## Abstract

Starting from a discrete linear approximation process that has the ability to turn polynomials into polynomials of the same degree, we introduce an integral generalization by using Beta-type bases.

Some properties of this new sequence of operators are investigated in unweighted and weighted spaces of functions defined on unbounded interval. In our construction particular cases are outlined.

## Authors

**Octavian Agratini
**Babeş-Bolyai University, Faculty of Mathematics and Computer Science, Cluj-Napoca, Romania

Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania

## Keywords

### References

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## Paper coordinates

O. Agratini, *Linear positive operators constructed by using Beta-type bases*, Hacettepe Journal of Mathematics & Statistics, 49 (2020) no. 3, pp. 1030-1038, doi: 10.15672/hujms.549015

## About this paper

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Hacettepe Journal of Mathematics & Statistics

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Hacettepe Üniversitesi

##### Print ISSN

2651-477X

##### Online ISSN

2651-477X

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## Paper (preprint) in HTML form

# Linear positive operators constructed

by using Beta-type bases

###### Abstract.

Starting from a discrete linear approximation process that has the ability to turn polynomials into polynomials of the same degree, we introduce an integral generalization by using Beta-type bases. Some properties of this new sequence of operators are investigated in unweighted and weighted spaces of functions defined on unbounded interval. In our construction particular cases are outlined.

Keywords and phrases: Korovkin theorem, modulus of smoothness, weighted space, rate of convergence.

Mathematics Subject Classification: 41A36, 41A25.

## 1. Introduction

An old area of mathematical research called Approximation Theory has a great potential for applications to a wide variety of issues. The study of the linear methods of approximation, which are given by sequences of linear and positive operators, has become a firmly rooted part of the domain mentioned above.

The starting point of this note is a general positive approximation process of discrete type and expressed by series. The functions that are approximated are defined on unbounded intervals. Since a linear substitution maps the interior of such intervals onto $(0,\mathrm{\infty})$ or $\mathbb{R}$, we will choose the benchmark interval $I=[0,\mathrm{\infty})={\mathbb{R}}_{+}$, as it accumulates the problems caused by a finite endpoint and by the unboundedness of the other endpoint. Also, this class uses equidistant network of the form ${\mathrm{\Delta}}_{n}={(k{\lambda}_{n})}_{n\ge 1}$, where ${({\lambda}_{n})}_{n\ge 1}$ is a strictly decreasing positive sequence of real numbers with the property

$$\underset{n\to \mathrm{\infty}}{lim}{\lambda}_{n}=0.$$ | (1) |

The operators we are referring to are designed as follows

$$({L}_{n}f)(x)=\sum _{k=0}^{\mathrm{\infty}}{a}_{k}({\lambda}_{n};x)f(k{\lambda}_{n}),n\in \mathbb{N},x\in {\mathbb{R}}_{+},$$ | (2) |

where ${a}_{k}({\lambda}_{n};\cdot )\in {\mathbb{R}}_{+}^{{\mathbb{R}}_{+}}$ is continuous function for each $(n,k)\in \mathbb{N}\times {\mathbb{N}}_{0}$. Here ${\mathbb{N}}_{0}=\{0\}\cup \mathbb{N}$ and $f$ belongs to a function space, say $\mathcal{S}({\mathbb{R}}_{+})$, for which the right-hand side of relation (2) is well defined.

Using a Beta functions base, we introduce and explore an integral generalization of the operators defined by (2). The construction of the new class is present in the next section. Their approximation properties such as convergence towards identity operator and the rate of convergence are investigated in different function spaces.

## 2. The operators

Set ${e}_{j}$ the monomial of $j$-th degree, ${e}_{0}(x)=1$ and ${e}_{j}(x)={x}^{j}$, $j\ge 1$. Let ${\mathrm{\Pi}}_{j}^{\ast}$ be the set of all algebraic polynomials of degree $j$, $j\in {\mathbb{N}}_{0}$. Regarding ${L}_{n}$, $n\in \mathbb{N}$, operators we assume $\mathrm{\Pi}:={\displaystyle \bigcup _{j\ge 0}}{\mathrm{\Pi}}_{j}^{\ast}\subset \mathcal{S}({\mathbb{R}}_{+})$,

$${L}_{n}{e}_{0}={e}_{0}\text{and}{L}_{n}{e}_{j}\in {\mathrm{\Pi}}_{j}^{\ast},j\in \mathbb{N}.$$ | (3) |

