the convention that

${\mathcal{𝒜}}_{n}^{D\left(\overline{N}\right)}\left(x\right)=\frac{1}{{S}_{n}}\sum _{m={a}_{n}+1}^{{b}_{n}}\sum _{k=1}^{\mathrm{\infty }}{s}_{m}{a}_{m,k}{x}_{k}.$

#### Definition 1.4.

Let $\mathcal{𝒜}=\left({a}_{n,k}\right)$ be an infinite matrix and let $\left({a}_{n}\right)$ and $\left({b}_{n}\right)$ be sequences of non-negative integers. The matrix $\mathcal{𝒜}=\left({a}_{n,k}\right)$ is said to be a deferred weighted regular matrix (or deferred weighted regular method) if

with

${c}^{D\left(\overline{N}\right)}-\underset{n\to \mathrm{\infty }}{lim}\mathcal{𝒜}x=\mathcal{𝒜}-lim\left({x}_{n}\right).$

We denote it by $\mathcal{𝒜}\in \left(c:{c}^{D\left(\overline{N}\right)}\right)$. This means that ${\mathcal{𝒜}}_{n}^{D\left(\overline{N}\right)}\left(x\right)$ exists for each $n\in ℕ$ and $x\in c$, and

${\mathcal{𝒜}}_{n}^{D\left(\overline{N}\right)}\left(x\right)\to \mathrm{\ell }\mathit{ }\left(n\to \mathrm{\infty }\right)\mathit{ }\text{whenever}\mathit{ }{x}_{n}\to \mathrm{\ell }\mathit{ }\left(n\to \mathrm{\infty }\right).$

We denote the set of all such regular matrices (methods) by ${\mathcal{ℛ}}_{w}^{+}$.

As a characterization of the deferred weighted regular methods, we present the following theorem.

#### Theorem 1.5.

Let $\mathcal{A}\mathrm{=}\mathrm{\left(}{a}_{n\mathrm{,}k}\mathrm{\right)}$ be an infinite matrix. Also let $\mathrm{\left(}{a}_{n}\mathrm{\right)}$ and $\mathrm{\left(}{b}_{n}\mathrm{\right)}$ be sequences of non-negative integers. Then $\mathcal{A}\mathrm{\in }\mathrm{\left(}c\mathrm{:}{c}^{D\mathit{}\mathrm{\left(}\overline{N}\mathrm{\right)}}\mathrm{\right)}$ if and only if

$\underset{n}{sup}\sum _{k=1}^{\mathrm{\infty }}\frac{1}{{S}_{n}}|\sum _{m={a}_{n}+1}^{{b}_{n}}{s}_{m}{a}_{m,k}|<\mathrm{\infty },$(1.1)(1.2)$\underset{n}{lim}\frac{1}{{S}_{n}}\sum _{m={a}_{n}+1}^{{b}_{n}}\sum _{k=1}^{\mathrm{\infty }}{s}_{m}{a}_{m,k}=1.$(1.3)

#### Proof.

Assume that (1.1), (1.2) and (1.3) hold true and that ${x}_{k}\to \mathrm{\ell }$ ($k\to \mathrm{\infty }$). Then, for each $ϵ>0$, there exists ${m}_{0}\in ℕ$ such that $|{x}_{k}-\mathrm{\ell }|\leqq ϵ$ ($m>{m}_{0}$). Thus, we have

$|{\mathcal{𝒜}}_{n}^{D\left(\overline{N}\right)}\left(x\right)-\mathrm{\ell }|=|\frac{1}{{S}_{n}}\sum _{m={a}_{n}+1}^{{b}_{n}}\sum _{k=1}^{\mathrm{\infty }}{s}_{m}{a}_{m,k}{x}_{k}-\mathrm{\ell }|$$=|\frac{1}{{S}_{n}}\sum _{m={a}_{n}+1}^{{b}_{n}}\sum _{k=1}^{\mathrm{\infty }}{s}_{m}{a}_{m,k}\left({x}_{k}-\mathrm{\ell }\right)+\mathrm{\ell }\left(\frac{1}{{S}_{n}}\sum _{m={a}_{n}+1}^{{b}_{n}}\sum _{k=1}^{\mathrm{\infty }}{s}_{m}{a}_{m,k}-1\right)|$$\leqq |\frac{1}{{S}_{n}}\sum _{m={a}_{n}+1}^{{b}_{n}}\sum _{k=1}^{\mathrm{\infty }}{s}_{m}{a}_{m,k}\left({x}_{k}-\mathrm{\ell }\right)|+|\mathrm{\ell }||\frac{1}{{S}_{n}}\sum _{m={a}_{n}+1}^{{b}_{n}}\sum _{k=1}^{\mathrm{\infty }}{s}_{m}{a}_{m,k}-1|$$\leqq |\frac{1}{{S}_{n}}\sum _{m={a}_{n}+1}^{{b}_{n}}\sum _{k=1}^{{b}_{n-2}}{s}_{m}{a}_{m,k}\left({x}_{k}-\mathrm{\ell }\right)|+|\frac{1}{{S}_{n}}\sum _{m={a}_{n}+1}^{{b}_{n}}\sum _{k={b}_{n-1}}^{\mathrm{\infty }}{s}_{m}{a}_{m,k}\left({x}_{k}-1\right)|$$+|\mathrm{\ell }||\frac{1}{{S}_{n}}\sum _{m={a}_{n}+1}^{{b}_{n}}\sum _{k=1}^{\mathrm{\infty }}{s}_{m}{a}_{m,k}-1|$$\leqq \underset{k}{sup}|{x}_{k}-\mathrm{\ell }|\sum _{k=1}^{{b}_{n-2}}\frac{1}{{S}_{n}}\sum _{m={a}_{n}+1}^{{b}_{n}}{s}_{m}{a}_{m,k}+ϵ\frac{1}{{S}_{n}}\sum _{m={a}_{n}+1}^{{b}_{n}}\sum _{k=1}^{\mathrm{\infty }}{s}_{m}{a}_{m,k}+|\mathrm{\ell }||\frac{1}{{S}_{n}}\sum _{m={a}_{n}+1}^{{b}_{n}}\sum _{k=1}^{\mathrm{\infty }}{s}_{m}{a}_{m,k}-1|.$

Taking $n\to \mathrm{\infty }$ and using (1.2) and (1.3), we get

$|\frac{1}{{S}_{n}}\sum _{m={a}_{n}+1}^{{b}_{n}}\sum _{k=1}^{\mathrm{\infty }}{s}_{m}{a}_{m,k}{x}_{k}-\mathrm{\ell }|\leqq ϵ,$

which implies that

$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{{S}_{n}}\sum _{m={a}_{n}+1}^{{b}_{n}}\sum _{k=1}^{\mathrm{\infty }}{s}_{m}{a}_{m,k}{x}_{k}=\mathrm{\ell }=lim\left({x}_{n}\right)$

since $ϵ>0$ is arbitrary.

Conversely, let $\mathcal{𝒜}\in \left(c:{c}^{D\left(\overline{N}\right)}\right)$ and ${x}_{k}\in c$. Then, since $\mathcal{𝒜}x$ exists, we have the inclusion given by

$\left(c:{c}^{D\left(\overline{N}\right)}\right)\subset \left(c:{\mathrm{\ell }}_{\mathrm{\infty }}\right).$

Clearly, there exists a constant M such that

and the corresponding series given by

$\frac{1}{{S}_{n}}\sum _{m={a}_{n}+1}^{{b}_{n}}\sum _{k=1}^{\mathrm{\infty }}{s}_{m}{a}_{m,k}$

converges absolutely for each n. Therefore, (1.1) is valid.

We now consider the sequence ${x}^{\left(n\right)}=\left({x}_{k}^{\left(n\right)}\right)\in {c}_{0}$ defined by

${x}_{k}^{\left(n\right)}=\left\{\begin{array}{cc}1,\hfill & n=k,\hfill \\ 0,\hfill & n\ne k,\hfill \end{array}$

for all $n\in ℕ$ and $y=\left({y}_{k}\right)=\left(1,1,1,\mathrm{\dots }\right)\in c$. Then, since $\mathcal{𝒜}{x}^{\left(n\right)}$ and $\mathcal{𝒜}y$ belong to ${c}^{D\left(\overline{N}\right)}$, equations (1.2) and (1.3) are fairly obvious. ∎

Next, for the statistical version, we present Definition 1.6 below.

#### Definition 1.6.

Let $\mathcal{𝒜}=\left({a}_{n,k}\right)$ be a non-negative deferred weighted regular matrix and let $K=\left({k}_{i}\right)\subset ℕ$ (${k}_{i}<{k}_{i+1}$). Also let $\left({a}_{n}\right)$ and $\left({b}_{n}\right)$ be sequences of non-negative integers. Then the deferred weighted $\mathcal{𝒜}$-density of K is defined by

${d}_{D\left(\overline{N}\right)}^{\mathcal{𝒜}}\left(K\right)=\underset{n\to \mathrm{\infty }}{lim}\frac{1}{{S}_{n}}\sum _{m={a}_{n}+1}^{{b}_{n}}\sum _{k\in K}{s}_{m}{a}_{m,k},$

provided that the limit exists.

We say that the sequence $\left({x}_{n}\right)$ is deferred weighted $\mathcal{𝒜}$-statistical convergent to the number $\mathrm{\ell }$ if, for every $ϵ>0$, we have

${d}_{D\left(\overline{N}\right)}^{\mathcal{𝒜}}\left({K}_{ϵ}\right)=0,$

where

In this case, we write

${\mathrm{stat}}_{\mathcal{𝒜}}^{D\left(\overline{N}\right)}\underset{n\to \mathrm{\infty }}{lim}\left({x}_{n}\right)=\mathrm{\ell }.$

## 2 Construction of a family of positive linear operators

In this section, we construct a sequence of new positive linear operators by means of $\left(p,q\right)$-Lagrange polynomials in order to establish some new approximation results via deferred weighted $\mathcal{𝒜}$-statistical convergence.

