A Stancu-Schurer type extension of higher order of the Cheney-Sharma operators

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(original), Approximation Theory, JCR/ISI, paper, positive linear operators

Abstract

In this paper we introduce a Stancu-Schurer type extension of higher order of the Cheney-Sharma operators. Starting from the operators studied by Bostanci and Ba\c scanbaz-Tunca, respectively by C\u atina\c s and Buda (in the form of a Stancu operator with generalized Bernstein polynomials) we extend the convex combination of two terms which appears in the expression of the operator to a convex combination of $m$ terms, where $m\in\mathbb{N}$, with $m\geq 1$. We called these new operators the Stancu-Schurer type extension of order $m$ of the Cheney-Sharma operators (of first, respectively second kind). For these operators we study some approximation and convexity properties, modulus of continuity and Korovkin-type theorems.

Authors

Eduard Stefan Grigoriciuc
Faculty of Mathematics and Computer Science, Babeş-Bolyai University, Cluj-Napoca, Romania
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania

Andra Malina
Faculty of Mathematics and Computer Science, Babeş-Bolyai University, Cluj-Napoca, Romania
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania

Keywords

Cheney-Sharma operator, Stancu-Schurer operator, Jensen inequality, modulus of continuity, Korovkin theorem

Paper coordinates

E.S. Grigoriciuc, A. Malina, A Stancu-Schurer type extension of higher order of the Cheney-Sharma operators, Math. Slovaca (2026), to appear (DOI: 10.1515/ms-2025-1155)

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Mathematica Slovaca

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De Gruyter Brill

DOI

https://doi.org/10.1515/ms-2025-1155

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1337-2211

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A Stancu-Schurer type extension of higher order of the Cheney-Sharma operators

Eduard Ştefan Grigoriciuc^∗,∗∗ and Andra Malina^∗,∗∗ * Faculty of Mathematics and Computer Science
Babeş-Bolyai University
1 M. Kogălniceanu Street
400084 Cluj-Napoca
ROMANIA ** Tiberiu Popoviciu Institute of Numerical Analysis
Romanian Academy
P.O. Box 68-1
400110 Cluj-Napoca
ROMANIA eduard.grigoriciuc@ubbcluj.ro & andra.malina@ubbcluj.ro

Abstract.

In this paper we introduce a Stancu-Schurer type extension of higher order of the Cheney-Sharma operators. Starting from the operators studied by Bostanci and Başcanbaz-Tunca, respectively by Cătinaş and Buda (in the form of a Stancu operator with generalized Bernstein polynomials) we extend the convex combination of two terms which appears in the expression of the operator to a convex combination of $m$ terms, where $m\in\mathbb{N}$ , with $m\geq 1$ . We called these new operators the Stancu-Schurer type extension of order $m$ of the Cheney-Sharma operators (of first, respectively second kind). For these operators we study some approximation and convexity properties, modulus of continuity and Korovkin-type theorems.

Key words and phrases:

Cheney-Sharma operator, Stancu-Schurer operator, Jensen inequality, modulus of continuity, Korovkin theorem

2010 Mathematics Subject Classification:

Primary 41A35, 41A36, 47A58

Grigoriciuc E.S. was supported by a grant of the Ministry of Research, Innovation and Digitization, CNCS/CCCDI-UEFISCDI, project number PN-IV-P8-8.1-PRE-HE-ORG-2023-0118, within PNCDI IV

1. Introduction

For $f\in C[0,1]$ and $n\in\mathbb{N}$ , Bernstein introduced in [6] the following linear and positive operators

B_{n}f(x)=\sum_{k=0}^{n}b_{n,k}(x)f\bigg(\dfrac{k}{n}\bigg),

(1.1)

with the basis polynomials $b_{n,k}$ given by

b_{n,k}(x)={n\choose k}x^{k}(1-x)^{n-k},

(1.2)

for $k=0,\ldots,n$ , and $x\in[0,1]$ . Later on, using a probabilistic approach, Stancu introduced in [29] another type of linear and positive operators, defined as

S_{n}f(x)=\sum_{k=0}^{n-r}b_{n-r,k}(x)\bigg[(1-x)f\bigg(\dfrac{k}{n}\bigg)+xf% \bigg(\dfrac{k+r}{n}\bigg)\bigg],

(1.3)

with the polynomials $b_{n-r,k}$ given by (1.2), $f\in C[0,1]$ , $x\in[0,1]$ and $r$ a non-negative integer such that $n>2r$ , for every $n\in\mathbb{N}$ .

Over time, various generalizations of this operator have been made. Among the generalized operators that have been introduced and studied, we mention here only the Bernstein-Stancu operator (see e.g. [28]), the Stancu-Schurer operator (see e.g [21], [25]), the Brass-Stancu operator (see e.g. [8] and [10]), the Stancu-Kantorovich operator (see e.g. [18]). In each of these generalizations (as well as in others studied but not mentioned here), the fundamental polynomials $b_{n,k}$ are multiplied by values of the function $f$ , respectively linear combinations of values of the function $f$ . The novelty of the generalization presented in this paper consists in the use of higher-order combinations, as can be seen in the following definition.

Definition 1.

Consider $m\in\mathbb{N}$ , $m\geq 1$ . Also, let $n,\ r_{1},\ldots,r_{m},\ R\in\mathbb{N}$ be such that $n>2R$ , where $R=r_{1}+\ldots+r_{m}$ . Then, for every function $f:[0,1]\rightarrow\mathbb{R}$ , we define the generalized Stancu operator of order $m$ by

S^{m}_{n}f(x)=\sum_{k=0}^{n-R}b_{n-R,k}(x)\bigg[\bigg(1-\sum_{j=1}^{m}x^{j}% \bigg)f\bigg(\dfrac{k}{n}\bigg)+\sum_{j=1}^{m}x^{j}f\bigg(\dfrac{k+r_{j}}{n}% \bigg)\bigg],

(1.4)

where the polynomials $b_{n-R,k}(x)$ are given by (1.2).

In order to generalize the previous operator, a small modification can be made in its definition. Similarly to what was done in [25] (see also [16], [24]), we consider $p\in\mathbb{N}$ with $p\geq 0$ and $C[0,1+p]$ the space of continuous functions in the interval $[0,1+p]$ .

Remark 1.

Let $p\in\mathbb{N}$ . If we replace $n$ by $n+p$ in the first part of relation (1.4), then we obtain the generalized Stancu-Schurer operator of order $m$ , i.e.,

S^{m}_{n,p}f(x)=\sum_{k=0}^{n+p-R}b_{n+p-R,k}(x)\bigg[\bigg(1-\sum_{j=1}^{m}x^% {j}\bigg)f\bigg(\dfrac{k}{n}\bigg)+\sum_{j=1}^{m}x^{j}f\bigg(\dfrac{k+r_{j}}{n% }\bigg)\bigg],

(1.5)

for all $x\in[0,1]$ , where the polynomials $b_{n+p-R,k}(x)$ are given by (1.2) and $f\in C[0,1+p]$ .

It is not difficult to observe that $S_{n,0}^{1}=S_{n}:C[0,1]\rightarrow C[0,1]$ is the classical Stancu operator (of order one) defined by (1.3) and $S^{1}_{n,p}:C[0,1+p]\rightarrow C[0,1]$ is the classical Stancu-Schurer operator defined in [25].

Remark 2.

The Bernstein polynomials $b_{n,k}$ defined in (1.2) have been generalized by Cheney and Sharma (see [19]) based on Jensen’s formulas

(u+v+m\beta)^{m}=\sum_{k=0}^{m}{m\choose k}u(u+k\beta)^{k-1}[v+(m-k)\beta]^{m-k}

(1.6)

and

(u+v)(u+v+m\beta)^{m-1}=\sum_{k=0}^{m}{m\choose k}u(u+k\beta)^{k-1}v[v+(m-k)% \beta]^{m-k-1},

(1.7)

where $u,v,\beta\in\mathbb{R}$ and $m\in\mathbb{N}$ . Hence, for $\beta\geq 0$ and $n\in\mathbb{N}$ , we denote by

a^{\beta}_{n,k}(x)={n\choose k}\dfrac{x(x+k\beta)^{k-1}[1-x+(n-k)\beta]^{n-k}}% {(1+n\beta)^{n}},

(1.8)

respectively by

b^{\beta}_{n,k}(x)={n\choose k}\dfrac{x(x+k\beta)^{k-1}(1-x)[1-x+(n-k)\beta]^{% n-k-1}}{(1+n\beta)^{n-1}}

(1.9)

the generalizations of the polynomial $b_{n,k}$ . It is clear that $a^{0}_{n,k}=b^{0}_{n,k}=b_{n,k}$ .

Using the above notations, the Cheney-Sharma operator of the first kind is given as (see [19])

P_{n}^{\beta}f(x)=\sum\limits_{k=0}^{n}a_{n,k}^{\beta}(x)f\left(\frac{k}{n}% \right),

(1.10)

while the Cheney-Sharma operator of the second kind is defined as (see [19])

Q_{n}^{\beta}f(x)=\sum\limits_{k=0}^{n}b_{n,k}^{\beta}(x)f\left(\frac{k}{n}% \right),

(1.11)

for any function $f:[0,1]\to\mathbb{R}$ . Note that, recently, the Cheney-Sharma operators have been studied on different domains in higher dimensions. In [11] and [12] the authors present some extensions of these operators on a triangle with straight sides, respectively with one curved side. Other properties on such domains are studied in [14] and [15] in terms of iterates of multivariate Cheney-Sharma type operators.

Lemma 1.1.

According to [19], [30] and [7], respectively, we have the following relations for the operator $Q_{n}^{\beta}$ given in (1.11):

a)

$Q_{n}^{\beta}(e_{0};x)=1$ ;
b)

$Q_{n}^{\beta}(e_{1};x)=x$ ;
c)

$Q_{n}^{\beta}(e_{2};x)=\dfrac{x}{n}+\dfrac{n-1}{n}\cdot\dfrac{x(1-x)}{(1+n% \beta)^{n-1}}\displaystyle\sum\limits_{i=0}^{n-2}{n-2\choose i}\left(x+(i+2)% \beta\right)^{i+1}\left(1-x+\left(n-i-2\right)\beta\right)^{n-i-3}$ , for all $\beta\geq 0$ and $x\in[0,1]$ .

We end this introductory section by bringing together already known results and innovative ideas that open the way for the study of the operators introduced in this paper. The primary elements that we will use in the construction of the new operators are the generalized Stancu-Schurer operator of order $m$ given by (1.5) and the Cheney-Sharma polynomials $a^{\beta}_{n,k}$ and $b^{\beta}_{n,k}$ given by (1.8), respectively by (1.9).

Using these types of operators, respectively, basis polynomials, we can construct a new generalization of the Stancu-Schurer and Cheney-Sharma operators. Taking into account that Cheney and Sharma defined two types of operators, i.e., of the first, respectively the second kind, we can define the Stancu-Schurer type extension of order $m$ for both cases. These two operators will be defined and rigorously studied in the following sections.