Due to the linearity of ${L}_{n}$, the above conditions imply that these operators preserve the constants and map polynomials into polynomials of the same degree. Throughout the paper, based on relations (3), we consider the following representations of the polynomials ${L}_{n}{e}_{j}$

$$({L}_{n}{e}_{j})(x)=\sum _{i=0}^{j}{\alpha}_{i,j}(n){x}^{i},{\alpha}_{j,j}(n)\ne 0,x\in {\mathbb{R}}_{+}\text{and}j\in \mathbb{N}.$$ | (4) |

Since we are working on the assumption that the sequence ${({L}_{n})}_{n\ge 1}$ is an approximation process, this will involve at least the pointwise convergence of ${({L}_{n}{e}_{j})}_{n\ge 1}$ to ${e}_{j}$ for each $j\in \mathbb{N}$, in other words,

$$\underset{n\to \mathrm{\infty}}{lim}{\alpha}_{i,j}(n)={\delta}_{i,j},0\le i\le j\text{and}j\in \mathbb{N},$$ | (5) |

where ${\delta}_{i,j}$ is Kronecker symbol.

Our requirements in (3) are not very restrictive. Many classical operators enjoy these properties. For instance:

iii) Mastroianni operators [10] representing a generalization of Baskakov approximation process. For comparison, [1, pages 344, 350] may be consulted. Relations (3) can easily be deduced from the cited works.

In all the above examples ${\lambda}_{n}=1/n$, $n\ge 1$, takes place.

iv) Bernstein-Chlodovsky operators introduced by Chlodovsky [2] can be found in [9, p. 36] or in [1, pages 347, 351]. In this case ${\lambda}_{n}={b}_{n}/n$, where ${({b}_{n})}_{n\ge 1}$ is a sequence of strictly positive real numbers such that $\underset{n\to \mathrm{\infty}}{lim}{b}_{n}=\mathrm{\infty}$ and (1) holds. Also, in this case, the requirements (3) are obviously satisfied. We propose to study the following integral variant of ${L}_{n}$, $n\ge 1$, operators, while meeting the assumption (3).

$$({D}_{n}f)(x)={a}_{0}({\lambda}_{n};x)f(0)+\sum _{k=1}^{\mathrm{\infty}}\frac{{a}_{k}({\lambda}_{n};x)}{B(n+1,k)}{\int}_{0}^{\mathrm{\infty}}f(n{\lambda}_{n}t)\frac{{t}^{k-1}}{{(1+t)}^{n+k+1}}\mathit{d}t,$$ | (6) |

$x\in {\mathbb{R}}_{+}$ and $B$ stands for Beta function,

$$B(p,q)={\int}_{0}^{\mathrm{\infty}}{t}^{p-1}{(1+t)}^{-(p+q)}\mathit{d}t,(p,q)\in {\mathbb{R}}_{+}^{\ast}\times {\mathbb{R}}_{+}^{\ast}.$$ |

In the above $f\in \mathcal{F}({\mathbb{R}}_{+})$, the space of all real valued functions defined on ${\mathbb{R}}_{+}$ with the property that the series from the right hand side of relation (6) is convergent. We infer that $f$ must be a Lebesgue measurable function on ${\mathbb{R}}_{+}$.

Clearly, ${D}_{n}$, $n\in \mathbb{N}$, are linear and positive operators. Moreover,

$$({D}_{n}{e}_{0})(x)=({L}_{n}{e}_{0})(x)=1,x\in {\mathbb{R}}_{+},n\ge 1,$$ | (7) |

see (3).

Using relation (2), by a straightforward calculation, we obtain the following identities

$$({D}_{n}{e}_{1})(x)=({L}_{n}{e}_{1})(x),x\in {\mathbb{R}}_{+},n\ge 1,$$ | (8) |

$$({D}_{n}{e}_{2})(x)=\frac{n}{n-1}({L}_{n}{e}_{2})(x)+\frac{n{\lambda}_{n}}{n-1}({L}_{n}{e}_{1})(x),x\in {\mathbb{R}}_{+},n\ge 2.$$ | (9) |

Currently, in approximation by linear positive operators, particular attention is paid to those which reproduce affine functions, property implied by the relations ${L}_{n}{e}_{j}={e}_{j}$, $j\in \{0,1\}$. From (7) and (8) we notice that if ${L}_{n}$ operators have this property, also ${D}_{n}$ operators enjoy it.

We specify that particular cases of operators described by formula (6) have been investigated by various authors. To illustrate this, we present two examples.