Let us first recall some basic definitions and notations about the $\left(p,q\right)$-integers: For any $\left(n\in ℕ\right)$, the $\left(p,q\right)$-integer ${\left[n\right]}_{p,q}$ is defined by

${\left[n\right]}_{p,q}=\left\{\begin{array}{cc}\frac{{p}^{n}-{q}^{n}}{p-q},\hfill & n\geqq 1,\hfill \\ 0,\hfill & n=0,\hfill \end{array}$

where $0.

The $\left(p,q\right)$-factorial is defined by

${\left[n\right]!}_{p,q}=\left\{\begin{array}{cc}{\left[1\right]}_{p,q}{\left[2\right]}_{p,q}\mathrm{\cdots }{\left[n\right]}_{p,q},\hfill & n\ge 1,\hfill \\ 1,\hfill & n=0.\hfill \end{array}$

Also the $\left(p,q\right)$-binomial coefficient is defined by

In the year 2001, Chan, Chyan and Srivastava [8] introduced and studied the following multivariable Lagrange polynomials [8, p. 140, (6)]:

${g}_{n}^{\left({\alpha }_{1},\mathrm{\dots },{\alpha }_{r}\right)}\left({x}_{1},\mathrm{\dots },{x}_{r}\right)=\sum _{{k}_{1}+{k}_{2}+\mathrm{\cdots }+{k}_{r}=n}{\left({\alpha }_{1}\right)}_{{k}_{1}}\mathrm{\cdots }{\left({\alpha }_{r}\right)}_{{k}_{r}}\frac{{x}_{1}^{{k}_{1}}}{{k}_{1}!}\mathrm{\cdots }\frac{{x}_{r}^{{k}_{r}}}{{k}_{r}!},$(2.1)

where

${\left(\lambda \right)}_{k}=\lambda \left(\lambda +1\right)\mathrm{\cdots }\left(\lambda +k-1\right)\mathit{ }\text{and}\mathit{ }{\left(\lambda \right)}_{0}=1.$

By using the above (Chan–Chyan–Srivastava) polynomials, Erkuş, Duman and Srivastava [10, p. 215, (2.6)] introduced the following family of positive linear operators on $C\left[0,1\right]$:

${L}_{n}^{{u}^{\left(1\right)},\mathrm{\dots },{u}^{\left(r\right)}}\left(f;x\right)$$=\left(\prod _{j=1}^{r}{\left(1-x{u}_{n}^{\left(j\right)}\right)}^{n}\right)\cdot \sum _{m=0}^{\mathrm{\infty }}\left[\sum _{{k}_{1}+{k}_{2}+\mathrm{\cdots }+{k}_{r}=m}f\left(\frac{{k}_{r}}{n+{k}_{r}-1}\right)\frac{{\left({u}_{n}^{\left(1\right)}\right)}^{{k}_{1}}}{{k}_{1}!}\mathrm{\cdots }\frac{{\left({u}_{n}^{\left(r\right)}\right)}^{{k}_{r}}}{{k}_{r}!}{\left(n\right)}_{{k}_{1}}\mathrm{\cdots }{\left(n\right)}_{{k}_{r}}\right]{x}^{m},$(2.2)

where ${u}^{\left(j\right)}={\left({u}_{n}^{\left(j\right)}\right)}_{n\in ℕ}$ are sequences of real numbers such that

$0<{u}_{n}^{\left(j\right)}<1\mathit{ }\left(j=0,1,2,\mathrm{\dots },r,n\in ℕ\right)\mathit{ }\text{and}\mathit{ }f\in C\left[0,1\right]\mathit{ }\left(x\in \left[0,1\right]\right).$

In the year 2008, Duman [9, p. 540, (2.2)] introduced the q-polynomials ${g}_{m,q}^{\left({\alpha }_{1},\mathrm{\dots },{\alpha }_{r}\right)}\left({x}_{1},\mathrm{\dots },{x}_{r}\right)$, the q-analogue of Chan–Chyan–Srivastava polynomials in (2.1), which are generated by

$\left(\prod _{i=1}^{r}\frac{1}{{\left({x}_{i}t;q\right)}_{{\alpha }_{i}}}\right)=\prod _{i=1}^{r}\left\{{\left(1-t{x}_{i}{q}^{k}\right)}^{-{\alpha }_{i}}\right\}=\sum _{m=0}^{\mathrm{\infty }}{g}_{m,q}^{\left({\alpha }_{1},\mathrm{\dots },{\alpha }_{r}\right)}\left({x}_{1},\mathrm{\dots },{x}_{r}\right){t}^{m}$$=\sum _{m=0}^{\mathrm{\infty }}\left[\sum _{{k}_{1}+{k}_{2}+\mathrm{\cdots }+{k}_{r}=m}{\left({q}^{{\alpha }_{{k}_{1}}};q\right)}_{{k}_{1}}{\left({q}^{{\alpha }_{{k}_{2}}};q\right)}_{{k}_{2}}\mathrm{\cdots }{\left({q}^{{\alpha }_{{k}_{r}}};q\right)}_{{k}_{r}}\frac{{x}_{1}^{{k}_{1}}}{{\left(q;q\right)}_{{k}_{1}}}\mathrm{\cdots }\frac{{x}_{r}^{{k}_{r}}}{{\left(q;q\right)}_{{k}_{r}}}\right]{t}^{m},$

where $|t|<\mathrm{min}\left(|{x}_{1}|,|{x}_{2}|\right)$.

Very recently, Mursaleen, Khan, Srivastava and Nisar [25, p. 6912, (2.4)] investigated the following family of positive linear operators on $C\left[0,1\right]$, which happens to be a q-analogue of (2.2):

${L}_{n,q}^{{u}^{\left(1\right)},\mathrm{\dots },{u}^{\left(r\right)}}\left(f;x\right)=\left(\prod _{j=1}^{r}{\left(1-x{u}_{n}^{\left(j\right)}{q}^{r}\right)}^{n}\right)\cdot \sum _{m=0}^{\mathrm{\infty }}\left[\sum _{{k}_{1}+{k}_{2}+\mathrm{\cdots }+{k}_{r}=m}{\left({q}^{{\alpha }_{1}};q\right)}_{{k}_{1}}\mathrm{\cdots }{\left({q}^{{\alpha }_{r}};q\right)}_{{k}_{r}}$$\cdot f\left(\frac{{\left(q;q\right)}_{{k}_{r}}}{n+{\left(q;q\right)}_{{k}_{r}}-1}\right)\frac{{\left({u}_{n}^{\left(1\right)}\right)}^{{k}_{1}}}{{\left(q;q\right)}_{{k}_{1}}}\mathrm{\cdots }\frac{{\left({u}_{n}^{\left(r\right)}\right)}^{{k}_{r}}}{{\left(q;q\right)}_{{k}_{r}}}{\left(n;q\right)}_{{k}_{1}}\mathrm{\cdots }{\left(n;q\right)}_{{k}_{r}}\right]x{}^{m},$(2.3)

where ${u}^{\left(j\right)}={\left({u}_{n}^{\left(j\right)}\right)}_{n\in ℕ}$ are sequences of real numbers such that

$0<{u}_{n}^{\left(j\right)}<1\mathit{ }\left(j=0,1,2,\mathrm{\dots },r,n\in ℕ\right)\mathit{ }\text{and}\mathit{ }f\in C\left[0,1\right]\mathit{ }\left(x\in \left[0,1\right]\right).$

Here, as before, q ($|q|<1$) is a real number, the number ${\left(\lambda ;q\right)}_{n}$ is defined by

${\left(\lambda ;q\right)}_{n}=\frac{{\left(\lambda ;q\right)}_{\mathrm{\infty }}}{{\left(\lambda {q}^{n};q\right)}_{\mathrm{\infty }}},$

and ${\left(\cdot ;\cdot \right)}_{\mathrm{\infty }}$ is the infinite version of the so-called q-shifted factorial defined by

${\left(\lambda ;q\right)}_{\mathrm{\infty }}=\prod _{k=0}^{\mathrm{\infty }}\left(1-\lambda {q}^{k}\right).$

As usual, let $C\left[0,1\right]$ denote the space of all real-valued continuous functions defined on the closed interval $\left[0,1\right]$. We also consider the usual supremum norm $\parallel \cdot \parallel$ on $C\left[0,1\right]$. Then, by definition, a linear operator L defined on $C\left[0,1\right]$ is said to be positive if

$L\left(f\right)\geqq 0\mathit{ }\text{whenever}\mathit{ }f\geqq 0.$

We denote the value of $L\left(f\right)$ at a point $t\in \left[0,1\right]$ by $L\left(f\left(y\right);t\right)$ or, simply, by $L\left(f;t\right)$.