In Section 2 we present the definitions of our new operators, denoted by $\mathcal{S}^{m,\beta}_{n,p}$ , respectively $\mathcal{T}^{m,\beta}_{n,p}$ , where $m,n,p\in\mathbb{N}$ with $m\geq 1$ and $\beta\geq 0$ . The next two sections are dedicated to the study of the operator $\mathcal{T}^{m,\beta}_{n,p}$ . In Section 3 we include some properties related to the moments, approximation and convexity of the operator $\mathcal{T}^{m,\beta}_{n,p}$ . On the other hand, in Section 4 we discuss Korovkin-type theorems for the same operator $\mathcal{T}^{m,\beta}_{n,p}$ . Finally, Section 5 contains some results related to the operator $\mathcal{S}^{m,\beta}_{n,p}$ , less studied in this paper, but which will be addressed in a subsequent study. The paper ends with a section of conclusions and further research directions.

2. Extensions of higher orders of the Cheney-Sharma operators

In this section, based on the ideas presented above, we construct new operators of Cheney-Sharma type, considering a Stancu-Schurer extension of higher order. Particular cases of the operators introduced in this section have been studied by Bostanci and Başcanbaz-Tunca in [7], respectively by Cătinaş and Buda in [13]. Hence, inspired by the results obtained in the foundational work of Bostanci and Başcanbaz-Tunca and also Cătinaş and Buda we study some general properties of our newly formulated operators.

For these operators, moment calculations, approximation and convexity properties are studied (see Section 3, respectively Section 5). In addition, for the Stancu-Schurer type extension $\mathcal{T}^{m,\beta}_{n,p}$ of order $m$ of the second Cheney-Sharma operator we study the modulus of continuity and prove some relevant Korovkin-type results (see Section 4).

Definition 2.

Consider $m\in\mathbb{N}$ , $m\geq 1$ and $\beta\geq 0$ . Let $n,p,r_{1},\ldots,r_{m},R\in\mathbb{N}$ be such that $n>2R$ , where $R=r_{1}+\ldots+r_{m}$ . Then, for every continuous function $f:[0,1+p]\rightarrow\mathbb{R}$ , we define:

a)

the Stancu-Schurer type extension of order $m$ of the first Cheney-Sharma operator as

$\mathcal{S}^{m,\beta}_{n,p}f(x)=\sum_{k=0}^{n+p-R}a^{\beta}_{n+p-R,k}(x)\bigg[% \bigg(1-\sum_{j=1}^{m}x^{j}\bigg)f\bigg(\dfrac{k}{n}\bigg)+\sum_{j=1}^{m}x^{j}% f\bigg(\dfrac{k+r_{j}}{n}\bigg)\bigg],$ (2.1)

with the polynomials $a^{\beta}_{n+p-R,k}$ given by (1.8), for all $x\in[0,1]$ ;
b)

the Stancu-Schurer type extension of order $m$ of the second Cheney-Sharma operator as

$\mathcal{T}^{m,\beta}_{n,p}f(x)=\sum_{k=0}^{n+p-R}b^{\beta}_{n+p-R,k}(x)\bigg[% \bigg(1-\sum_{j=1}^{m}x^{j}\bigg)f\bigg(\dfrac{k}{n}\bigg)+\sum_{j=1}^{m}x^{j}% f\bigg(\dfrac{k+r_{j}}{n}\bigg)\bigg],$ (2.2)

with the polynomials $b^{\beta}_{n+p-R,k}$ given by (1.9), for all $x\in[0,1]$ .

It is clear that $\mathcal{S}_{n,p}^{m,0}=\mathcal{T}^{m,0}_{n,p}=S^{m}_{n,p}:C\left[0,1+p\right% ]\rightarrow C[0,1]$ is the generalized Stancu-Schurer operator of order $m$ defined by (1.5). Moreover, if $m=1$ and $p=0$ , then $\mathcal{S}_{n,0}^{1,\beta}$ and $\mathcal{T}_{n,0}^{1,\beta}$ are the Stancu type extensions of the Cheney-Sharma operators of the first (respectively, the second) kind studied by Cătinaş and Buda in [13], respectively by Bostanci and Başcanbaz-Tunca in [7].

In addition, one can easily see that if $m=1,\ r_{1}=0$ and $p=0$ , then $\mathcal{S}_{n,0}^{1,\beta}$ and $\mathcal{T}_{n,0}^{1,\beta}$ are the Cheney-Sharma operators of the first and second kind, respectively, given in (1.10) and (1.11), respectively, both reducing to the Bernstein operator (1.2) when $\beta=0$ .

Example 1.

To graphically illustrate the operators $\mathcal{S}_{n,p}^{m,\beta}$ and $\mathcal{T}_{n,p}^{m,\beta}$ , given in (2.1) and (2.2), respectively, we consider the functions (see e.g. [8])

f_{1}(x)=x(1-x)\quad\text{and}\quad f_{2}(x)=\cos(2\pi x),\quad x\in[0,1].

The graphs of the functions $f_{1}$ and $f_{2}$ , together with those of the corresponding operators $\mathcal{S}_{n,p}^{m,\beta}$ and $\mathcal{T}_{n,p}^{m,\beta}$ , are presented in Figures 1 and 2, respectively, for several choices of the parameters appearing in the definitions of the operators.

Refer to caption — Figure 1. Functions $f_{1}(x)$ and $f_{2}(x)$ and the corresponding operators $\mathcal{S}_{n,p}^{m,\beta}(f_{1};x)$ and $\mathcal{S}_{n,p}^{m,\beta}(f_{2};x)$ .

3. Properties of the operator $\mathcal{T}_{{n,p}}^{{m,\beta}}$

In this section we focus our attention on the properties of the second-type extension operator $\mathcal{T}_{n,p}^{m,\beta}$ . We study the expressions of the moments of $\mathcal{T}_{n,p}^{m,\beta}$ and some properties related to them. Finally, we study approximation and convexity results. The proofs of our results follow the ideas presented in [30] (see also [7], [19] and [22]) for the particular case $m=1$ .

3.1. Moments of $\mathcal{T}_{{n,p}}^{{m,\beta}}$ and their properties

Our first result is a generalization of Lemma 1.1 (see [19]).

Theorem 3.1.

Consider $m\in\mathbb{N}$ , $m\geq 1$ and $\beta\geq 0$ . Let $n,p,r_{1},\ldots,r_{m},R\in\mathbb{N}$ be such that $n>2R$ , where $R=r_{1}+\ldots+r_{m}$ . Also, let $e_{j}(x)=x^{j}$ , for every $j\in\mathbb{N}$ and $x\in[0,1]$ . Then

a)

$\mathcal{T}^{m,\beta}_{n,p}(e_{0};x)=1$ ;
b)

$\mathcal{T}^{m,\beta}_{n,p}(e_{1};x)=\frac{n+p}{n}x+\frac{r_{2}}{n}x(x-1)+% \frac{r_{3}}{n}x(x^{2}-1)+\ldots+\frac{r_{m}}{n}x(x^{m-1}-1)$ , for $m\geq 2$ .

In particular, $\mathcal{T}^{1,\beta}_{n,p}(e_{1};x)=\frac{n+p}{n}x$ , for all $x\in[0,1]$ .

Proof.

Following the ideas presented in [30] (see also [7] and [22]), we can give a complete proof of the previous two relations.

Let $e_{0}(x)=1$ , for all $x\in[0,1]$ . Then

	$\displaystyle\mathcal{T}^{m,\beta}_{n,p}(e_{0};x)$	$\displaystyle=\sum_{k=0}^{n+p-R}b^{\beta}_{n+p-R,k}(x)\big[(1-x-\ldots-x^{m})+% (x+x^{2}+\ldots+x^{m})\big]$
		$\displaystyle=\sum_{k=0}^{\lambda}{\lambda\choose k}\dfrac{x(x+k\beta)^{k-1}(1% -x)[1-x+(\lambda-k)\beta]^{\lambda-k-1}}{(1+\lambda\beta)^{\lambda-1}},$

where $\lambda=n+p-R$ . If we replace in (1.7) $m$ by $\lambda$ , $u=x$ and $v=1-x$ , then

	$\displaystyle\mathcal{T}^{m,\beta}_{n,p}(e_{0};x)$	$\displaystyle=(1+\lambda\beta)^{1-\lambda}\sum_{k=0}^{\lambda}{\lambda\choose k% }u(u+k\beta)^{k-1}v[v+(\lambda-k)\beta]^{\lambda-k-1}$
		$\displaystyle=(1+\lambda\beta)^{1-\lambda}(1+\lambda\beta)^{\lambda-1}$
		$\displaystyle=1.$

Hence, we obtain that $\mathcal{T}^{m,\beta}_{n,p}(e_{0};x)=1$ .

For the second case, let $e_{1}(x)=x$ , for all $x\in[0,1]$ . Also, for simplicity, let us denote $n+p-R$ by $\lambda$ . Then

	$\displaystyle\mathcal{T}^{m,\beta}_{n,p}(e_{1};x)$	$\displaystyle=\sum_{k=0}^{\lambda}b^{\beta}_{\lambda,k}(x)\bigg[\big(1-x-x^{2}% -\ldots-x^{m}\big)\dfrac{k}{n}+x\dfrac{k+r_{1}}{n}+\ldots+x^{m}\dfrac{k+r_{m}}% {n}\bigg]$
		$\displaystyle=\sum_{k=0}^{\lambda}b^{\beta}_{\lambda,k}(x)\bigg(\dfrac{k}{n}+x% \dfrac{r_{1}}{n}+x^{2}\dfrac{r_{2}}{n}+\ldots+x^{m}\dfrac{r_{m}}{n}\bigg)$
		$\displaystyle=\sum_{k=0}^{\lambda}\dfrac{k}{n}b^{\beta}_{\lambda,k}(x)+\bigg(x% \dfrac{r_{1}}{n}+x^{2}\dfrac{r_{2}}{n}+\ldots+x^{m}\dfrac{r_{m}}{n}\bigg)\sum_% {k=0}^{\lambda}b^{\beta}_{\lambda,k}(x)$
		$\displaystyle=\sum_{k=0}^{\lambda}\dfrac{k}{n}b^{\beta}_{\lambda,k}(x)+\bigg(x% \dfrac{r_{1}}{n}+x^{2}\dfrac{r_{2}}{n}+\ldots+x^{m}\dfrac{r_{m}}{n}\bigg)% \mathcal{T}^{m,\beta}_{n,p}(e_{0};x)$
		$\displaystyle=x\dfrac{r_{1}}{n}+x^{2}\dfrac{r_{2}}{n}+\ldots+x^{m}\dfrac{r_{m}% }{n}+B_{\lambda}(x),$

in view of the first part of this result, where we denoted by

B_{\lambda}(x)=\sum_{k=0}^{\lambda}\dfrac{k}{n}b^{\beta}_{\lambda,k}(x).