Starting from Jain operators [6] defined as follows

$$({P}_{n}^{[\beta ]}f)(x)={e}^{-nx}+nx\sum _{k=1}^{\mathrm{\infty}}\frac{{(nx+k\beta )}^{k-1}}{k!}{e}^{-(nx+k\beta )}f\left(\frac{k}{n}\right),x\ge 0,$$ | (10) |

where the parameter $\beta \in [0,1)$, in [13] an integral version with Beta bases functions has been investigated. If in (10) we choose $\beta =0$, ${P}_{n}^{[0]}$, $n\in \mathbb{N}$, turn into well known Szász-Mirakjan operators. An integral extension of ${P}_{n}^{[0]}$ operators following the model shown in (7), was achieved in [5].

The second example is taken from the recent paper [7]. Acar and Kumar introduced and studied the following generalization of Baskakov-Durrmeyer operators in the sense of D.D. Stancu

$$({L}_{n,a}^{\alpha ,\beta}f)(x)=\sum _{k=1}^{\mathrm{\infty}}\frac{{W}_{n,k}^{a}(x)}{B(k,n)}{\int}_{0}^{\mathrm{\infty}}\frac{{t}^{k-1}}{{(1+t)}^{n+k}}f\left(\frac{nt+\alpha}{n+\beta}\right)\mathit{d}t+{W}_{n,0}^{a}(x)f\left(\frac{\alpha}{n+\beta}\right),$$ |

where

$${W}_{n,k}^{a}(x)={e}^{-\frac{ax}{1+x}}\frac{{P}_{k}(n,a)}{k!}\frac{{x}^{k}}{{(1+x)}^{n+k}},{P}_{k}(n,a)=\sum _{i=0}^{k}\left(\genfrac{}{}{0pt}{}{k}{i}\right){(n)}_{i}{a}^{k-i},$$ |

$0\le \alpha \le \beta $, $a>0$ and ${(n)}_{i}$ is the Pochammer symbol given by ${(n)}_{0}=1$, ${(n)}_{i}=n(n+1)\mathrm{\dots}(n+i-1)$ for $i\in \mathbb{N}$.

At this point we introduce the $j$-th central moment of ${D}_{n}$ operators, $n\ge j\in \mathbb{N}$, i.e.,

$${\mathcal{M}}_{j}({D}_{n};x)=({D}_{n}{\phi}_{x}^{j})(x),\text{where}{\phi}_{x}(t)=t-x,(t,x)\in {\mathbb{R}}_{+}\times {\mathbb{R}}_{+}.$$ |

Examining identities (7)-(9) and taking into account (4), we get the first two central moments

$${\mathcal{M}}_{j}({D}_{n};x)=\sum _{l=0}^{j}{\beta}_{l,j}(n){x}^{j},x\ge 0,n\ge 2,j\in \{1,2\},$$ |

where

$${\beta}_{0,1}(n)={\alpha}_{0,1}(n),{\beta}_{1,1}(n)={\alpha}_{1,1}(n)-1,$$ | (11) |

$$\{\begin{array}{ccc}{\beta}_{0,2}(n)=\frac{n}{n-1}({\alpha}_{0,2}(n)+{\lambda}_{n}{\alpha}_{0,1}(n)),\hfill & & \\ {\beta}_{1,2}(n)=\frac{n}{n-1}({\alpha}_{1,2}(n)+{\lambda}_{n}{\alpha}_{1,1}(n))-2{\alpha}_{0,1}(n),\hfill & & \\ {\beta}_{2,2}(n)=\frac{n}{n-1}{\alpha}_{2,2}(n)-2{\alpha}_{1,1}(n)+1.\hfill & & \end{array}$$ | (12) |

Considering the notation $c(n)=\underset{0\le l\le 2}{\mathrm{max}}|{\beta}_{l,2}(n)|$, $n\ge 2$, we can write

$${\mathcal{M}}_{2}({D}_{n};x)\le c(n)(1+x+{x}^{2}),x\ge 0.$$ | (13) |

Taking in view (1) and (5), $\underset{n\to \mathrm{\infty}}{lim}c(n)=0$ takes place.

## 3. Approximation properties

Our goal is to show how the approximation properties are transferred from ${L}_{n}$ to ${D}_{n}$, $n\in \mathbb{N}$, operators. As usual, $B({\mathbb{R}}_{+})$ and $C({\mathbb{R}}_{+})$ stand for bounded, respectively continuous real valued functions defined on ${\mathbb{R}}_{+}$. Set ${C}_{B}({\mathbb{R}}_{+})=B({\mathbb{R}}_{+})\cap C({\mathbb{R}}_{+})$ which is endowed with the norm of the uniform convergence (sup-norm) denoted by $\parallel \cdot {\parallel}_{\mathrm{\infty}}$.