We now define the following (presumably new) family of positive linear operators on $C\left[0,1\right]$, that is, a $\left(p,q\right)$-analogue of (2.3) given by

${L}_{n,p,q}^{{x}^{\left(1\right)},\mathrm{\dots },{x}^{\left(r\right)}}\left(f;t\right)=\left(\prod _{j=1}^{r}{\left({p}^{r}-t{x}_{n}^{\left(j\right)}{q}^{r}\right)}^{n}\right)\cdot \sum _{m=0}^{\mathrm{\infty }}\left[\sum _{{k}_{1}+{k}_{2}+\mathrm{\cdots }+{k}_{r}=m}{\left({p}^{{\alpha }_{1}};{q}^{{\alpha }_{1}}\right)}_{{k}_{1}}\mathrm{\cdots }{\left({p}^{{\alpha }_{r}};{q}^{{\alpha }_{r}}\right)}_{{k}_{r}}$$\cdot f\left(\frac{{\left(p;q\right)}_{{k}_{r}}}{n+{\left(p;q\right)}_{{k}_{r}}-1}\right)\frac{{\left({x}_{n}^{\left(1\right)}\right)}^{{k}_{1}}}{{\left(p;q\right)}_{{k}_{1}}}\mathrm{\cdots }\frac{{\left({x}_{n}^{\left(r\right)}\right)}^{{k}_{r}}}{{\left(p;q\right)}_{{k}_{r}}}{\left(p;q\right)}_{{k}_{1}}\mathrm{\cdots }{\left(p;q\right)}_{{k}_{r}}\right]t{}^{m},$(2.4)

where

$|t|<\mathrm{min}\left(|{x}_{n}^{\left(1\right)}|,|{x}_{n}^{\left(2\right)}|\right),f\in C\left[0,1\right],t\in \left[0,1\right],0<{x}_{n}^{\left(j\right)}<1\mathit{ }\left(j=0,1,2,\mathrm{\dots },r,n\in ℕ\right).$

Here p and q ($0) are real numbers, the number ${\left(p;q\right)}_{n}$ is defined by

${\left(p;q\right)}_{n}=\frac{{\left(p;q\right)}_{p,q}^{\mathrm{\infty }}}{{\left({p}^{n+1};{q}^{n+1}\right)}_{p,q}^{\mathrm{\infty }}}$

and ${\left(\cdot ;\cdot \right)}_{\mathrm{\infty }}$ is the infinite $\left(p,q\right)$-shifted factorial defined by

${\left(\alpha ;\beta \right)}_{p,q}^{\mathrm{\infty }}=\prod _{k=0}^{\mathrm{\infty }}\left(\alpha {p}^{k}-\beta {q}^{k}\right)\mathit{ }\left(0<\beta <\alpha \leqq 1\right).$

#### Remark 2.1.

If we take $p=1$, then the $\left(p,q\right)$-Chan–Chyan–Srivastava polynomial reduces to the q-Chan–Chyan–Srivastava polynomial (see [25]).

## 3 Properties of the deferred weighted $\mathcal{𝒜}$-statistical convergence

In this section, we investigate some approximation properties of the positive linear operators ${L}_{n,p,q}^{{x}^{\left(1\right)},\mathrm{\dots },{x}^{\left(r\right)}}\left(f;t\right)$ given by (2.4) with respect to the notion of our newly-defined deferred weighted $\mathcal{𝒜}$-statistical convergence.

We begin by considering the case when $r=2$ in equation (2.4), and we have

${L}_{n,p,q}^{{x}^{\left(1\right)},{x}^{\left(2\right)}}\left(f;t\right)=\left(\prod _{j=1}^{2}{\left({p}^{2}-t{x}_{n}^{\left(j\right)}{q}^{2}\right)}^{n}\right)\cdot \sum _{m=0}^{\mathrm{\infty }}\left[\sum _{{k}_{1}+{k}_{2}=m}{\left({p}^{{\alpha }_{1}};{q}^{{\alpha }_{1}}\right)}_{{k}_{1}}{\left({p}^{{\alpha }_{2}};{q}^{{\alpha }_{2}}\right)}_{{k}_{2}}$$\cdot f\left(\frac{{\left(p;q\right)}_{{k}_{2}}}{n+{\left(p;q\right)}_{{k}_{2}}-1}\right)\frac{{\left({x}_{n}^{\left(1\right)}\right)}^{{k}_{1}}}{{\left(p;q\right)}_{{k}_{1}}}\frac{{\left({x}_{n}^{\left(2\right)}\right)}^{{k}_{2}}}{{\left(p;q\right)}_{{k}_{2}}}{\left(p;q\right)}_{{k}_{1}}{\left(p;q\right)}_{{k}_{2}}\right]t{}^{m},$(3.1)

where

$|t|<\mathrm{min}\left(|{x}_{n}^{\left(1\right)}|,|{x}_{n}^{\left(2\right)}|\right),f\in C\left[0,1\right],t\in \left[0,1\right],0<{x}_{n}^{\left(1\right)},{x}_{n}^{\left(2\right)}<1\mathit{ }\left(n\in ℕ\right).$

We now state and prove some preliminary properties in the form of the following lemmas.

#### Lemma 3.1.

For each $t\mathrm{\in }\mathrm{\left[}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{\right]}$ and $n\mathrm{\in }\mathrm{N}$,

${L}_{n,p,q}^{{x}^{\left(1\right)},{x}^{\left(2\right)}}\left({f}_{0};t\right)=1\mathit{ }\left({f}_{0}\left(t\right)=1\right).$

#### Proof.

In (2.4), we first set $r=2$ and ${\alpha }_{1}={\alpha }_{2}=n$.

We observe that, since

$0<{x}_{n}^{\left(1\right)},{x}_{n}^{\left(2\right)}<1\mathit{ }\left(n\in ℕ\right),$

the condition of (2.4) is satisfied for each $t\in \left[0,1\right]$. Then, by (3.1), we get

${L}_{n,p,q}^{{x}^{\left(1\right)},{x}^{\left(2\right)}}\left({f}_{0};t\right)=\prod _{j=1}^{2}{\left({p}^{2}-t{x}_{n}^{\left(j\right)}{q}^{2}\right)}^{n}\sum _{m=0}^{\mathrm{\infty }}{g}_{m,p,q}^{\left(n,n\right)}\left({x}_{n}^{\left(1\right)},{x}_{n}^{\left(2\right)}\right){t}^{m}=1,$

where

${g}_{m,p,q}^{\left(n,n\right)}\left({x}_{n}^{\left(1\right)},{x}_{n}^{\left(2\right)}\right){t}^{m}=\sum _{{k}_{1}+{k}_{2}=m}{\left({p}^{n};{q}^{n}\right)}_{{k}_{1}}{\left({p}^{n};{q}^{n}\right)}_{{k}_{2}}\frac{{\left({x}_{n}^{\left(1\right)}\right)}^{{k}_{1}}}{{\left(p;q\right)}_{{k}_{1}}}\frac{{\left({x}_{n}^{\left(2\right)}\right)}^{{k}_{2}}}{{\left(p;q\right)}_{{k}_{2}}}{\left(p;q\right)}_{{k}_{1}}{\left(p;q\right)}_{{k}_{2}}{t}^{m}.$

This evidently completes the proof of Lemma 3.1. ∎

#### Lemma 3.2.

For each $t\mathrm{\in }\mathrm{\left[}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{\right]}$ and $n\mathrm{\in }\mathrm{N}$,

${L}_{n,p,q}^{{x}^{\left(1\right)},{x}^{\left(2\right)}}\left({f}_{1};t\right)=t{x}_{n}^{\left(2\right)}\mathit{ }\left({f}_{1}\left(t\right)=t\right).$

#### Proof.

Let $t\in \left[0,1\right]$ be fixed. Then we easily find from (3.1) that

${L}_{n,p,q}^{{x}^{\left(1\right)},{x}^{\left(2\right)}}\left({f}_{1};t\right)=\left(\prod _{j=1}^{2}{\left({p}^{2}-t{x}_{n}^{\left(j\right)}{q}^{2}\right)}^{n}\right)\cdot \sum _{m=0}^{\mathrm{\infty }}\left[\sum _{{k}_{1}+{k}_{2}=m}{\left({p}^{{\alpha }_{1}};{q}^{{\alpha }_{1}}\right)}_{{k}_{1}}{\left({p}^{{\alpha }_{2}};{q}^{{\alpha }_{2}}\right)}_{{k}_{2}}$$\cdot f\left(\frac{{\left(p;q\right)}_{{k}_{2}}}{n+{\left(p;q\right)}_{{k}_{2}}-1}\right)\frac{{\left({x}_{n}^{\left(1\right)}\right)}^{{k}_{1}}}{{\left(p;q\right)}_{{k}_{1}}}\frac{{\left({x}_{n}^{\left(2\right)}\right)}^{{k}_{2}}}{{\left(p;q\right)}_{{k}_{2}}}{\left(p;q\right)}_{{k}_{1}}{\left(p;q\right)}_{{k}_{2}}\right]t{}^{m}$$=\left(\prod _{j=1}^{2}{\left({p}^{2}-t{x}_{n}^{\left(j\right)}{q}^{2}\right)}^{n}\right)\cdot \sum _{m=1}^{\mathrm{\infty }}\left[\sum _{k=1}^{m}{\left({p}^{{\alpha }_{1}};{q}^{{\alpha }_{1}}\right)}_{m-k}{\left({p}^{{\alpha }_{2}};{q}^{{\alpha }_{2}}\right)}_{k}$$\cdot f\left(\frac{{\left(p;q\right)}_{k}}{n+{\left(p;q\right)}_{k}-1}\right)\frac{{\left({x}_{n}^{\left(1\right)}\right)}^{m-k}}{{\left(p;q\right)}_{m-k}}\frac{{\left({x}_{n}^{\left(2\right)}\right)}^{k}}{{\left(p;q\right)}_{k}}{\left(p;q\right)}_{m-k}{\left(p;q\right)}_{k}\right]t{}^{m}$$=t{x}_{n}^{\left(2\right)}\left(\prod _{j=1}^{2}{\left({p}^{2}-t{x}_{n}^{\left(j\right)}{q}^{2}\right)}^{n}\right)\cdot \sum _{m=1}^{\mathrm{\infty }}\left[\sum _{k=1}^{m}{\left({p}^{{\alpha }_{1}};{q}^{{\alpha }_{1}}\right)}_{m-k}{\left({p}^{{\alpha }_{2}};{q}^{{\alpha }_{2}}\right)}_{k-1}$$\cdot \frac{{\left({x}_{n}^{\left(1\right)}\right)}^{m-k}}{{\left(p;q\right)}_{m-k}}\frac{{\left({x}_{n}^{\left(2\right)}\right)}^{k-1}}{{\left(p;q\right)}_{k-1}}{\left(p;q\right)}_{m-k}{\left(p;q\right)}_{k-1}\right]t{}^{m-1}$$=t{x}_{n}^{\left(2\right)}\left(\prod _{j=1}^{2}{\left({p}^{2}-t{x}_{n}^{\left(j\right)}{q}^{2}\right)}^{n}\right)\cdot \sum _{m=1}^{\mathrm{\infty }}\left[\sum _{{k}_{1}+{k}_{2}=m-1}{\left({p}^{{\alpha }_{1}};{q}^{{\alpha }_{1}}\right)}_{{k}_{1}}{\left({p}^{{\alpha }_{2}};{q}^{{\alpha }_{2}}\right)}_{{k}_{2}}$$\cdot \frac{{\left({x}_{n}^{\left(1\right)}\right)}^{{k}_{1}}}{{\left(p;q\right)}_{{k}_{1}}}\frac{{\left({x}_{n}^{\left(2\right)}\right)}^{{k}_{2}}}{{\left(p;q\right)}_{{k}_{2}}}{\left(p;q\right)}_{{k}_{1}}{\left(p;q\right)}_{{k}_{2}}\right]t{}^{m-1}.$