(3.1)

Since the first term of the sum is zero, we obtain that

	$\displaystyle B_{\lambda}(x)$	$\displaystyle=\dfrac{\lambda}{n}\sum_{k=1}^{\lambda}\dfrac{k}{\lambda}b^{\beta% }_{\lambda,k}(x)$
		$\displaystyle=\dfrac{\lambda}{n}\sum_{k=1}^{\lambda}\dfrac{k}{\lambda}{\lambda% \choose k}\dfrac{x(x+k\beta)^{k-1}(1-x)[1-x+(\lambda-k)\beta]^{\lambda-k-1}}{(% 1+\lambda\beta)^{\lambda-1}}$
		$\displaystyle=\dfrac{\lambda}{n}\sum_{k=1}^{\lambda}{\lambda-1\choose k-1}% \dfrac{x(x+k\beta)^{k-1}(1-x)\left[1-x+(\lambda-k)\beta\right]^{\lambda-k-1}}{% (1+\lambda\beta)^{\lambda-1}}.$

Next, let us denote $k-1=j$ . Then B_λ(x)=λn ∑_j=0^λ-1(λ-1 j) x[x+(j+1)β]j(1-x)[1-x+(λ-j-1) β]λ-j-2(1+ λβ)λ-1. If we denote the index of summation by $k$ and take into account that x+(k+1)β=(1+λβ)-[1-x+(λ-k-1)β], then

	$\displaystyle B_{\lambda}(x)$	$\displaystyle=\dfrac{\lambda}{n}\sum_{k=0}^{\lambda-1}{\lambda-1\choose k}% \dfrac{x[x+(k+1)\beta]^{k}(1-x)\left[1-x+(\lambda-k-1)\beta\right]^{\lambda-k-% 2}}{(1+\lambda\beta)^{\lambda-1}}$
		$\displaystyle=\dfrac{\lambda}{n}\sum_{k=0}^{\lambda-1}{\lambda-1\choose k}% \dfrac{x\left[x+(k+1)\beta\right]^{k-1}(1-x)\left[1-x+(\lambda-k-1)\beta\right% ]^{\lambda-k-2}}{(1+\lambda\beta)^}$
		$\displaystyle\;\;\;\;-\dfrac{\lambda}{n}\sum_{k=0}^{\lambda-1}{\lambda-1% \choose k}\dfrac{x\left[x+(k+1)\beta\right]^{k-1}(1-x)\left[1-x+(\lambda-k-1)% \beta\right]^{\lambda-k-1}}{(1+\lambda\beta)^}$
		$\displaystyle=\dfrac{\lambda}{n}\cdot\dfrac{x}{x+\beta}(1+\beta)-\dfrac{% \lambda}{n}\cdot\dfrac{x}{x+\beta}(1-x)$
		$\displaystyle=\dfrac{\lambda x}{n},$

according to relation (1.7) for $m=\lambda-1$ , $u=x+\beta$ and $v=1-x$ . Finally, we obtain that

	$\displaystyle\mathcal{T}^{m,\beta}_{n,p}(e_{1};x)$	$\displaystyle=x\dfrac{r_{1}}{n}+x^{2}\dfrac{r_{2}}{n}+\ldots+x^{m}\dfrac{r_{m}% }{n}+B_{\lambda}(x)$
		$\displaystyle=x\dfrac{r_{1}}{n}+x^{2}\dfrac{r_{2}}{n}+\ldots+x^{m}\dfrac{r_{m}% }{n}+x\dfrac{\lambda}{n}$
		$\displaystyle=\dfrac{n+p}{n}x+\frac{r_{2}}{n}x(x-1)+\dfrac{r_{3}}{n}x(x^{2}-1)% +\ldots+\dfrac{r_{m}}{n}x(x^{m-1}-1),$

where $\lambda=n+p-(r_{1}+r_{2}+\ldots+r_{m})$ and this completes the proof.

∎

Remark 3.

It is clear that $B_{\lambda}(x)=Q_{\lambda}^{\beta}(e_{1};x)$ , for every $x\in[0,1]$ , where $n$ is replaced by $\lambda$ , respectively $f$ by $e_{1}$ in formula (1.11). Hence, the previous proof can be particularized for the Cheney-Sharma operator (see [19]). Indeed, if $p=R=0$ , then Theorem 3.1 reduces to the results obtained by Cheney and Sharma in [19], respectively by Bostanci and Başcanbaz-Tunca in [7] (see Lemma 1.1).

Remark 4.

For all $x\in[0,1]$ and any $m\in\mathbb{N},\ m\geq 1$ , we can prove the following immediate properties:

a)

$\displaystyle\lim_{n\to\infty}\mathcal{T}^{m,\beta}_{n,p}(e_{1};x)=x$ ;
b)

$\mathcal{T}^{m,\beta}_{n,p}(e_{1};x)\in\left[0,\frac{n+p}{n}\right]$ . In particular, if $p=0$ , then $\mathcal{T}^{m,\beta}_{n,p}(e_{1};x)\in\left[0,1\right]$ .

Lemma 3.1.

Consider $m\in\mathbb{N}$ , $m\geq 1$ and $\beta\geq 0$ . Let $n,p,r_{1},\ldots,r_{m},R\in\mathbb{N}$ be such that $n>2R$ , where $R=r_{1}+\ldots+r_{m}$ . If $p\geq R^{*}=r_{2}+\ldots+r_{m}$ , then

\mathcal{T}^{m,\beta}_{n,p}(e_{1};x)\geq x,

(3.2)

where $e_{1}(x)=x$ , for all $x\in[0,1]$ . The equality holds if and only if $p=R^{*}=0$ .

Proof.

According to Theorem 3.1, we know that

\mathcal{T}^{m,\beta}_{n,p}(e_{1};x)=x\dfrac{r_{1}}{n}+x^{2}\dfrac{r_{2}}{n}+% \ldots+x^{m}\dfrac{r_{m}}{n}+x\dfrac{\lambda}{n},

where $\lambda=n+p-(r_{1}+r_{2}+\ldots+r_{m})$ . If $m=1$ , then the inequality (3.2) is obvious, since $x\in[0,1]$ and $p\in\mathbb{N}$ .

Next, let us consider $m\in\mathbb{N}$ such that $m\geq 2$ . Simple computations show then that (3.2) is equivalent to

h(x)=n\left(\mathcal{T}^{m,\beta}_{n,p}(e_{1};x)-x\right)=r_{m}x^{m}+\ldots+r_% {2}x^{2}+(p-R^{*})x\geq 0,

(3.3)

where $R^{*}=r_{2}+\ldots+r_{m}$ , $p\in\mathbb{N}$ and $x\in[0,1]$ . In order to obtain the inequality (3.3), it is sufficient that all the coefficients of $h$ be non-negative, i.e., $r_{2},\ldots,r_{m}\geq 0$ and $p\geq R^{*}$ , but this is true in view of our hypothesis. Hence, $h(x)\geq 0$ , for all $x\in[0,1]$ and this completes the proof. ∎

Lemma 3.2.

Consider $m\in\mathbb{N}$ , $m\geq 1$ and $\beta\geq 0$ . Let $n,p,r_{1},\ldots,r_{m},R\in\mathbb{N}$ be such that $n>2R$ , where $R=r_{1}+\ldots+r_{m}$ . If $p\geq R^{*}=r_{2}+\ldots+r_{m}$ and $\alpha\in\mathbb{N}$ , then

\left(\mathcal{T}^{m,\beta}_{n,p}(e_{1};x)\right)^{\alpha}\geq x^{\alpha},

(3.4)

where $e_{1}(x)=x$ , for all $x\in[0,1]$ .

Proof.

Relation (3.4) is equivalent to showing that

h(x)=\left(\mathcal{T}^{m,\beta}_{n,p}(e_{1};x)\right)^{\alpha}-x^{\alpha}\geq 0.

(3.5)

We have that

h(x)=\left(\mathcal{T}^{m,\beta}_{n,p}(e_{1};x)-x\right)\cdot\left[\left(% \mathcal{T}^{m,\beta}_{n,p}(e_{1};x)\right)^{\alpha-1}+x\left(\mathcal{T}^{m,% \beta}_{n,p}(e_{1};x)\right)^{\alpha-2}+\ldots+x^{\alpha-2}\mathcal{T}^{m,% \beta}_{n,p}(e_{1};x)+x^{\alpha-1}\right].

Since we proved in Lemma 3.1 that the first term is positive and we have for $x\in[0,1]$ that $\mathcal{T}^{m,\beta}_{n,p}(e_{1};x)\geq 0$ , we get the desired conclusion. ∎

Remark 5.

It is important to mention here that relations (3.2) and (3.4) hold in the case $m=1$ independent of $r_{1}$ (without any other condition on $p$ ), since $\mathcal{T}^{1,\beta}_{n,p}(e_{1};x)=\frac{n+p}{n}x$ , for all $x\in[0,1]$ and $n,p\in\mathbb{N}$ . Hence, the assumption $p\geq R^{*}$ imposed in Lemmas 3.1 and 3.2 is necessary only if $m\geq 2$ .

Next, we state an auxiliary result (see [7, Lemma 2.1]), which we shall use in the sequel.

Lemma 3.3.

[7, Lemma 2.1] Considering $x,y\in[0,1]$ , $N\in\mathbb{N}$ , $k=0,1,\ldots,N$ and

T(k,N,x,y):=\sum\limits_{i=0}^{N}{N\choose i}\left(x+\beta i\right)^{i+k-1}% \left(y+(N-i)\beta\right)^{N-i-1},

(3.6)

one has

T(k,N,x,y)=xT\left(k-1,N,x,y\right)+N\beta T\left(k,N-1,x+\beta,y\right).

(3.7)

Moreover, we have that

yT(2,N,x,y)=\int_{0}^{\infty}e^{-s}\left[x\left(x+y+N\beta+s\beta\right)^{N}+% Ns\beta^{2}\left(x+y+N\beta+s\beta\right)^{N-1}\right]\ ds.

(3.8)

Following the ideas presented by Cheney and Sharma in [19] (see also [7]), we can prove the following result related to the expression of $\mathcal{T}^{m,\beta}_{n,p}(e_{2};x)$ on $[0,1]$ .

Theorem 3.2.

\mathcal{T}^{m,\beta}_{n,p}(e_{2};x)=\frac{1}{n^{2}}\sum_{j=1}^{m}x^{j}r_{j}^{% 2}+\frac{2\lambda x}{n^{2}}\sum_{j=1}^{m}x^{j}r_{j}+C_{\lambda}(x),

where $C_{\lambda}$ is given by the formula

C_{\lambda}(x)=\dfrac{\lambda(\lambda-1)}{n^{2}}\bigg[x(x+2\beta)M_{\lambda}+x% (\lambda-2)\beta^{2}N_{\lambda}\bigg]+\dfrac{\lambda x}{n^{2}},

(3.9)

for all $x\in[0,1]$ and $\lambda=n+p-R$ with

M_{\lambda}=\dfrac{1}{1+\lambda\beta}\int_{0}^{\infty}e^{-s}\bigg(1+\dfrac{s% \beta}{1+\lambda\beta}\bigg)^{\lambda-2}ds

(3.10)

and

N_{\lambda}=\dfrac{1}{(1+\lambda\beta)^{2}}\int_{0}^{\infty}se^{-s}\bigg(1+% \dfrac{s\beta}{1+\lambda\beta}\bigg)^{\lambda-3}ds.

(3.11)

Next, we give a detailed proof of the previous result. A sketch of the ideas used in this proof has been presented in [19] (see also [7]).

Proof.

Let $e_{2}(x)=x^{2}$ , for all $x\in[0,1]$ . Then

\mathcal{T}^{m,\beta}_{n,p}(e_{2};x)=\sum_{k=0}^{n+p-R}b^{\beta}_{n+p-R,k}(x)% \bigg[\bigg(1-\sum_{j=1}^{m}x^{j}\bigg)\dfrac{k^{2}}{n^{2}}+\sum_{j=1}^{m}x^{j% }\dfrac{(k+r_{j})^{2}}{n^{2}}\bigg].

For simplicity, let us denote $n+p-R$ by $\lambda$ and $b^{\beta}_{n+p-R,k}$ by $b$ . Then

	$\displaystyle\mathcal{T}^{m,\beta}_{n,p}(e_{2};x)$	$\displaystyle=\sum_{k=0}^{\lambda}b(x)\bigg[\bigg(1-\sum_{j=1}^{m}x^{j}\bigg)% \dfrac{k^{2}}{n^{2}}+\sum_{j=1}^{m}x^{j}\dfrac{(k+r_{j})^{2}}{n^{2}}\bigg]$
		$\displaystyle=\sum_{k=0}^{\lambda}b(x)\bigg[\dfrac{k^{2}}{n^{2}}-\dfrac{xk^{2}% }{n^{2}}-\ldots-\dfrac{x^{m}k^{2}}{n^{2}}+\dfrac{x(k+r_{1})^{2}}{n^{2}}+\ldots% +\dfrac{x^{m}(k+r_{m})^{2}}{n^{2}}\bigg]$
		$\displaystyle=\sum_{k=0}^{\lambda}b(x)\bigg[\dfrac{k^{2}}{n^{2}}+\dfrac{2k}{n^% {2}}\bigg(xr_{1}+\ldots+x^{m}r_{m}\bigg)+\dfrac{1}{n^{2}}\bigg(xr_{1}^{2}+% \ldots+x^{m}r_{m}^{2}\bigg)\bigg]$
		$\displaystyle=\dfrac{1}{n^{2}}\sum_{j=1}^{m}x^{j}r^{2}_{j}\sum_{k=0}^{\lambda}% b(x)+\dfrac{2}{n}\sum_{j=1}^{m}x^{j}r_{j}\sum_{k=0}^{\lambda}\dfrac{k}{n}b(x)+% \sum_{k=0}^{\lambda}\dfrac{k^{2}}{n^{2}}b(x).$

According to the proof of Theorem 3.1 we have that

	$\displaystyle\mathcal{T}^{m,\beta}_{n,p}(e_{2};x)$	$\displaystyle=\dfrac{1}{n^{2}}\sum_{j=1}^{m}x^{j}r^{2}_{j}\cdot\mathcal{T}^{m,% \beta}_{n,p}(e_{0};x)+\dfrac{2}{n}\sum_{j=1}^{m}x^{j}r_{j}\cdot B_{\lambda}(x)% +C_{\lambda}(x)$
		$\displaystyle=\dfrac{1}{n^{2}}\sum_{j=1}^{m}x^{j}r^{2}_{j}+\dfrac{2\lambda x}{% n^{2}}\sum_{j=1}^{m}x^{j}r_{j}+C_{\lambda}(x),$

where $\mathcal{T}^{m,\beta}_{n,p}(e_{0};x)=1$ (see the first part of Theorem 3.1), $B_{\lambda}$ is given by (3.1) and

C_{\lambda}(x)=\sum_{k=0}^{\lambda}\dfrac{k^{2}}{n^{2}}b(x),

(3.12)

for all $x\in[0,1]$ . The next important step in our proof is to compute the expression of $C_{\lambda}$ given by (3.12). We have that

	$\displaystyle C_{\lambda}(x)$	$\displaystyle=\dfrac{\lambda^{2}}{n^{2}}\sum_{k=0}^{\lambda}\dfrac{k^{2}}{% \lambda^{2}}b^{\beta}_{\lambda,k}(x)$
		$\displaystyle=\dfrac{\lambda^{2}}{n^{2}}\sum_{k=0}^{\lambda}\bigg[\dfrac{k(k-1% )}{\lambda^{2}}+\dfrac{k}{\lambda^{2}}\bigg]b^{\beta}_{\lambda,k}(x)$
		$\displaystyle=\dfrac{\lambda^{2}}{n^{2}}\sum_{k=0}^{\lambda}\dfrac{k(k-1)}{% \lambda^{2}}b^{\beta}_{\lambda,k}(x)+\dfrac{\lambda}{n^{2}}\sum_{k=0}^{\lambda% }\dfrac{k}{\lambda}b^{\beta}_{\lambda,k}(x).$

It is not difficult to observe that the second sum reduces to $\frac{nB_{\lambda}}{\lambda}$ , where $B_{\lambda}$ is given by (3.1) and then

C_{\lambda}(x)=\dfrac{\lambda^{2}}{n^{2}}\sum_{k=0}^{\lambda}\dfrac{k(k-1)}{% \lambda^{2}}b^{\beta}_{\lambda,k}(x)+\dfrac{\lambda x}{n^{2}}.

Moreover,

	$\displaystyle\dfrac{\lambda^{2}}{n^{2}}\sum_{k=0}^{\lambda}\dfrac{k(k-1)}{% \lambda^{2}}b^{\beta}_{\lambda,k}(x)$	$\displaystyle=\dfrac{\lambda(\lambda-1)}{n^{2}}\sum_{k=0}^{\lambda}\dfrac{k(k-% 1)}{\lambda(\lambda-1)}b^{\beta}_{\lambda,k}(x)$
		$\displaystyle=\dfrac{\lambda(\lambda-1)x(1-x)}{n^{2}(1+\lambda\beta)^{\lambda-% 1}}\sum_{k=0}^{\lambda}\dfrac{k(k-1)}{\lambda(\lambda-1)}b^{\beta}_{\lambda,k}% (x).$

If we compute separately the sum in the last member of the previous equality, we obtain

	$\displaystyle\sum_{k=0}^{\lambda}\dfrac{k(k-1)}{\lambda(\lambda-1)}b^{\beta}_{% \lambda,k}(x)$	$\displaystyle=\sum_{k=0}^{\lambda}\dfrac{k(k-1)}{\lambda(\lambda-1)}{\lambda% \choose k}(x+k\beta)^{k-1}[1-x+(\lambda-k)\beta]^{\lambda-k-1}$
		$\displaystyle=\sum_{k=2}^{\lambda}{\lambda-2\choose k-2}(x+k\beta)^{k-1}[1-x+(% \lambda-k)\beta]^{\lambda-k-1}$
		$\displaystyle=\sum_{k=0}^{\lambda-2}{\lambda-2\choose k}(x+2\beta+k\beta)^{k+1% }[1-x+(\lambda-k-2)\beta]^{\lambda-k-3},$

where we replaced $k-2$ by $k$ . According to Lemma 3.3, the expression on the right hand side of the previous equality reduces to $T(2,\lambda-2,x+2\beta,1-x)$ given by (3.6). Hence,

C_{\lambda}(x)=\dfrac{\lambda(\lambda-1)}{n^{2}}\cdot\dfrac{x(1-x)}{(1+\lambda% \beta)^{\lambda-1}}\cdot T(2,\lambda-2,x+2\beta,1-x)+\dfrac{\lambda x}{n^{2}}.

Based on relation (3.8) for $u=x+2\beta$ , $v=1-x$ and $N=\lambda-2$ , we have that

	$\displaystyle C_{\lambda}(x)$	$\displaystyle=\dfrac{\lambda(\lambda-1)x}{n^{2}(1+\lambda\beta)^{\lambda-1}}% \int_{0}^{\infty}e^{-s}\big[(x+2\beta)(1+\lambda\beta+s\beta)^{\lambda-2}+(% \lambda-2)s\beta^{2}(1+\lambda\beta+s\beta)^{\lambda-3}\big]ds+\dfrac{\lambda x% }{n^{2}}$
		$\displaystyle=\dfrac{\lambda(\lambda-1)}{n^{2}}\bigg[x(x+2\beta)\int_{0}^{% \infty}e^{-s}\dfrac{(1+\lambda\beta+s\beta)^{\lambda-2}}{(1+\lambda\beta)^{% \lambda-1}}ds$
		$\displaystyle\;\;\;\;+x(\lambda-2)\beta^{2}\int_{0}^{\infty}se^{-s}\dfrac{(1+% \lambda\beta+s\beta)^{\lambda-3}}{(1+\lambda\beta)^{\lambda-1}}ds\bigg]+\dfrac% {\lambda x}{n^{2}}$
		$\displaystyle=\dfrac{\lambda(\lambda-1)}{n^{2}}\bigg[x(x+2\beta)\dfrac{1}{1+% \lambda\beta}\int_{0}^{\infty}e^{-s}\bigg(1+\dfrac{s\beta}{1+\lambda\beta}% \bigg)^{\lambda-2}ds$
		$\displaystyle\;\;\;\;+x(\lambda-2)\beta^{2}\dfrac{1}{(1+\lambda\beta)^{2}}\int% _{0}^{\infty}se^{-s}\bigg(1+\dfrac{s\beta}{1+\lambda\beta}\bigg)^{\lambda-3}ds% \bigg]+\dfrac{\lambda x}{n^{2}}.$

Finally, if we denote

\int_{0}^{\infty}e^{-s}\bigg(1+\dfrac{s\beta}{1+\lambda\beta}\bigg)^{\lambda-2% }ds=M_{\lambda}

and

\int_{0}^{\infty}se^{-s}\bigg(1+\dfrac{s\beta}{1+\lambda\beta}\bigg)^{\lambda-% 3}ds=N_{\lambda},

then

C_{\lambda}(x)=\dfrac{\lambda(\lambda-1)}{n^{2}}\bigg[x(x+2\beta)M_{\lambda}+x% (\lambda-2)\beta^{2}N_{\lambda}\bigg]+\dfrac{\lambda x}{n^{2}}

and this completes the proof. ∎

Remark 6.

If $\beta\to 0$ , then $\lim\limits_{n\to\infty}M_{\lambda}=\lim\limits_{n\to\infty}N_{\lambda}=1$ , hence $\lim\limits_{n\to\infty}\mathcal{T}_{n,p}^{m,\beta}(e_{2};x)=x^{2}$ , for any $x\in[0,1]$ .