Thus, clearly, we have

${L}_{n,p,q}^{{x}^{\left(1\right)},{x}^{\left(2\right)}}\left({f}_{1};t\right)=t{x}_{n}^{\left(2\right)}\prod _{j=1}^{2}{\left({p}^{2}-t{x}_{n}^{\left(j\right)}{q}^{2}\right)}^{n}\cdot \sum _{m=1}^{\mathrm{\infty }}{g}_{m-1,p,q}^{\left({\alpha }_{1},{\alpha }_{2}\right)}\left({x}_{n}^{\left(1\right)},{x}_{n}^{\left(2\right)}\right){t}^{m-1},$

where

${g}_{m-1,p,q}^{\left({\alpha }_{1},{\alpha }_{2}\right)}\left({x}_{n}^{\left(1\right)},{x}_{n}^{\left(2\right)}\right){t}^{m-1}=\sum _{{k}_{1}+{k}_{2}=m-1}{\left({p}^{{\alpha }_{1}};{q}^{{\alpha }_{1}}\right)}_{{k}_{1}}{\left({p}^{{\alpha }_{2}};{q}^{{\alpha }_{2}}\right)}_{{k}_{2}}\frac{{\left({x}_{n}^{\left(1\right)}\right)}^{{k}_{1}}}{{\left(p;q\right)}_{{k}_{1}}}\frac{{\left({x}_{n}^{\left(2\right)}\right)}^{{k}_{2}}}{{\left(p;q\right)}_{{k}_{2}}}{\left(p;q\right)}_{{k}_{1}}{\left(p;q\right)}_{{k}_{2}}{t}^{m-1}.$

Now, by taking

$r=2,{\alpha }_{1}={\alpha }_{2}=n,{x}_{1}={x}_{n}^{\left(1\right)},{x}_{2}={x}_{n}^{\left(2\right)}$

in (2.4), we have

${L}_{n,p,q}^{{x}^{\left(1\right)},{x}^{\left(2\right)}}\left({f}_{1};t\right)=t{x}_{n}^{\left(2\right)},$

which completes the proof of Lemma 3.2. ∎

#### Lemma 3.3.

For each $t\mathrm{\in }\mathrm{\left[}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{\right]}$ and $n\mathrm{\in }\mathrm{N}$,

$|{L}_{n,p,q}^{{x}^{\left(1\right)},{x}^{\left(2\right)}}\left({f}_{2};t\right)-{f}_{2}\left(t\right)|\leqq 2{t}^{2}\left(1-{x}_{n}^{\left(2\right)}\right)+\frac{t{x}_{n}^{\left(2\right)}}{n}\mathit{ }\left({f}_{2}\left(t\right)={t}^{2}\right).$

#### Proof.

Let $t\in \left[0,1\right]$ be fixed. Then we easily find from (3.1) that

${L}_{n,p,q}^{{x}^{\left(1\right)},{x}^{\left(2\right)}}\left({f}_{2};t\right)=\left(\prod _{j=1}^{2}{\left({p}^{2}-t{x}_{n}^{\left(j\right)}{q}^{2}\right)}^{n}\right)\cdot \sum _{m=0}^{\mathrm{\infty }}\left[\sum _{{k}_{1}+{k}_{2}=m}{\left({p}^{{\alpha }_{1}};{q}^{{\alpha }_{1}}\right)}_{{k}_{1}}{\left({p}^{{\alpha }_{2}};{q}^{{\alpha }_{2}}\right)}_{{k}_{2}}$$\cdot f{\left(\frac{{\left(p;q\right)}_{{k}_{2}}}{n+{\left(p;q\right)}_{{k}_{2}}-1}\right)}^{2}\frac{{\left({x}_{n}^{\left(1\right)}\right)}^{{k}_{1}}}{{\left(p;q\right)}_{{k}_{1}}}\frac{{\left({x}_{n}^{\left(2\right)}\right)}^{{k}_{2}}}{{\left(p;q\right)}_{{k}_{2}}}{\left(p;q\right)}_{{k}_{1}}{\left(p;q\right)}_{{k}_{2}}\right]t{}^{m}$$=\left(\prod _{j=1}^{2}{\left({p}^{2}-t{x}_{n}^{\left(j\right)}{q}^{2}\right)}^{n}\right)\cdot \sum _{m=1}^{\mathrm{\infty }}\left[\sum _{k=1}^{m}{\left({p}^{{\alpha }_{1}};{q}^{{\alpha }_{1}}\right)}_{m-k}{\left({p}^{{\alpha }_{2}};{q}^{{\alpha }_{2}}\right)}_{k}$$\cdot f{\left(\frac{{\left(p;q\right)}_{k}}{n+{\left(p;q\right)}_{k}-1}\right)}^{2}\frac{{\left({x}_{n}^{\left(1\right)}\right)}^{m-k}}{{\left(p;q\right)}_{m-k}}\frac{{\left({x}_{n}^{\left(2\right)}\right)}^{k}}{{\left(p;q\right)}_{k}}{\left(p;q\right)}_{m-k}{\left(p;q\right)}_{k}\right]t{}^{m},$

which yields

${L}_{n,p,q}^{{x}^{\left(1\right)},{x}^{\left(2\right)}}\left({f}_{2};t\right)=t{x}_{n}^{\left(2\right)}\left(\prod _{j=1}^{2}{\left({p}^{2}-t{x}_{n}^{\left(j\right)}{q}^{2}\right)}^{n}\right)\cdot \sum _{m=1}^{\mathrm{\infty }}\left[\sum _{k=1}^{m}{\left({p}^{{\alpha }_{1}};{q}^{{\alpha }_{1}}\right)}_{m-k}{\left({p}^{{\alpha }_{2}};{q}^{{\alpha }_{2}}\right)}_{k-1}$$\cdot f\left(\frac{{\left(p;q\right)}_{k}}{n+{\left(p;q\right)}_{k}-1}\right)\frac{{\left({x}_{n}^{\left(1\right)}\right)}^{m-k}}{{\left(p;q\right)}_{m-k}}\frac{{\left({x}_{n}^{\left(2\right)}\right)}^{k-1}}{{\left(p;q\right)}_{k-1}}{\left(p;q\right)}_{m-k}{\left(p;q\right)}_{k-1}\right]t{}^{m}$$={t}^{2}{\left({x}_{n}^{\left(2\right)}\right)}^{2}\left(\prod _{j=1}^{2}{\left({p}^{2}-t{x}_{n}^{\left(j\right)}{q}^{2}\right)}^{n}\right)\cdot \sum _{m=2}^{\mathrm{\infty }}\left[\sum _{k=2}^{m}{\left({p}^{{\alpha }_{1}};{q}^{{\alpha }_{1}}\right)}_{m-k}{\left({p}^{{\alpha }_{2}};{q}^{{\alpha }_{2}}\right)}_{k-2}$$\cdot f\left(\frac{n+{\left(p;q\right)}_{k}-2}{n+{\left(p;q\right)}_{k}-1}\right)\frac{{\left({x}_{n}^{\left(1\right)}\right)}^{m-k}}{{\left(p;q\right)}_{m-k}}\frac{{\left({x}_{n}^{\left(2\right)}\right)}^{k-2}}{{\left(p;q\right)}_{k-2}}{\left(p;q\right)}_{m-k}{\left(p;q\right)}_{k-2}\right]t{}^{m-2}$$+t{x}_{n}^{\left(2\right)}\left(\prod _{j=1}^{2}{\left({p}^{2}-t{x}_{n}^{\left(j\right)}{q}^{2}\right)}^{n}\right)\cdot \sum _{m=1}^{\mathrm{\infty }}\left[\sum _{k=1}^{m}{\left({p}^{{\alpha }_{1}};{q}^{{\alpha }_{1}}\right)}_{m-k}{\left({p}^{{\alpha }_{2}};{q}^{{\alpha }_{2}}\right)}_{k-1}$$\cdot f\left(\frac{1}{n+{\left(p;q\right)}_{k}-1}\right)\frac{{\left({x}_{n}^{\left(1\right)}\right)}^{m-k}}{{\left(p;q\right)}_{m-k}}\frac{{\left({x}_{n}^{\left(2\right)}\right)}^{k-1}}{{\left(p;q\right)}_{k-1}}{\left(p;q\right)}_{m-k}{\left(p;q\right)}_{k-1}\right]t{}^{m-1}.$