Another important result presented in this section explores the norm of the operator $\mathcal{T}_{n,p}^{m,\beta}$ applied on a continuous function defined on the interval $[0,1+p]$ .

Proposition 3.3.

For every $f\in C[0,1+p]$ , we have that

\|\mathcal{T}_{n,p}^{m,\beta}f\|_{C[0,1]}\leq\|f\|_{C[0,1+p]},

where $\|\cdot\|_{X}$ denotes the uniform norm on the space $X$ .

Proof.

Considering the form of the operator $\mathcal{T}_{n,p}^{m,\beta}$ given in (2.2) and Theorem 3.1, we have that:

	$\displaystyle\left\|\mathcal{T}_{n,p}^{m,\beta}f(x)\right\|$	$\displaystyle=\left\|\sum_{k=0}^{n+p-R}b^{\beta}_{n+p-R,k}(x)\bigg[\bigg(1-\sum% _{j=1}^{m}x^{j}\bigg)f\bigg(\dfrac{k}{n}\bigg)+\sum_{j=1}^{m}x^{j}f\bigg(% \dfrac{k+r_{j}}{n}\bigg)\bigg]\right\|$
		$\displaystyle\leq\sum_{k=0}^{n+p-R}b^{\beta}_{n+p-R,k}(x)\left\|\bigg(1-\sum_{j% =1}^{m}x^{j}\bigg)f\bigg(\dfrac{k}{n}\bigg)+\sum_{j=1}^{m}x^{j}f\bigg(\dfrac{k% +r_{j}}{n}\bigg)\right\|$
		$\displaystyle\leq\sum_{k=0}^{n+p-R}b^{\beta}_{n+p-R,k}(x)\bigg[\bigg(1-\sum_{j% =1}^{m}x^{j}\bigg)\left\|f\bigg(\dfrac{k}{n}\bigg)\right\|+\sum_{j=1}^{m}x^{j}% \left\|f\bigg(\dfrac{k+r_{j}}{n}\bigg)\right\|\bigg]$
		$\displaystyle\leq\\|f\\|_{C[0,1+p]}\sum\limits_{k=0}^{n+p-R}b_{n+p-R,k}(x)\left(% 1-\sum_{j=1}^{m}x^{j}+\sum_{j=1}^{m}x^{j}\right)$
		$\displaystyle=\\|f\\|_{C[0,1+p]}\mathcal{T}_{n,p}^{m,\beta}(e_{0};x)$
		$\displaystyle=\\|f\\|_{C[0,1+p]}.$

∎

3.2. Convexity properties of the operator $\mathcal{T}_{{n,p}}^{{m,\beta}}$

Inspired by the property presented in [26, Theorem 3] (see also [5, Theorem 3] and [22, Proposition 12]), we can prove an important result related to the operator $\mathcal{T}_{n,p}^{m,\beta}$ . Although the inequality relation does not apply to $f(x)$ , it nevertheless generalizes the results obtained for other operators (for example, if $p=0$ and $m=1$ , then Proposition 3.4 reduces to [22, Proposition 12] for the extremal case $\lambda_{n}=1$ ).

Proposition 3.4.

If $f$ is convex on $[0,1+p]$ , then $\mathcal{T}_{n,p}^{m,\beta}f(x)\geq f\left(\mathcal{T}_{n,p}^{m,\beta}(e_{1};x% )\right)$ , for all $x\in[0,1]$ .

Proof.

Based on (2.2), we have that:

\displaystyle\mathcal{T}_{n,p}^{m,\beta}f(x)

\displaystyle=\sum_{k=0}^{n+p-R}b^{\beta}_{n+p-R,k}(x)\bigg[\bigg(1-\sum_{j=1}% ^{m}x^{j}\bigg)f\bigg(\dfrac{k}{n}\bigg)+\sum_{j=1}^{m}x^{j}f\bigg(\dfrac{k+r_% {j}}{n}\bigg)\bigg].

For simplicity of notation, we denote by $y_{i}=\frac{k+r_{i}}{n},\ i=0,\ldots,m,$ with $r_{0}=0$ and by $\alpha_{0}=1-\sum\limits_{j=1}^{m}x^{j},\ \alpha_{i}=x^{i},\ i=1,\ldots,m$ . The operator $\mathcal{T}_{n,p}^{m,\beta}$ can now be written as

	$\displaystyle\mathcal{T}_{n,p}^{m,\beta}f(x)$	$\displaystyle=\sum_{k=0}^{n+p-R}b^{\beta}_{n+p-R,k}(x)\sum\limits_{i=0}^{m}% \alpha_{i}f(y_{i})=\sum\limits_{i=0}^{m}\alpha_{i}\sum_{k=0}^{n+p-R}b^{\beta}_% {n+p-R,k}(x)f(y_{i})$
		$\displaystyle\geq\sum\limits_{i=0}^{m}\alpha_{i}f\left(\sum_{k=0}^{n+p-R}b^{% \beta}_{n+p-R,k}(x)y_{i}\right)\geq f\left(\sum\limits_{i=0}^{m}\alpha_{i}\sum% _{k=0}^{n+p-R}b^{\beta}_{n+p-R,k}(x)y_{i}\right)$
		$\displaystyle=f\left(\sum_{k=0}^{n+p-R}b^{\beta}_{n+p-R,k}(x)\sum\limits_{i=0}% ^{m}\alpha_{i}y_{i}\right)=f\left(\mathcal{T}_{n,p}^{m,\beta}(e_{1};x)\right).$

Hence,

\mathcal{T}_{n,p}^{m,\beta}f(x)\geq f\left(\mathcal{T}_{n,p}^{m,\beta}(e_{1};x% )\right),

(3.13)

for all $x\in[0,1]$ . ∎

Remark 7.

In particular, if $p=0$ and $m=1$ , then $\mathcal{T}_{n,0}^{1,\beta}(e_{1};x)=x$ and

\mathcal{T}_{n,0}^{1,\beta}f(x)\geq f\left(\mathcal{T}_{n,p}^{m,\beta}(e_{1};x% )\right)=f(x),

for all $x\in[0,1]$ , where $\mathcal{T}_{n,0}^{1,\beta}$ is the Stancu type extension of the Cheney-Sharma operator of the second kind studied by Bostanci and Başcanbaz-Tunca in [7] (see also [5] and [26]).

Proposition 3.5.

If $p\geq R^{*}=r_{2}+\ldots+r_{m}$ and $f$ is an increasing convex function on $[0,1+p]$ , then $\mathcal{T}_{n,p}^{m,\beta}f(x)\geq f(x)$ , for all $x\in[0,1]$ .

Proof.

In view of Propostion 3.4 we know that if $f$ is convex on $[0,1+p]$ , then

\mathcal{T}_{n,p}^{m,\beta}f(x)\geq f\left(\mathcal{T}_{n,p}^{m,\beta}(e_{1};x% )\right),\quad x\in[0,1].

(3.14)

Since $f$ is increasing on $[0,1+p]$ , it follows that for any two points $A,B\in[0,1+p]$ with $A\leq B$ we have that $f(A)\leq f(B)$ . In particular, for $A=x\in[0,1]\subseteq[0,1+p]$ and $B=\mathcal{T}_{n,p}^{m,\beta}(e_{1};x)\in\left[0,\frac{n+p}{n}\right]\subseteq% [0,1+p]$ , we know (see Lemma 3.1) that

A=x\leq\mathcal{T}_{n,p}^{m,\beta}(e_{1};x)=B,

for all $x\in[0,1]$ . Hence,

f(x)=f(A)\leq f(B)=f\left(\mathcal{T}_{n,p}^{m,\beta}(e_{1};x)\right),\quad x% \in[0,1].

(3.15)

In view of relations (3.14) and (3.15) we deduce that $\mathcal{T}_{n,p}^{m,\beta}f(x)\geq f(x)$ , for all $x\in[0,1]$ and this completes the proof. ∎

Remark 8.

We mention again that in Proposition 3.5 the assumption $p\geq R^{*}$ is necessary only if $m\geq 2$ (see Remark 5). In the case $m=1$ , the operator $\mathcal{T}_{n,p}^{1,\beta}$ reduces to the classical Stancu-Schurer type extension of the Cheney-Sharma operator (of the second kind) and the result holds for every $p\in\mathbb{N}$ .

It is important to observe that the convexity property imposed in Proposition 3.5 is essential in order to obtain the inequality $\mathcal{T}_{n,p}^{m,\beta}f(x)\geq f(x)$ , for all $x\in[0,1]$ . To illustrate this fact, let us consider the following example in which the function $f$ is not convex on its domain.

Example 2.

Let $f:[0,1+p]\rightarrow\mathbb{R}$ be given by $f(x)=\frac{\sqrt{x}}{x+1}$ , for all $x\in[0,1+p]$ . Since $f$ is not convex on $[0,1+p]$ , we obtain the following counterexamples of Proposition 3.5:

•

Let $m=1$ . According to Lemma 3.1 and Remark 5 we know that $\mathcal{T}_{n,p}^{m,\beta}(e_{1};x)\geq x$ , for all $x\in[0,1]$ . However, since $f$ is not convex on $[0,1+p]$ , we have that $\mathcal{T}_{n,p}^{m,\beta}f(x)\not\geq f(x)$ , for all $x\in[0,1]$ , as it can be observed in Figure 4 and Figure 4.

Figure 3. $m=1$ , $n=10$ , $p=4$ , $r=3$ and $\beta=5$ .

Figure 4. $m=1$ , $n=5$ , $p=4$ , $r=2$ and $\beta=0$ .
•

Let $m\geq 2$ . In this case, $\mathcal{T}_{n,p}^{m,\beta}(e_{1};x)\geq x$ , for all $x\in[0,1]$ only if $p\geq R^{*}=r_{2}+\ldots+r_{m}$ (see Lemma 3.1). Although this condition can be easily fulfilled by choosing a suitable number $p$ , since the function $f$ is not convex, we do not obtain the inequality from Proposition 3.5. This important remark is illustrated in Figure 6, respectively in Figure 6. In each case we consider $m=4$ , $n=17$ , $r_{1}=2$ , $r_{2}=1$ , $r_{3}=3$ , $r_{4}=2$ and $\beta=5$ .

Figure 5. $p=3<r_{2}+r_{3}+r_{4}=6$ .

Figure 6. $p=8>r_{2}+r_{3}+r_{4}=6$ .

Similar examples can be obtained for different values of constants $m,n,p,r_{1},\ldots,r_{m}\in\mathbb{N}$ , $\beta\geq 0$ and non-convex functions $f$ defined on $[0,1+p]$ .

On the other hand, it is also important to know that the operator $\mathcal{S}^{m,\beta}_{n,p}$ does not satisfy an inequality of the form $\mathcal{S}^{m,\beta}_{n,p}f(x)\geq f(x)$ , for all $x\in[0,1]$ . For details, one may consult Remark 10 in Section 5.