Using this last representation, we get

${L}_{n,p,q}^{{x}^{\left(1\right)},{x}^{\left(2\right)}}\left({f}_{2};t\right)\leqq {t}^{2}\left({x}_{n}^{\left(2\right)}\right)\prod _{j=1}^{2}{\left({p}^{2}-t{x}_{n}^{\left(j\right)}{q}^{2}\right)}^{n}\cdot \sum _{m=2}^{\mathrm{\infty }}{g}_{m-2,p,q}^{\left(n,n\right)}{t}^{m-2}+\frac{t{x}_{n}^{\left(2\right)}}{n}\prod _{j=1}^{2}{\left({p}^{2}-t{x}_{n}^{\left(j\right)}{q}^{2}\right)}^{n}\cdot \sum _{m=2}^{\mathrm{\infty }}{g}_{m-1,p,q}^{\left(n,n\right)}{t}^{m-1}$$={t}^{2}{\left({x}_{n}^{\left(2\right)}\right)}^{2}+\frac{t{x}_{n}^{\left(2\right)}}{n},$

where

${g}_{m-2,p,q}^{\left(n,n\right)}\left({x}_{n}^{\left(1\right)},{x}_{n}^{\left(2\right)}\right){t}^{m-2}=\sum _{{k}_{1}+{k}_{2}=m-2}{\left({p}^{n};{q}^{n}\right)}_{{k}_{1}}{\left({p}^{n};{q}^{n}\right)}_{{k}_{2}}\frac{{\left({x}_{n}^{\left(1\right)}\right)}^{{k}_{1}}}{{\left(p;q\right)}_{{k}_{1}}}\frac{{\left({x}_{n}^{\left(2\right)}\right)}^{{k}_{2}}}{{\left(p;q\right)}_{{k}_{2}}}{\left(p;q\right)}_{{k}_{1}}{\left(p;q\right)}_{{k}_{2}}{t}^{m-2}$

and

${g}_{m-1,p,q}^{\left(n,n\right)}\left({x}_{n}^{\left(1\right)},{x}_{n}^{\left(2\right)}\right){t}^{m-1}=\sum _{{k}_{1}+{k}_{2}=m-1}{\left({p}^{n};{q}^{n}\right)}_{{k}_{1}}{\left({p}^{n};{q}^{n}\right)}_{{k}_{2}}\frac{{\left({x}_{n}^{\left(1\right)}\right)}^{{k}_{1}}}{{\left(p;q\right)}_{{k}_{1}}}\frac{{\left({x}_{n}^{\left(2\right)}\right)}^{{k}_{2}}}{{\left(p;q\right)}_{{k}_{2}}}{\left(p;q\right)}_{{k}_{1}}{\left(p;q\right)}_{{k}_{2}}{t}^{m-1},$

which implies that

${L}_{n,p,q}^{{x}^{\left(1\right)},{x}^{\left(2\right)}}\left({f}_{2};t\right)-{f}_{2}\left(t\right)\leqq {t}^{2}\left({\left({x}_{n}^{\left(2\right)}\right)}^{2}-1\right)+\frac{t{x}_{n}^{\left(2\right)}}{n}.$(3.2)

On the other hand, since

$0\leqq {L}_{n,p,q}^{{x}^{\left(1\right)},{x}^{\left(2\right)}}\left({\left(y-t\right)}^{2};t\right)={L}_{n,p,q}^{{x}^{\left(1\right)},{x}^{\left(2\right)}}\left({f}_{2};t\right)-2t{L}_{n,p,q}^{{x}^{\left(1\right)},{x}^{\left(2\right)}}\left({f}_{1};t\right)+{t}^{2},$

it follows from Lemmas 3.1 and 3.2 that

${L}_{n,p,q}^{{x}^{\left(1\right)},{x}^{\left(2\right)}}\left({f}_{2};t\right)-{f}_{2}\left(t\right)\geqq 2{t}^{2}\left({x}_{n}^{\left(2\right)}-1\right).$(3.3)

Combining (3.2) with (3.3), we have

$|{L}_{n,p,q}^{{x}^{\left(1\right)},{x}^{\left(2\right)}}\left({f}_{2};t\right)-{f}_{2}\left(t\right)|\leqq 2{t}^{2}\left(1-{x}_{n}^{\left(2\right)}\right)+\frac{t{x}_{n}^{\left(2\right)}}{n},$

which is precisely the inequality asserted by Lemma 3.3. ∎

We now present below a new Korovkin-type approximation theorem via the deferred weighted $\mathcal{𝒜}$-statistical convergence based upon our operators defined in (3.1).

#### Theorem 3.4.

Let $\mathcal{A}\mathrm{=}\mathrm{\left(}{a}_{n\mathrm{,}k}\mathrm{\right)}$ be a non-negative deferred weighted regular matrix. Also let $\mathrm{\left(}{a}_{n}\mathrm{\right)}$ and $\mathrm{\left(}{b}_{n}\mathrm{\right)}$ be sequences of non-negative integers. Then

${\mathrm{stat}}_{\mathcal{𝒜}}^{D\left(\overline{N}\right)}\underset{n}{lim}{x}_{n}^{\left(2\right)}=1$(3.4)

if and only if, for all $f\mathrm{\in }C\mathit{}\mathrm{\left[}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{\right]}$,

${\mathrm{stat}}_{\mathcal{𝒜}}^{D\left(\overline{N}\right)}\underset{n}{lim}\parallel {L}_{n,p,q}^{{x}^{\left(1\right)},{x}^{\left(2\right)}}\left(f\right)-f\parallel =0.$(3.5)

#### Proof.

First of all, we assume that assertion (3.5) holds true for all $f\in C\left[0,1\right]$. Then we have

${\mathrm{stat}}_{\mathcal{𝒜}}^{D\left(\overline{N}\right)}\underset{n}{lim}\parallel {L}_{n,p,q}^{{x}^{\left(1\right)},{x}^{\left(2\right)}}\left({f}_{1}\right)-{f}_{1}\parallel =0\mathit{ }\left({f}_{1}\in C\left[0,1\right]\right).$(3.6)

By applying Lemma 3.2, since

$\parallel {L}_{n,p,q}^{{x}^{\left(1\right)},{x}^{\left(2\right)}}\left({f}_{1}\right)-{f}_{1}\parallel =1-{x}_{n}^{\left(2\right)},$(3.7)

from (3.6) and (3.7) we immediately get

${\mathrm{stat}}_{\mathcal{𝒜}}^{D\left(\overline{N}\right)}\underset{n}{lim}{x}_{n}^{\left(2\right)}=1.$

Conversely, we assume that (3.4) holds true. Then it is easy to verify from Lemma 3.1 that

$\underset{n}{lim}\parallel {L}_{n,p,q}^{{x}^{\left(1\right)},{x}^{\left(2\right)}}\left({f}_{0}\right)-{f}_{0}\parallel =0\mathit{ }\left(\text{in the ordinary sense}\right).$

Moreover, since every convergent sequence is deferred weighted $\mathcal{𝒜}$-statistical convergent to the same value for any non-negative regular matrix $\mathcal{𝒜}=\left({a}_{n,k}\right)$, we get

${\mathrm{stat}}_{\mathcal{𝒜}}^{D\left(\overline{N}\right)}\underset{n}{lim}\parallel {L}_{n,p,q}^{{x}^{\left(1\right)},{x}^{\left(2\right)}}\left({f}_{0}\right)-{f}_{0}\parallel =0\mathit{ }\left({f}_{0}\left(t\right)=1\right).$(3.8)

It also follows from Lemma 3.2 that

$\parallel {L}_{n,p,q}^{{x}^{\left(1\right)},{x}^{\left(2\right)}}\left({f}_{1}\right)-{f}_{1}\parallel =1-{x}_{n}^{\left(2\right)}.$

Since

${\mathrm{stat}}_{\mathcal{𝒜}}^{D\left(\overline{N}\right)}\underset{n}{lim}{x}_{n}^{\left(2\right)}=1,$

we may write

${\mathrm{stat}}_{\mathcal{𝒜}}^{D\left(\overline{N}\right)}\underset{n}{lim}\parallel {L}_{n,p,q}^{{x}^{\left(1\right)},{x}^{\left(2\right)}}\left({f}_{1}\right)-{f}_{1}\parallel =0\mathit{ }\left({f}_{1}\left(t\right)=t\right).$(3.9)

We now claim that

${\mathrm{stat}}_{\mathcal{𝒜}}^{D\left(\overline{N}\right)}\underset{n}{lim}\parallel {L}_{n,p,q}^{{x}^{\left(1\right)},{x}^{\left(2\right)}}\left({f}_{2}\right)-{f}_{2}\parallel =0\mathit{ }\left({f}_{2}\left(t\right)={t}^{2}\right).$(3.10)

Indeed, in view of Lemma 3.3, we have

$\underset{n}{lim}\parallel {L}_{n,p,q}^{{x}^{\left(1\right)},{x}^{\left(2\right)}}\left({f}_{2}\right)-{f}_{2}\parallel \leqq 2\left(1-{x}_{n}^{\left(2\right)}\right)+\frac{{x}_{n}^{\left(2\right)}}{n}.$(3.11)

For a given $ϵ>0$, we now define the following sets:

$B=\left\{n:\parallel {L}_{n,p,q}^{{x}^{\left(1\right)},{x}^{\left(2\right)}}\left({f}_{2}\right)-{f}_{2}\parallel \geqq ϵ\right\},{B}_{1}=\left\{n:1-{x}_{n}^{\left(2\right)}\geqq \frac{ϵ}{4}\right\},{B}_{2}=\left\{n:\frac{{x}_{n}^{\left(2\right)}}{n}\geqq \frac{ϵ}{2}\right\}.$