4. Korovkin type theorems for the operator $\mathcal{T}_{n,p}^{m,\beta}$

In this section we study some approximation results based on the modulus of continuity and show that the operator $\mathcal{T}_{n,p}^{m,\beta}$ satisfies the conditions of the Korovkin-type theorem (see [3], [20], [27], [31]). In order to prove the first result of this section, we introduce the following auxiliary lemma.

Lemma 4.1.

[31] Considering $f\in C[0,1+p]$ and the modulus of continuity associated to $f$ ,
$\omega(f,\delta)=\sup\limits_{x,x+h\in[0,1]}\{\left|f(x+h)-f(x)\right|:\ 0\leq h% \leq\delta\}$ , there exists a positive constant $C>0$ such that

C^{-1}\omega(f,\delta)\leq K_{1}(f,\delta)\leq C\omega(f,\delta),

(4.1)

where $K_{1}(f,\delta)$ is given as

K_{1}(f,\delta)=\inf\limits_{g\in W^{1}}\{\|f-g\|+\delta\|g^{\prime}\|\},

with $W^{1}=\{g\in C[0,1+p]:\ g^{\prime}\in C[0,1+p]\}$ and $\delta>0$ .

We can now prove the following result.

Theorem 4.1.

Consider $m\in\mathbb{N}$ , $m\geq 1$ and $\beta\geq 0$ . Let $n,p,r_{1},\ldots,r_{m},R\in\mathbb{N}$ be such that $n>2R$ , where $R=r_{1}+\ldots+r_{m}$ , with $p\geq R$ . If we denote by

\delta_{n,p}^{r_{2},\ldots,r_{m}}(x)=\mathcal{T}_{n,p}^{m,\beta}(e_{1};x)-x=% \frac{p}{n}x+\frac{r_{2}}{n}x(x-1)+\frac{r_{3}}{n}x(x^{2}-1)+\ldots+\frac{r_{m% }}{n}x(x^{m-1}-1),

and by

\tilde{\delta}_{n,p}^{r_{2},\ldots,r_{m}}(x)=\frac{1}{2}\delta_{n,p}^{r_{2},% \ldots,r_{m}}(x),

then

\left|\mathcal{T}_{n,p}^{m,\beta}f(x)-f(x)\right|\leq C\omega(f,\tilde{\delta}% _{n,p}^{r_{2},\ldots,r_{m}}(x)).

Proof.

Since the operator $\mathcal{T}_{n,p}^{m,\beta}$ is linear, we have that

	$\displaystyle\left\|\mathcal{T}_{n,p}^{m,\beta}f(x)-f(x)\right\|$	$\displaystyle=\left\|\mathcal{T}_{n,p}^{m,\beta}(f-g)(x)-(f-g)(x)+\mathcal{T}_{% n,p}^{m,\beta}g(x)-g(x)\right\|$
		$\displaystyle\leq\left\|\mathcal{T}_{n,p}^{m,\beta}(f-g)(x)-(f-g)(x)\right\|+% \left\|\mathcal{T}_{n,p}^{m,\beta}g(x)-g(x)\right\|$

Now, using Proposition 3.3, we obtain

\left|\mathcal{T}_{n,p}^{m,\beta}f(x)-f(x)\right|\leq 2\|f-g\|+\left|\mathcal{% T}_{n,p}^{m,\beta}g(x)-g(x)\right|

(4.2)

Since $g\in W^{1}$ , applying Lagrange theorem, we have that there exists a point $\xi$ between $x$ and $t$ such that

g(t)-g(x)=g^{\prime}(\xi)(t-x).

Applying the linear operator $\mathcal{T}_{n,p}^{m,\beta}$ , we get

\displaystyle\mathcal{T}_{n,p}^{m,\beta}(g;t)-g(x)\mathcal{T}_{n,p}^{m,\beta}(% e_{0};t)=g^{\prime}(\xi)\left(\mathcal{T}_{n,p}^{m,\beta}(e_{1};t)-x\mathcal{T% }_{n,p}^{m,\beta}(e_{0};t)\right),

and for $t:=x$

	$\displaystyle\mathcal{T}_{n,p}^{m,\beta}g(x)-g(x)$	$\displaystyle=g^{\prime}(\xi)\left(\mathcal{T}_{n,p}^{m,\beta}(e_{1};x)-x% \mathcal{T}_{n,p}^{m,\beta}(e_{0};x)\right)$
		$\displaystyle=g^{\prime}(\xi)\left[\frac{p}{n}x+\frac{r_{2}}{n}x(x-1)+\frac{r_% {3}}{n}x(x^{2}-1)+\ldots+\frac{r_{m}}{n}x(x^{m-1}-1)\right]$
		$\displaystyle=g^{\prime}(\xi)\delta_{n,p}^{r_{2},\ldots,r_{m}}(x).$

Substituting this into (4.2), we obtain

\displaystyle\left|\mathcal{T}_{n,p}^{m,\beta}f(x)-f(x)\right|\leq 2\|f-g\|+\|% g^{\prime}\|\cdot\delta_{n,p}^{r_{2},\ldots,r_{m}}(x)\leq 2\left(\|f-g\|+\|g^{% \prime}\|\tilde{\delta}_{n,p}^{r_{2},\ldots,r_{m}}(x)\right)

Taking the infimum over all $g\in W^{1}$ , by relation (4.1), we get that

\left|\mathcal{T}_{n,p}^{m,\beta}f(x)-f(x)\right|\leq 2K_{1}(f,\tilde{\delta}_% {n,p}^{r_{2},\ldots,r_{m}})\leq C\omega(f,\tilde{\delta}_{n,p}^{r_{2},\ldots,r% _{m}}(x)),

which completes the proof. ∎

Theorem 4.2.

If $\beta\to 0$ , the operator $\mathcal{T}_{n,p}^{m,\beta}$ satisfies the following relation

\lim\limits_{n\to\infty}\mathcal{T}_{n,p}^{m,\beta}f=f\mbox{ uniformly on }[0,% 1],\mbox{ for any }f\in C[0,1+p].

(4.3)

Proof.

According to Theorem 3.1, we have that $\mathcal{T}_{n,p}^{m,\beta}(e_{0};x)=1$ and $\lim\limits_{n\to\infty}\mathcal{T}_{n,p}^{m,\beta}(e_{1};x)=x$ , for $x\in[0,1]$ . Using Remark 6 when $\beta\to 0$ , we obtain $\lim\limits_{n\to\infty}\mathcal{T}_{n,p}^{m,\beta}(e_{2};x)=x^{2}$ , for $x\in[0,1]$ . Hence, the linear and positive operator $\mathcal{T}_{n,p}^{m,\beta}$ satisfies the assumptions of the Korovkin-type theorem (see [31]), which gives us the result in relation (4.3). ∎

5. Properties of the operator $\mathcal{S}_{{n,p}}^{{m,\beta}}$

In this section we direct our attention to the first-type extension operator $\mathcal{S}_{{n,p}}^{{m,\beta}}$ . As for the previous operator, we study the expressions of the moments and some approximation results. Although not all the properties of the operator $\mathcal{T}_{{n,p}}^{{m,\beta}}$ carry over directly to $\mathcal{S}_{{n,p}}^{{m,\beta}}$ , as shown in this section through some examples, several others can be established for $\mathcal{S}_{{n,p}}^{{m,\beta}}$ .

Theorem 5.1.

Consider $m\in\mathbb{N}$ , $m\geq 1$ and $\beta\geq 0$ . Let $n,p,r_{1},\ldots,r_{m},R\in\mathbb{N}$ be such that $n>2R$ , where $R=r_{1}+\ldots+r_{m}$ and denote by $\lambda:=n+p-R$ . Also, let $e_{j}(x)=x^{j}$ , for every $j\in\mathbb{N}$ and $x\in[0,1]$ . Then

a)

$\mathcal{S}^{m,\beta}_{n,p}(e_{0};x)=1$ ;
b)

$\mathcal{S}^{m,\beta}_{n,p}(e_{1};x)=\dfrac{1}{n}\left(r_{1}x+r_{2}x^{2}+% \ldots+r_{m}x^{m}\right)+\dfrac{\lambda}{n}\tilde{M}_{\lambda}x$ , where

$\tilde{M}_{\lambda}=\dfrac{1}{1+\lambda\beta}_{0}^{\infty}e^{-s}\bigg(% 1+\dfrac{s\beta}{1+\lambda\beta}\bigg)^{\lambda-1}ds;$ (5.1)
c)

$\mathcal{S}^{m,\beta}_{n,p}(e_{2};x)=\dfrac{1}{n^{2}}\left(r_{1}^{2}x+r_{2}^{2% }x^{2}+\ldots+r_{m}^{2}x^{m}\right)+\dfrac{2\lambda\left(r_{1}x+r_{2}x^{2}+% \ldots r_{m}x^{m}\right)}{n^{2}}\tilde{M}_{\lambda}x+\tilde{C}_{\lambda}(x)$ , where $\tilde{C}_{\lambda}(x)$ is given by

$\tilde{C}_{\lambda}(x)=\frac{\lambda(\lambda-1)}{n^{2}}\left[x\left(x+2\beta% \right)\tilde{A}_{\lambda}+x(\lambda-2)\beta^{2}\tilde{B}_{\lambda}\right]+% \frac{\lambda}{n^{2}}\tilde{M}_{\lambda}x,$ (5.2)

with

$\tilde{A}_{\lambda}=\frac{1}{(1+\lambda\beta)^{2}}\int\limits_{0}^{\infty}e^{-% t}dt\int\limits_{0}^{\infty}e^{-s}\left(1+\frac{\beta\left(t+s\right)}{1+% \lambda\beta}\right)^{\lambda-2}ds,$ (5.3)

$\tilde{B}_{\lambda}=\frac{1}{(1+\lambda\beta)^{3}}\int\limits_{0}^{\infty}e^{-% t}dt\int\limits_{0}^{\infty}se^{-s}\left(1+\frac{\beta\left(t+s\right)}{1+% \lambda\beta}\right)^{\lambda-3}ds,$ (5.4)

and $\tilde{M}_{\lambda}$ given by (5.1).

Proof.

a)

The proof is similar to the one in Theorem 3.1, using Jensen’s formula (1.6) with $m=\lambda$ , $u=x$ and $v=1-x$ .
b)

By similar computations as in Theorem 3.1, one gets S^m,β_n,p(e_1;x)= 1n ( r_1 x + r_2 x^2 + …+ r_m x^m) + A_λ(x), where we have denoted by

$A_{\lambda}(x)=\sum\limits_{k=0}^{\lambda}\dfrac{k}{n}a_{\lambda,k}^{\beta}(x).$ (5.5)

Since A_λ(x) = λn ∑_k=0^λ kλ a_λ, k^β(x), using a result given in [19], we obtain that A_λ(x) = λn ~M_λx , where ~M_λ=11+λβ∫ _0^∞e^-s(1+sβ1+λβ)^λ-1ds.
c)

Following the ideas of the proof of Theorem 3.2, we have that S^m,β_n,p(e_2;x) = 1n2 (r_1^2 x + r_2^2 x^2 + …+ r_m^2 x^m ) + 2 λ(r1x + r2x2+ …rmxm)n2 ~M_λx + ~C_λ(x), with

$\tilde{C}_{\lambda}(x)=\sum\limits_{k=0}^{\lambda}\frac{k^{2}}{n^{2}}a_{% \lambda.k}^{\beta}(x)=\frac{\lambda^{2}}{n^{2}}\sum\limits_{k=0}^{\lambda}% \frac{k^{2}}{\lambda^{2}}a_{\lambda.k}^{\beta}(x).$ (5.6)

Applying now a result presented in [13, Lemma 3], we have that ∑_k=0^λ k2λ2 a_λ.k^β(x) = λ-1λ [x(x+2 β) ~A_λ + x(λ-2)β^2 ~B_λ ] + 1λ ~M_λx, hence ~C_λ(x) = λ(λ-1)n2 [x(x+2 β) ~A_λ + x(λ-2)β^2 ~B_λ ] + λn2 ~M_λx, with $\tilde{A}_{\lambda}$ , $\tilde{B}_{\lambda}$ and $\tilde{M}_{\lambda}$ given in (5.3), (5.4) and (5.1), respectively.

∎

Remark 9.

As pointed out in Lemma 3.1, the inequality $\mathcal{T}_{n,p}^{m,\beta}(e_{1};x)\geq x$ , for all $x\in[0,1]$ is satisfied if $n>2R$ , where $R=r_{1}+\ldots+r_{m}$ and $p\geq R^{*}=r_{2}+\ldots+r_{m}$ . However, in the case of the operator $\mathcal{S}_{n,p}^{m,\beta}$ this is not always true, as illustrated in Figure 7.

Remark 10.

As mentioned at the end of Section 4, the operator $\mathcal{S}_{n,p}^{m,\beta}$ does not satisfy an inequality of the form $\mathcal{S}_{n,p}^{m,\beta}f(x)\geq f(x),$ for all $x\in[0,1]$ , with $f$ being an increasing convex function on $[0,1+p]$ , as was proved in Proposition 3.5 for the operator $\mathcal{T}_{n,p}^{m,\beta}$ . This is illustrated in Figure 8, considering the function $f(x)=x^{2}$ on $[0,1]$ and various values for the operator’s parameters.

Remark 11.

If $\beta\to 0$ , then $\lim\limits_{n\to\infty}\tilde{M}_{\lambda}=\tilde{A}_{\lambda}=\tilde{B}_{% \lambda}=1$ , hence $\lim\limits_{n\to\infty}\mathcal{S}^{m,\beta}_{n,p}(e_{j};x)=x^{j},\ j=0,1,2,$ and we can apply a Korovkin-type result as in Theorem 4.2 for the operator $\mathcal{S}^{m,\beta}_{n,p}$ .

Proposition 5.2.

For every $f\in C[0,1+p]$ , we have that

\|\mathcal{S}_{n,p}^{m,\beta}f\|_{C[0,1]}\leq\|f\|_{C[0,1+p]},

where $\|\cdot\|_{X}$ denotes the uniform norm on the space $X$ .

Proof.

It is similar to the result proven in Proposition 3.3. ∎

6. Conclusions and future work

We end this paper with a short comparative analysis of the operators $\mathcal{S}_{n,p}^{m,\beta}$ and $\mathcal{T}_{n,p}^{m,\beta}$ with analogous operators presented in the literature. We will point out the main distinctions and possible advantages of the operators introduced in this paper. It is clear that the closest family of operators is that of shifted-knot Bernstein-Stancu operators (the classical generalization where the evaluation nodes and weights are shifted by parameters). However, in our case, the operators use multiple shifts $r_{j}$ with different weights $x^{j}$ . Moreover, the weighted combination depends on powers of $x$ , and this gives us more flexibility and local adaptation. Our operators remain in the Bernstain-Stancu family, but enrich it via multiple shifts and weighted combination of evaluations, without changing to an integral-based or multivariate framework. In contrast with Bernstein-Stancu operators with shifted knots (see e.g. [1], [4], [23], [32]), Baskakov-Kantorovich operators (see e.g. [9], [33]), or Baskakov-Schurer-Stancu generalizations that use single shifts (see e.g. [2], [9]), the operators introduced in this paper stand out by using multiple shifted evaluations per basis node, weighted by powers of $x$ . Again, this particular property offers more local flexibility and adjusatable approximation at different regions through $x^{j}$ .

It is clear that other important properties can be obtained for the operators $\mathcal{T}_{n,p}^{m,\beta}$ , respectively $\mathcal{S}_{n,p}^{m,\beta}$ in $\mathbb{R}$ (e.g., the preservation of the Lipschitz property, extensions of Chlodovsky type, the case of three parameters, the multivariate case of the Cheney-Sharma operators). Different extensions and results were recently obtained by Başcanbaz-Tunca in [4], Başcanbaz-Tunca, Erençin and Taşdelen in [5] and [32], Miclăuş in [24], or Söylemez and Taşdelen in [26] and [27]. An interesting problem would be to study how such results can be adapted or generalized to the operators introduced in our paper.

On the other hand, Çetin and Mutlu have recently obtained extensions of the classical results in the complex plane (see [16] and [17]). Taking into account their ideas, it will be of interest to prove similar results for $\mathcal{T}_{n,p}^{m,\beta}$ and $\mathcal{S}_{n,p}^{m,\beta}$ in $\mathbb{C}$ (extension results and preservation of geometric properties of the original function such as univalence, starlikeness, convexity and spirallikeness).

Acknowledgments

The authors thank the referees for carefully reading the manuscript and providing helpful suggestions.

References

[1] AGRATINI, O.: Stancu modified operators revisited, Rev. Anal. Numér. Théor. Approx. 31(1), 9–16, 2002.
[2] AGRAWALA, P.N.—GOYALA, M.: Generalized Baskakov Kantorovich Operators Filomat 31(19), 6131–6151, 2017.
[3] ALTOMARE, F.—CAMPITI, M.: Korovkin-type approximaton theory and its applications, Walter de Gruyter, Berlin-New York, 1994.
[4] BAŞCANBAZ-TUNCA, G.: A note on Stancu operators with three parameters, Bull. Transilv. Univ. Braşov Ser. III. Math. Comput. Sci. 5(67), 55–70, 2025.
[5] BAŞCANBAZ-TUNCA, G.—ERENÇIN, A.—TAŞDELEN, F.: Some properties of Bernstein type Cheney and Sharma operators, General Mathematics 24(1-2), 17–25, 2016.
[6] BERNSTEIN, S.N.: Démonstration du théoréme de Weierstrass fondée sur le calcul des probabilités, Commun. Kharkov Math. Soc. 13, 1–2, 1912/1913.
[7] BOSTANCI, T.—BAŞCANBAZ-TUNCA, G.: A Stancu type extension of Cheney and Sharma operator, J. Numer. Anal. Approx. Theory 47(2), 124–134, 2018.
[8] BODUR, M.—BOSTANCI, T., BAŞCANBAZ-TUNCA, G.: On Kantorovich variant of Brass-Stancu operators Demonstratio Mathematica 57(1), pp. 20240007, 2024.
[9] BUSTAMANTE, J.: Baskakov-Kantorovich operators reproducing affine functions Stud. Univ. Babeş-Bolyai Math. 66(4), 739–756, 2021.
[10] BRASS, H.: Eine Verallgemeinerung der Bernsteinschen Operatoren Abh. Math. Sem. Univ. Hamburg 38, 111–122. 1971.
[11] CĂTINAŞ, T.: Extension of some Cheney-Sharma type operators to a triangle with one curved side, Miskolc Math. Notes 21, 101–111, 2020.
[12] CĂTINAŞ, T.: Cheney–Sharma type operators on a triangle with straight sides, Symmetry 14(11), 2446, 2022.
[13] CĂTINAŞ, T.—BUDA, I.: An extension of the Cheney-Sharma operator of the first kind, J. Numer. Anal. Approx. Theory 52(2), 172–181, 2023.
[14] CĂTINAŞ, T.—OTROCOL, D.: Iterates of multivariate Cheney-Sharma operators, J. Comput. Anal. Appl. 15(7), 1240–1246, 2013.
[15] CĂTINAŞ, T.—OTROCOL, D.: Iterates of Cheney-Sharma type operators on a triangle with curved side, J. Comput. Anal. Appl. 28(4), 737–744, 2020.
[16] ÇETIN, N.: A new complex generalized Bernstein-Schurer operator, Carpathian J. Math. 37(1), 81–89, 2021.
[17] ÇETIN, N.—MUTLU, N.M.: Complex generalized Stancu-Schurer operators, Math. Slovaca 74(5), 1215–1232, 2024.
[18] ÇETIN, N.—MUCURTAY, A.—BOSTANCI, T.: A new generalization of Kantorovich operators depending on a non-negative integer, 2nd International E-Conference on Mathematical and Statistical Science: A Selçuk Meeting (ICOMMS’23 https://icomss23.selcuk.edu.tr).
[19] CHENEY, E.W.—SHARMA, A.: On a generalization of Bernstein polynomials, Riv. Mat. Univ. Parma 2, 77–84, 1964.
[20] GADJIEV, A.D.: The convergence problem for a sequence of positive linear operators on unbounded sets and theorems analogues to that of P.P. Korovkin, Dokl. Akad. Nauk 218, 1974.
[21] GONSKA, H.H.—MEIER, J.: Quantitative theorems on approximation by Bernstein Stancu operators, Calcolo, 21, pp. 317335, 1984.
[22] GRIGORICIUC, E.Ş.: A Stancu type extension of the Cheney-Sharma Chlodovsky operators, J. Numer. Anal. Approx. Theory 53(1), 103–117, 2024.
[23] HOLHOŞ, A.: Uniform approximation of functions by Bernstein-Stancu operator, Carpathian J. Math. 31(2), 205–212, 2015.
[24] MICLĂUŞ, D.: The generalization of some results for Schurer and Schurer-Stancu operators, Rev. Anal. Numer. Theor. Approx. 40(1), 52–63, 2011.
[25] SCHURER, F.: Linear positive operators in approximation theory, Math. Inst. Techn. Univ. Delft Report, 1962.
[26] SÖYLEMEZ, D.—TAŞDELEN, F.: On Cheney-Sharma Chlodovsky operators, Bull. Math. Anal. Appl. 11(1), 36–43, 2019.
[27] SÖYLEMEZ, D.—TAŞDELEN, F.: Approximation by Cheney-Sharma Chlodovsky operators, Hacet. J. Math. Stat. 49(2), 512–522, 2020.
[28] STANCU, D.D.: Approximation of functions by a new class of linear polynomial operators, Rev. Roumaine Math. Pures Appl. 8, 1173–1194, 1968.
[29] STANCU, D.D.: Quadrature formulas constructed by using certain linear positive operators, In: Hämmerlin, G. (eds) Numerical Integration. ISNM 57: International Series of Numerical Mathematics 57, Birkhäuser, Basel, 241–251, 1981.
[30] STANCU, D.D.—CISMAŞIU, C.: On an approximating linear positive operator of Cheney Sharma, Rev. Anal. Numér. Théor. Approx. 26, 221–227, 1997.
[31] STANCU, D.D.—COMAN, GH.—AGRATINI, O.—TRÎMBIŢAŞ, R.T.—BLAGA, P.—CHIOREAN, I.: Analiză numerică și teoria aproximării, Presa Universitară Clujeană, 2001 (in Romanian).
[32] TAŞDELEN, F.—BAŞCANBAZ-TUNCA, G.—ERENÇIN, A.: On a new type Bernstein-Stancu operators, Fasc. Math. 48, 119–128, 2012.
[33] ZHANG, C.—ZHU, Z.: Preservation properties of the Baskakov-Kantorovich operators, Comput. Math. Appl. 57(9), 1450–1455, 2009.