Thus, from (3.11) it is easy to see that $B\subseteq {B}_{1}\cup {B}_{2}$. Then, for each $j\in ℕ$, we get

$\sum _{n\in B}{a}_{j,n}\leqq \sum _{n\in {B}_{1}}{a}_{j,n}+\sum _{n\in {B}_{2}}{a}_{j,n}.$(3.12)

Since

${\mathrm{stat}}_{\mathcal{𝒜}}^{D\left(\overline{N}\right)}\underset{n}{lim}{x}_{n}^{\left(2\right)}=1,$

we observe that

${\mathrm{stat}}_{\mathcal{𝒜}}^{D\left(\overline{N}\right)}\underset{n}{lim}\left({x}_{n}^{\left(2\right)}-1\right)=0$

and

${\mathrm{stat}}_{\mathcal{𝒜}}^{D\left(\overline{N}\right)}\underset{n}{lim}\frac{{x}_{n}^{\left(2\right)}}{n}=0.$

Using these facts and also by taking the limit when $\left(n\to \mathrm{\infty }\right)$ in (3.12), we conclude that

$\underset{j}{lim}\sum _{n\in B}{a}_{j,n}=0\mathit{ }\left(j=1,2\right),$

which yields (3.10). Now, by combining (3.8), (3.9) and (3.10), and using a known result [14, Theorem 1] (see also [10] and [25]), we complete the proof of Theorem 3.4. ∎

#### Remark 3.5.

The above-cited known results [14, 10, 25] are given for statistical convergence and $\mathcal{𝒜}$-statistical convergence, but the present proof works also for deferred weighted $\mathcal{𝒜}$-statistical convergence.

In a similar manner, we can extend Theorem 3.6 to the r-dimensional case for the operators ${L}_{n,p,q}^{{x}^{\left(1\right)},\mathrm{\dots },{x}^{\left(r\right)}}\left(f;t\right)$ given by (2.4) as follows.

#### Theorem 3.6.

Let $\mathcal{A}\mathrm{=}\mathrm{\left(}{a}_{n\mathrm{,}k}\mathrm{\right)}$ be a non-negative deferred weighted regular matrix. Also let $\mathrm{\left(}{a}_{n}\mathrm{\right)}$ and $\mathrm{\left(}{b}_{n}\mathrm{\right)}$ be sequences of non-negative integers. Then

${\mathrm{stat}}_{\mathcal{𝒜}}^{D\left(\overline{N}\right)}\underset{n}{lim}{x}_{n}^{\left(r\right)}=1$

if and only if, for all $f\mathrm{\in }C\mathit{}\mathrm{\left[}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{\right]}$,

${\mathrm{stat}}_{\mathcal{𝒜}}^{D\left(\overline{N}\right)}\underset{n}{lim}\parallel {L}_{n,p,q}^{{x}^{\left(1\right)},\mathrm{\dots },{x}^{\left(r\right)}}\left(f\right)-f\parallel =0.$

#### Remark 3.7.

If, in Theorem 3.6, we put ${a}_{n}=0$, ${b}_{n}=n$, ${s}_{n}=1$ and $p=1$, then it simply reduces to the $\mathcal{𝒜}$-statistical convergence case (see [25, p. 6915, Theorem 2]). Furthermore, for ${a}_{n}=0$, ${b}_{n}=n$ and ${s}_{n}=1$, and upon replacing $\mathcal{𝒜}=\left({a}_{n,k}\right)$ by the identity matrix I, we immediately get the following theorem, which is simply the ordinary (classical) convergence case of Theorem 3.6.

#### Theorem 3.8.

In the ordinary sense,

$\underset{n}{lim}{x}_{n}^{\left(r\right)}=1$

if and only if, for all $f\mathrm{\in }C\mathit{}\mathrm{\left[}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{\right]}$, the sequence of positive linear operators

$\underset{n}{lim}\parallel {L}_{n,p,q}^{{x}^{\left(1\right)},\mathrm{\dots },{x}^{\left(r\right)}}\left(f\right)-f\parallel =0$

is uniformly convergent to f on $\mathrm{\left[}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{\right]}$.

We now present below an illustrative example for the sequence of positive linear operators that neither satisfies the hypotheses of Theorem 3.8 nor the results of Erkuş et al. (see [10, p. 219, Theorem 2] and Mursaleen et al. [25, p. 6915, Theorem 2]), but it does satisfy the hypotheses of our Theorem 3.6.

#### Example 3.9.

Let us consider $\mathcal{𝒜}$ as a Cesàro matrix defined as follows:

${a}_{n,k}=\left\{\begin{array}{cc}\frac{1}{n},\hfill & 1\leqq k\leqq n,\hfill \\ 0,\hfill & k>n.\hfill \end{array}$

For ${a}_{n}=2n$, ${b}_{n}=4n$ and ${s}_{n}=1$, it can easily be seen that $\mathcal{𝒜}\in {\mathcal{ℛ}}_{w}^{+}$. Furthermore, we consider the sequence ${x}^{\left(j\right)}={\left({x}_{n}^{\left(j\right)}\right)}_{n\in ℕ}$ ($j=1,2,\mathrm{\dots },r-1$) such that $0<{x}_{n}^{\left(j\right)}<1$ is given by

${x}_{n}^{\left(r\right)}=\left\{\begin{array}{cc}\frac{1}{2},\hfill & n={m}^{2},m\in ℕ,\hfill \\ \frac{n}{n+1}\hfill & \text{otherwise}.\hfill \end{array}$(3.13)

Here, in this example, we observe that

$0<{x}_{n}^{\left(r\right)}<1\mathit{ }\left(n\in ℕ\right)$

and also that

${\mathrm{stat}}_{\mathcal{𝒜}}^{D\left(\overline{N}\right)}\underset{n}{lim}{x}_{n}^{\left(r\right)}=1.$

Therefore, by Theorem 3.6, for all $f\in C\left[0,1\right]$ we have

${\mathrm{stat}}_{\mathcal{𝒜}}^{D\left(\overline{N}\right)}\underset{n}{lim}\parallel {L}_{n,p,q}^{{x}^{\left(1\right)},\mathrm{\dots },{x}^{\left(r\right)}}\left(f\right)-f\parallel =0.$

However, since the sequence $\left({x}_{n}^{\left(r\right)}\right)$ defined by (3.13) is non-convergent (in the ordinary sense), Theorem 3.8 does not hold true in this case, while our Theorem 3.6 still works. Furthermore, $\left({x}_{n}^{\left(r\right)}\right)$ defined by (3.13) is not $\mathcal{𝒜}$-statistical convergent, that is, the results of Erkuş et al. [10, p. 219, Theorem 2] and Mursaleen et al. [25, p. 6915, Theorem 2] do not hold true in this case, while our Theorem 3.6 still works. Thus, clearly, our Theorem 3.6 is stronger than each of these earlier results.

## 4 Rates of the deferred weighted $\mathcal{𝒜}$-statistical convergence

In this section, we study the rates of the deferred weighted $\mathcal{𝒜}$-statistical convergence for our sequence of positive linear operators in Theorem 3.6 with the help of the modulus of continuity and the elements of the Lipschitz class.

Let $f\in C\left[0,1\right]$. The modulus of continuity of f denoted by $\omega \left(f,\delta \right)$ is defined by

$\omega \left(f,\delta \right)=\underset{\begin{array}{c}|y-t|<\delta \\ t,y\in \left[0,1\right]\end{array}}{sup}|f\left(y\right)-f\left(t\right)|.$

The modulus of continuity of the function f in $C\left[0,1\right]$ provides the maximum oscillation of f in any interval of length not exceeding $\delta >0$. A necessary and sufficient condition for a function $f\in C\left[0,1\right]$ is that

$\underset{\delta \to 0}{lim}\omega \left(f,\delta \right)=0.$

It is also well known that, for any $\delta >0$ and for each $t,y\in \left[0,1\right]$,

$|f\left(y\right)-f\left(t\right)|\leqq \left(\frac{|y-t|}{\delta }+1\right)\omega \left(f,\delta \right).$(4.1)

Now we state and prove a new result in the form of the following theorem.

#### Theorem 4.1.

For each $n\mathrm{\in }\mathrm{N}$ and for all $f\mathrm{\in }C\mathit{}\mathrm{\left[}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{\right]}$,

$\parallel {L}_{n,p,q}^{{x}^{\left(1\right)},\mathrm{\dots },{x}^{\left(r\right)}}\left(f\right)-f\parallel \leqq 2\omega \left(f,{\delta }_{n}\right)$(4.2)

holds, where

${\delta }_{n}={\left(\frac{{x}_{n}^{\left(r\right)}}{n}+4\left(1-{x}_{n}^{\left(r\right)}\right)\right)}^{\frac{1}{2}}.$(4.3)

#### Proof.

Let $f\in C\left[0,1\right]$ and let $t\in \left[0,1\right]$ be fixed. By linearity and monotonicity of ${L}_{n,p,q}^{{x}^{\left(1\right)},\mathrm{\dots },{x}^{\left(r\right)}}\left(f;t\right)$, and by using (4.1), we find for any $\delta >0$ that

$|{L}_{n,p,q}^{{x}^{\left(1\right)},\mathrm{\dots },{x}^{\left(r\right)}}\left(f;t\right)-f\left(t\right)|\leqq {L}_{n,p,q}^{{x}^{\left(1\right)},\mathrm{\dots },{x}^{\left(r\right)}}\left(|f\left(y\right)-f\left(t\right)|;t\right)$$\leqq \omega \left(f,\delta \right){L}_{n,p,q}^{{x}^{\left(1\right)},\mathrm{\dots },{x}^{\left(r\right)}}\left(1+\frac{|y-t|}{\delta };t\right)$$\leqq \omega \left(f,\delta \right)\left(1+\frac{1}{\delta }{L}_{n,p,q}^{{x}^{\left(1\right)},\mathrm{\dots },{x}^{\left(r\right)}}\left(|y-t|;t\right)\right).$