AGRATINI, O.: Stancu modified operators revisited, Rev. Anal. Numér. Théor. Approx. 31(1), 9–16, 2002.

AGRAWALA, P.N.—GOYALA, M.: Generalized Baskakov Kantorovich Operators Filomat 31(19), 6131–6151, 2017.

ALTOMARE, F.—CAMPITI, M.: Korovkin-type approximaton theory and its applications, Walter de Gruyter, Berlin-New York, 1994.

BAȘCANBAZ-TUNCA, G.: A note on Stancu operators with three parameters, Bull. Transilv. Univ. Brașov Ser. III. Math. Comput. Sci. 5(67), 55–70, 2025.

BAȘCANBAZ-TUNCA, G.—ERENČIN, A.—TAȘDELEN, F.: Some properties of Bernstein type Cheney and Sharma operators, General Mathematics 24(1-2), 17–25, 2016.

BERNSTEIN, S.N.: Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités, Commun. Kharkov Math. Soc. 13, 1–2, 1912/1913.

BOSTANCI, T.—BAȘCANBAZ-TUNCA, G.: A Stancu type extension of Cheney and Sharma operator, J. Numer. Anal. Approx. Theory 47(2), 124–134, 2018.

BODUR, M.—BOSTANCI, T., BAȘCANBAZ-TUNCA, G.: On Kantorovich variant of Brass-Stancu operators Demonstratio Mathematica 57(1), pp. 20240007, 2024.

BUSTAMANTE, J.: Baskakov-Kantorovich operators reproducing affine functions Stud. Univ. Babeș-Bolyai Math. 66(4), 739–756, 2021.

BRASS, H.: Eine Verallgemeinerung der Bernsteinschen Operatoren Abh. Math. Sem. Univ. Hamburg 38, 111–122. 1971.

CĂTINĂȘ, T.: Extension of some Cheney-Sharma type operators to a triangle with one curved side, Miskolc Math. Notes 21, 101–111, 2020.

CĂTINĂȘ, T.: Cheney–Sharma type operators on a triangle with straight sides, Symmetry 14(11), 2446, 2022.

CĂTINĂȘ, T.—BUDA, I.: An extension of the Cheney-Sharma operator of the first kind, J. Numer. Anal. Approx. Theory 52(2), 172–181, 2023.

CĂTINĂȘ, T.—OTROCOL, D.: Iterates of multivariate Cheney-Sharma operators, J. Comput. Anal. Appl. 15(7), 1240–1246, 2013.

CĂTINĂȘ, T.—OTROCOL, D.: Iterates of Cheney-Sharma type operators on a triangle with curved side, J. Comput. Anal. Appl. 28(4), 737–744, 2020.

ÇETIN, N.: A new complex generalized Bernstein-Schurer operator, Carpathian J. Math. 37(1), 81–89, 2021.

ÇETIN, N.—MUTLU, N.M.: Complex generalized Stancu-Schurer operators, Math. Slovaca 74(5), 1215–1232, 2024.

ÇETIN, N.—MUCURTAY, A.—BOSTANCI, T.: A new generalization of Kantorovich operators depending on a non-negative integer, 2nd International E-Conference on Mathematical and Statistical Science: A Selçuk Meeting (ICOMMS’23 https://icomss23.selcuk.edu.tr).

CHENEY, E.W.—SHARMA, A.: On a generalization of Bernstein polynomials, Riv. Mat. Univ. Parma 2, 77–84, 1964.

GADJIEV, A.D.: The convergence problem for a sequence of positive linear operators on unbounded sets and theorems analogues to that of P.P. Korovkin, Dokl. Akad. Nauk 218, 1974.

GONSKA, H.H.—MEIER, J.: Quantitative theorems on approximation by Bernstein Stancu operators, Calcolo, 21, pp. 317–335, 1984.

GRIGORICIUC, E.Ș.: A Stancu type extension of the Cheney-Sharma Chlodovsky operators, J. Numer. Anal. Approx. Theory 53(1), 103–117, 2024.

HOLHOȘ, A.: Uniform approximation of functions by Bernstein-Stancu operator, Carpathian J. Math. 31(2), 205–212, 2015.

MICLĂUȘ, D.: The generalization of some results for Schurer and Schurer-Stancu operators, Rev. Anal. Numer. Theor. Approx. 40(1), 52–63, 2011.

SCHURER, F.: Linear positive operators in approximation theory, Math. Inst. Techn. Univ. Delft Report, 1962.

SÖYLEMEZ, D.—TAȘDELEN, F.: On Cheney-Sharma Chlodovsky operators, Bull. Math. Anal. Appl. 11(1), 36–43, 2019.

SÖYLEMEZ, D.—TAȘDELEN, F.: Approximation by Cheney-Sharma Chlodovsky operators, Hacet. J. Math. Stat. 49(2), 512–522, 2020.

STANCU, D.D.: Approximation of functions by a new class of linear polynomial operators, Rev. Roumaine Math. Pures Appl. 8, 1173–1194, 1968.

STANCU, D.D.: Quadrature formulas constructed by using certain linear positive operators, In: Hämmerlin, G. (eds) Numerical Integration. ISNM 57: International Series of Numerical Mathematics 57, Birkhäuser, Basel, 241–251, 1981.

STANCU, D.D.—CIMAȘIU, C.: On an approximating linear positive operator of Cheney Sharma, Rev. Anal. Numér. Théor. Approx. 26, 221–227, 1997.

STANCU, D.D.—COMAN, GH.—AGRATINI, O.—TRÎMBIȚAȘ, R.T.—BLAGA, P.—CHIOREAN, I.: Analiză numerică și teoria aproximării, Presa Universitară Clujeană, 2001 (in Romanian).

TAȘDELEN, F.—BAȘCANBAZ-TUNCA, G.—ERENČIN, A.: On a new type Bernstein-Stancu operators, Fasc. Math. 48, 119–128, 2012.

ZHANG, C.—ZHU, Z.: Preservation properties of the Baskakov-Kantorovich operators, Comput. Math. Appl. 57(9), 1450–1455, 2009.

No results found.

	$\displaystyle\left\|\mathcal{T}_{n,p}^{m,\beta}f(x)\right\|$	$\displaystyle=\left\|\sum_{k=0}^{n+p-R}b^{\beta}_{n+p-R,k}(x)\bigg[\bigg(1-\sum% _{j=1}^{m}x^{j}\bigg)f\bigg(\dfrac{k}{n}\bigg)+\sum_{j=1}^{m}x^{j}f\bigg(% \dfrac{k+r_{j}}{n}\bigg)\bigg]\right\|$
		$\displaystyle\leq\sum_{k=0}^{n+p-R}b^{\beta}_{n+p-R,k}(x)\left\|\bigg(1-\sum_{j% =1}^{m}x^{j}\bigg)f\bigg(\dfrac{k}{n}\bigg)+\sum_{j=1}^{m}x^{j}f\bigg(\dfrac{k% +r_{j}}{n}\bigg)\right\|$
		$\displaystyle\leq\sum_{k=0}^{n+p-R}b^{\beta}_{n+p-R,k}(x)\bigg[\bigg(1-\sum_{j% =1}^{m}x^{j}\bigg)\left\|f\bigg(\dfrac{k}{n}\bigg)\right\|+\sum_{j=1}^{m}x^{j}% \left\|f\bigg(\dfrac{k+r_{j}}{n}\bigg)\right\|\bigg]$
		$\displaystyle\leq\\|f\\|_{C[0,1+p]}\sum\limits_{k=0}^{n+p-R}b_{n+p-R,k}(x)\left(% 1-\sum_{j=1}^{m}x^{j}+\sum_{j=1}^{m}x^{j}\right)$
		$\displaystyle=\\|f\\|_{C[0,1+p]}\mathcal{T}_{n,p}^{m,\beta}(e_{0};x)$
		$\displaystyle=\\|f\\|_{C[0,1+p]}.$

A Stancu-Schurer type extension of higher order of the Cheney-Sharma operators

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A Stancu-Schurer type extension of higher order of the Cheney-Sharma operators

Abstract.

Key words and phrases:

2010 Mathematics Subject Classification:

1. Introduction

Definition 1.

Remark 1.

Remark 2.

Lemma 1.1.

2. Extensions of higher orders of the Cheney-Sharma operators

Definition 2.

Example 1.

3. Properties of the operator 𝒯n,pm,β

3.1. Moments of 𝒯n,pm,β and their properties

Theorem 3.1.

Proof.

Remark 3.

Remark 4.

Lemma 3.1.

Proof.

Lemma 3.2.

Proof.

Remark 5.

Lemma 3.3.

Theorem 3.2.

Proof.

Remark 6.

Proposition 3.3.

Proof.

3.2. Convexity properties of the operator 𝒯n,pm,β

Proposition 3.4.

Proof.

Remark 7.

Proposition 3.5.

Proof.

Remark 8.

Example 2.

4. Korovkin type theorems for the operator 𝒯n,pm,β

Lemma 4.1.

Theorem 4.1.

Proof.

Theorem 4.2.

Proof.

5. Properties of the operator 𝒮n,pm,β

Theorem 5.1.

Proof.

Remark 9.

Remark 10.

Remark 11.

Proposition 5.2.

Proof.

6. Conclusions and future work

Acknowledgments

References

References

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3. Properties of the operator $\mathcal{T}_{{n,p}}^{{m,\beta}}$

3.1. Moments of $\mathcal{T}_{{n,p}}^{{m,\beta}}$ and their properties

3.2. Convexity properties of the operator $\mathcal{T}_{{n,p}}^{{m,\beta}}$

4. Korovkin type theorems for the operator $\mathcal{T}_{n,p}^{m,\beta}$

5. Properties of the operator $\mathcal{S}_{{n,p}}^{{m,\beta}}$