Furthermore, by the Cauchy–Schwarz inequality for positive linear operators, we have

$|{L}_{n,p,q}^{{x}^{\left(1\right)},\mathrm{\dots },{x}^{\left(r\right)}}\left(f;t\right)-f\left(t\right)|\leqq \omega \left(f,\delta \right)\left\{1+\frac{1}{\delta }{\left({L}_{n,p,q}^{{x}^{\left(1\right)},\mathrm{\dots },{x}^{\left(r\right)}}\left(\psi ;t\right)\right)}^{\frac{1}{2}}\right\},$(4.4)

where

$\psi =\psi \left(y\right)={\left(y-t\right)}^{2}.$

We also observe that

${L}_{n,p,q}^{{x}^{\left(1\right)},\mathrm{\dots },{x}^{\left(r\right)}}\left(\psi ;t\right)={L}_{n,p,q}^{{x}^{\left(1\right)},\mathrm{\dots },{x}^{\left(r\right)}}\left({f}_{2};t\right)-2x{L}_{n,p,q}^{{x}^{\left(1\right)},\mathrm{\dots },{x}^{\left(r\right)}}\left({f}_{1};t\right)+{t}^{2}$$\leqq |{L}_{n,p,q}^{{x}^{\left(1\right)},\mathrm{\dots },{x}^{\left(r\right)}}\left({f}_{2};t\right)-{f}_{2}\left(t\right)|+2x|{L}_{n,p,q}^{{x}^{\left(1\right)},\mathrm{\dots },{x}^{\left(r\right)}}\left({f}_{1};t\right)-{f}_{1}\left(t\right)|,$

which, by virtue of Lemmas 3.2 and 3.3, yields

${L}_{n,p,q}^{{x}^{\left(1\right)},\mathrm{\dots },{x}^{\left(r\right)}}\left(\psi ;t\right)\leqq 2{t}^{2}\left(1-{x}_{n}^{\left(r\right)}\right)+\frac{t{x}_{n}^{\left(r\right)}}{n}+2{t}^{2}\left(1-{x}_{n}^{\left(r\right)}\right)\leqq \frac{t{x}_{n}^{\left(r\right)}}{n}+4{t}^{2}\left(1-{x}_{n}^{\left(r\right)}\right).$

Now, combining this last inequality with (4.4), we get

$|{L}_{n,p,q}^{{x}^{\left(1\right)},\mathrm{\dots },{x}^{\left(r\right)}}\left(f;t\right)-f\left(t\right)|\leqq \omega \left(f,\delta \right)\left\{1+\frac{1}{\delta }{\left(\frac{t{x}_{n}^{\left(r\right)}}{n}+4{t}^{2}\left(1-{x}_{n}^{\left(r\right)}\right)\right)}^{\frac{1}{2}}\right\},$

which implies that

$\parallel {L}_{n,p,q}^{{x}^{\left(1\right)},\mathrm{\dots },{x}^{\left(r\right)}}\left(f;t\right)-f\left(t\right)\parallel \leqq \omega \left(f,\delta \right)\left\{1+\frac{1}{\delta }{\left(\frac{{x}_{n}^{\left(r\right)}}{n}+4\left(1-{x}_{n}^{\left(r\right)}\right)\right)}^{\frac{1}{2}}\right\}.$(4.5)

If we choose $\delta ={\delta }_{n}$, where ${\delta }_{n}$ is given as in (4.3), then assertion (4.2) of Theorem 3.8 follows immediately from (4.5). ∎

We now turn to our investigation of the rate of convergence of the positive linear operators ${L}_{n,p,q}^{{x}^{\left(1\right)},\mathrm{\dots },{x}^{\left(r\right)}}\left(f;t\right)$ given by (2.4) by means of the elements of the Lipschitz class ${\mathrm{Lip}}_{M}\left(\alpha \right)$ ($0<\alpha \leqq 1$). We first recall that a function $f\in C\left[0,1\right]$ belongs to the Lipschitz class ${\mathrm{Lip}}_{M}\left(\alpha \right)$ ($0<\alpha \leqq 1$) if the following inequality holds true:

$|f\left(y\right)-f\left(t\right)|\leqq M{|y-t|}^{\alpha }\mathit{ }\left(t,y\in \left[0,1\right], 0<\alpha \leqq 1,M>0\right).$(4.6)

We now state and prove a new result in the form of the following theorem.

#### Theorem 4.2.

For each $n\mathrm{\in }\mathrm{N}$ and for all $f\mathrm{\in }{\mathrm{Lip}}_{M}\mathit{}\mathrm{\left(}\alpha \mathrm{\right)}$,

$\parallel {L}_{n,p,q}^{{x}^{\left(1\right)},\mathrm{\dots },{x}^{\left(r\right)}}\left(f;t\right)-f\left(t\right)\parallel \leqq 2{\delta }_{n}^{\alpha },$(4.7)

where ${\delta }_{n}$ is the same as in Theorem 3.8.

#### Proof.

Let $f\in {\mathrm{Lip}}_{M}\left(\alpha \right)$ and suppose that $0\leqq t\leqq 1$. We then find from (4.6) that

$\parallel {L}_{n,p,q}^{{x}^{\left(1\right)},\mathrm{\dots },{x}^{\left(r\right)}}\left(f;t\right)-f\left(t\right)\parallel \leqq {L}_{n,p,q}^{{x}^{\left(1\right)},\mathrm{\dots },{x}^{\left(r\right)}}\left(|f\left(y\right)-f\left(t\right)|;t\right)\leqq M{L}_{n,p,q}^{{x}^{\left(1\right)},\mathrm{\dots },{x}^{\left(r\right)}}\left({|y-t|}^{\alpha };t\right).$

Now, by applying Hölder’s inequality with

$p=\frac{2}{\alpha }\mathit{ }\text{and}\mathit{ }q=\frac{2}{2-\alpha },$

we get

$\parallel {L}_{n,p,q}^{{x}^{\left(1\right)},\mathrm{\dots },{x}^{\left(r\right)}}\left(f;t\right)-f\left(t\right)\parallel \leqq M{\left({L}_{n,p,q}^{{x}^{\left(1\right)},\mathrm{\dots },{x}^{\left(r\right)}}\left(\psi ;t\right)\right)}^{\frac{\alpha }{2}},$(4.8)

where

$\psi =\psi \left(y\right)={\left(y-t\right)}^{2}.$

Just as in the proof of Theorem 4.1, since

${L}_{n,p,q}^{{x}^{\left(1\right)},\mathrm{\dots },{x}^{\left(r\right)}}\left(\psi ;t\right)\leqq \frac{t{x}_{n}^{\left(r\right)}}{n}+4{t}^{2}\left(1-{x}_{n}^{\left(r\right)}\right),$

we can deduce from (4.8) that

$|{L}_{n,p,q}^{{x}^{\left(1\right)},\mathrm{\dots },{x}^{\left(r\right)}}\left(f;t\right)-f\left(t\right)|\leqq M{\left(\frac{t{x}_{n}^{\left(r\right)}}{n}+4{t}^{2}\left(1-{x}_{n}^{\left(r\right)}\right)\right)}^{\frac{\alpha }{2}},$

$|{L}_{n,p,q}^{{x}^{\left(1\right)},\mathrm{\dots },{x}^{\left(r\right)}}\left(f;t\right)-f\left(t\right)|\leqq M{\left(\frac{{x}_{n}^{\left(r\right)}}{n}+4\left(1-{x}_{n}^{\left(r\right)}\right)\right)}^{\frac{\alpha }{2}}.$(4.9)

If we take $\left({\delta }_{n}\right)$ as in (4.3), then assertion (4.7) of Theorem 4.2 would follow from this last inequality (4.9). ∎

## 5 Concluding remarks and observations

In this concluding section of our investigation, we present several further remarks and observations concerning the various results which we have proved here.

#### Remark 5.1.

Let ${\left({x}_{n}^{\left(r\right)}\right)}_{n\in ℕ}$ be the sequence given in Example 3.9. Then, since

by applying Theorem 3.6, we can write

${\mathrm{stat}}_{\mathcal{𝒜}}^{D\left(\overline{N}\right)}\underset{n\to \mathrm{\infty }}{lim}{\parallel {L}_{n,p,q}^{{x}^{\left(1\right)},\mathrm{\dots },{x}^{\left(r\right)}}\left(f\right)-f\parallel }_{C\left[0,1\right]}=0,\left(f\in C\left[0,1\right]\right).$(5.1)

However, since $\left({x}_{n}^{\left(r\right)}\right)$ is not ordinarily convergent and so it does not also converge uniformly in the ordinary sense, the classical Korovkin Theorem (Theorem 3.8) does not work here for the operators defined by (2.4). Hence, this application clearly indicates that our Theorem 3.6 is a non-trivial generalization of the classical Korovkin-type theorem (see Theorem 3.8).

#### Remark 5.2.

Let ${\left({x}_{n}^{\left(r\right)}\right)}_{n\in ℕ}$ be the sequence as given in Example 3.9. Then, since

by applying our Theorem 3.6, condition (5.1) holds true. However, since $\left({x}_{n}^{\left(r\right)}\right)$ is not $\mathcal{𝒜}$-statistically convergent, we can say that the results of Erkuş et al. [10, p. 219, Theorem 2] and Mursaleen et al. [25, p. 6915, Theorem 2] do not hold true for our operators defined in (2.4). Thus, our Theorem 3.6 is also a non-trivial extension of the results of Erkuş et al. [10, p. 219, Theorem 2] and Mursaleen et al. [25, p. 6915, Theorem 2]. Based upon the above results, it is concluded here that our proposed method has successfully worked for the operators defined in (2.4), and therefore that our results are stronger than the classical and $\mathcal{𝒜}$-statistical versions of the Korovkin-type approximation theorem established earlier (see [10, 25]).

#### Remark 5.3.

For a non-negative deferred weighted regular matrix $A=\left({a}_{n,k}\right)$, let $\left({a}_{n}\right)$ and $\left({b}_{n}\right)$ be sequences of non-negative integers. Then, if we first set

${\mathrm{stat}}_{\mathcal{𝒜}}^{D\left(\overline{N}\right)}\underset{n\to \mathrm{\infty }}{lim}{x}_{n}^{\left(r\right)}=1,$

we find that

${\mathrm{stat}}_{\mathcal{𝒜}}^{D\left(\overline{N}\right)}\underset{n\to \mathrm{\infty }}{lim}{\delta }_{n}=0,$

where ${\delta }_{n}$ is given by (4.3). It also implies that

${\mathrm{stat}}_{\mathcal{𝒜}}^{D\left(\overline{N}\right)}\underset{n}{lim}\omega \left(f,{\delta }_{n}\right)=0\mathit{ }\left(f\in C\left[0,1\right]\right).$

Thus, clearly, Theorems 4.1 and 4.2 give us the rates of deferred weighted $\mathcal{𝒜}$-statistical convergence in Theorem 3.6. Of course, if we replace the matrix $\mathcal{𝒜}=\left({a}_{n,k}\right)$ by the identity matrix I and let $\left({a}_{n}\right)=0$ and $\left({b}_{n}\right)=n$, then we naturally get the corresponding rates of classical (ordinary) convergence in Theorem 3.8.

#### Remark 5.4.

In our present investigation, we have considered a number of interesting special cases and illustrative examples in support of our definitions and also of the results which we have established here.

## References

• [1]

R. P. Agnew, On deferred Cesàro means, Ann. of Math. (2) 33 (1932), no. 3, 413–421.

• [2]

A. Aral and O. Doğru, Bleimann, Butzer, and Hahn operators based on the q-integers, J. Inequal. Appl. 2007 (2007), Article ID 79410.

• [3]

A. Aral and V. Gupta, On the q analogue of Stancu-beta operators, Appl. Math. Lett. 25 (2012), no. 1, 67–71.

• [4]

C. Belen and S. A. Mohiuddine, Generalized weighted statistical convergence and application, Appl. Math. Comput. 219 (2013), no. 18, 9821–9826.

• [5]

J. Boos, Classical and Modern Methods in Summability, Oxford Math. Monogr., Oxford University Press, Oxford, 2000.  Google Scholar

• [6]

N. L. Braha, V. Loku and H. M. Srivastava, ${\mathrm{\Lambda }}^{2}$-weighted statistical convergence and Korovkin and Voronovskaya type theorems, Appl. Math. Comput. 266 (2015), 675–686.

• [7]

N. L. Braha, H. M. Srivastava and S. A. Mohiuddine, A Korovkin’s type approximation theorem for periodic functions via the statistical summability of the generalized de la Vallée Poussin mean, Appl. Math. Comput. 228 (2014), 162–169.  Google Scholar

• [8]

W.-C. C. Chan, C.-J. Chyan and H. M. Srivastava, The Lagrange polynomials in several variables, Integral Transforms Spec. Funct. 12 (2001), no. 2, 139–148.

• [9]

E. E. Duman, A q-extension of the Erkus–Srivastava polynomials in several variables, Taiwanese J. Math. 12 (2008), no. 2, 539–543.

• [10]

E. Erkuş, O. Duman and H. M. Srivastava, Statistical approximation of certain positive linear operators constructed by means of the Chan–Chyan–Srivastava polynomials, Appl. Math. Comput. 182 (2006), no. 1, 213–222.  Google Scholar

• [11]

S. Ersan and O. Doğru, Statistical approximation properties of q-Bleimann, Butzer and Hahn operators, Math. Comput. Model. 49 (2009), no. 7–8, 1595–1606.

• [12]

H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951), 241–244.

• [13]

A. R. Freedman and J. J. Sember, Densities and summability, Pacific J. Math. 95 (1981), no. 2, 293–305.

• [14]

A. D. Gadjiev and C. Orhan, Some approximation theorems via statistical convergence, Rocky Mountain J. Math. 32 (2002), no. 1, 129–138.

• [15]

V. Gupta and C. Radu, Statistical approximation properties of q-Baskakov–Kantorovich operators, Cent. Eur. J. Math. 7 (2009), no. 4, 809–818.

• [16]

U. Kadak, N. L. Braha and H. M. Srivastava, Statistical weighted $\mathcal{ℬ}$-summability and its applications to approximation theorems, Appl. Math. Comput. 302 (2017), 80–96.  Google Scholar

• [17]

V. Karakaya and T. A. Chishti, Weighted statistical convergence, Iran. J. Sci. Technol. Trans. A Sci. 33 (2009), no. 3, 219–223.  Google Scholar

• [18]

N. Mahmudov and P. Sabancigil, A q-analogue of the Meyer–König and Zeller operators, Bull. Malays. Math. Sci. Soc. (2) 35 (2012), no. 1, 39–51.  Google Scholar

• [19]

S. A. Mohiuddine, Statistical weighted A-summability with application to Korovkin’s type approximation theorem, J. Inequal. Appl. 2016 (2016), Article ID 101.

• [20]

M. Mursaleen, A. Alotaibi and K. J. Ansari, On a Kantorovich variant of $\left(p,q\right)$-Szász–Mirakjan operators, J. Funct. Spaces 2016 (2016), Article ID 1035253.

• [21]

M. Mursaleen, K. J. Ansari and A. Khan, On $\left(p,q\right)$-analogue of Bernstein operators, Appl. Math. Comput. 266 (2015), 874–882.

• [22]

M. Mursaleen, K. J. Ansari and A. Khan, Some approximation results by $\left(p,q\right)$-analogue of Bernstein–Stancu operators, Appl. Math. Comput. 264 (2015), 392–402.

• [23]

M. Mursaleen, V. Karakaya, M. Ertürk and F. Gürsoy, Weighted statistical convergence and its application to Korovkin type approximation theorem, Appl. Math. Comput. 218 (2012), no. 18, 9132–9137.

• [24]

M. Mursaleen and A. Khan, Statistical approximation properties of modified q-Stancu-beta operators, Bull. Malays. Math. Sci. Soc. (2) 36 (2013), no. 3, 683–690.  Google Scholar

• [25]

M. Mursaleen, A. Khan, H. M. Srivastava and K. S. Nisar, Operators constructed by means of q-Lagrange polynomials and A-statistical approximation, Appl. Math. Comput. 219 (2013), no. 12, 6911–6918.

• [26]

M. Mursaleen, F. Khan and A. Khan, Approximation by $\left(p,q\right)$-Lorentz polynomials on a compact disk, Complex Anal. Oper. Theory 10 (2016), no. 8, 1725–1740.

• [27]

M. Mursaleen, M. Nasiruzzaman and A. Nurgali, Some approximation results on Bernstein–Schurer operators defined by $\left(p,q\right)$-integers, J. Inequal. Appl. 2015 (2015), Article ID 49.

• [28]

M. Örkcü and O. Doğru, Weighted statistical approximation by Kantorovich type q-Szász–Mirakjan operators, Appl. Math. Comput. 217 (2011), no. 20, 7913–7919.

• [29]

M. A. Özarslan, O. Duman and H. M. Srivastava, Statistical approximation results for Kantorovich-type operators involving some special polynomials, Math. Comput. Model. 48 (2008), no. 3–4, 388–401.

• [30]

C. Radu, Statistical approximation properties of Kantorovich operators based on q-integers, Creat. Math. Inform. 17 (2008), no. 2, 75–84.  Google Scholar

• [31]

H. M. Srivastava and M. Et, Lacunary statistical convergence and strongly lacunary summable functions of order α, Filomat 31 (2017), no. 6, 1573–1582.

• [32]

H. M. Srivastava, B. B. Jena, S. K. Paikray and U. K. Misra, Generalized equi-statistical convergence of the deferred Nörlund summability and its applications to associated approximation theorems, Rev. Real Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM (2017), 10.1007/s13398-017-0442-3.  Google Scholar

• [33]

H. M. Srivastava, B. B. Jena, S. K. Paikray and U. K. Misra, A certain class of weighted statistical convergence and associated Korovkin-type approximation theorems involving trigonometric functions, Math. Methods Appl. Sci. 41 (2018), no. 2, 671–683.

• [34]

H. M. Srivastava, M. Mursaleen, A. M. Alotaibi, M. Nasiruzzaman and A. A. H. Al-Abied, Some approximation results involving the q-Szász–Mirakjan–Kantorovich type operators via Dunkl’s generalization, Math. Methods Appl. Sci. 40 (2017), no. 15, 5437–5452.

• [35]

H. M. Srivastava, M. Mursaleen and A. Khan, Generalized equi-statistical convergence of positive linear operators and associated approximation theorems, Math. Comput. Model. 55 (2012), no. 9–10, 2040–2051.

• [36]

H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math. 2 (1951), 73–74.  Google Scholar

Accepted: 2018-04-12

Published Online: 2018-05-03

Published in Print: 2018-06-01

Citation Information: Journal of Applied Analysis, Volume 24, Issue 1, Pages 1–16, ISSN (Online) 1869-6082, ISSN (Print) 1425-6908,

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