Remark 1. Clearly, ${C}_{B}({\mathbb{R}}_{+})\subset \mathcal{F}({\mathbb{R}}_{+})$. Since (7) takes place, for any function $f\in {C}_{B}({\mathbb{R}}_{+})$ we deduce ${\Vert {D}_{n}f\Vert}_{\mathrm{\infty}}\le {\Vert f\Vert}_{\mathrm{\infty}}$, in other words, the operators ${D}_{n}$, $n\in \mathbb{N}$, are non-expansive.

We investigate the summation integral operators in functions spaces characterized by the polynomial weight $\rho (x)=1+{x}^{2}$, $x\ge 0$. We indicate the following notations used throughout the paper.

$${B}_{\rho}({\mathbb{R}}_{+})=\{f:{\mathbb{R}}_{+}\to \mathbb{R}\mid |f(x)|\le {M}_{f}\rho (x),x\ge 0\},$$ |

where ${M}_{f}$ is a constant depending only on $f$. ${B}_{\rho}({\mathbb{R}}_{+})$ is a normed space with respect to the norm $\parallel \cdot {\parallel}_{\rho}$,

$${\Vert f\Vert}_{\rho}=\underset{x\ge 0}{sup}\frac{|f(x)|}{\rho (x)}.$$ |

Further, we denote by ${C}_{\rho}({\mathbb{R}}_{+})$, ${C}_{\rho}^{\ast}({\mathbb{R}}_{+})$ the following subspaces of ${B}_{\rho}({\mathbb{R}}_{+})$

${C}_{\rho}({\mathbb{R}}_{+})=\{f\in {B}_{\rho}({\mathbb{R}}_{+}):f\text{is continuous on}{\mathbb{R}}_{+}\},$ | ||

${C}_{\rho}^{\ast}({\mathbb{R}}_{+})=\{f\in {C}_{\rho}({\mathbb{R}}_{+}):\underset{x\to \mathrm{\infty}}{lim}{\displaystyle \frac{f(x)}{\rho (x)}}\text{exists and is finite}\},$ |

respectively.

Theorem 1. Let ${L}_{n}$, $n\in \mathbb{N}$, be defined by (2) such that conditions (3)-(5) are fulfilled. For any compact $K\subset {\mathbb{R}}_{+}$, the operators ${D}_{n}$ defined by (6) possess the property

$$\underset{n\to \mathrm{\infty}}{lim}{\Vert {D}_{n}f-f\Vert}_{K}=0,f\in {C}_{\rho}^{\ast}({\mathbb{R}}_{+}),$$ | (14) |

where, for a function $h$, ${\Vert h\Vert}_{K}=\underset{t\in K}{sup}|h(t)|$.

Proof. We consider the lattice homomorphism ${T}_{K}:C(\mathbb{R})\to C(K)$ defined by ${T}_{K}(f)={f|}_{K}$. Relations (7), (8), (9) and the requirements formulated in the hypothesis of the theorem ensure the identities

$$\underset{n\to \mathrm{\infty}}{lim}{T}_{K}({D}_{n}{e}_{j})={T}_{K}({e}_{j}),j\in \{0,1,2\},$$ |

uniformly on $K$. Since ${C}_{\rho}^{\ast}({\mathbb{R}}_{+})$ is isomorphic to $C(K)$ and $\{{e}_{0},{e}_{1},{e}_{2}\}$ is a Korovkin set in ${C}_{\rho}^{\ast}({\mathbb{R}}_{+})$, the universal Korovkin-type criterion, see [1, Theorem 4.1.4(vi)] implies

$$({D}_{n}f)(x)\to f(x)\text{uniformly on}K\text{as}n\to \mathrm{\infty}.$$ |

Consequently, the proof of (14) is completed. $\mathrm{\square}$

We recall the notion of modulus of smoothness associated to a function $f\in C({\mathbb{R}}_{+})$ on the compact interval $[0,a]$, $a>0$ fixed. Denoting by $\omega {(f;\cdot )}_{[0,a]}$, it is defined as follows

$$\omega {(f;\delta )}_{[0,a]}=sup\{|f(t)-f(x)|:|t-x|\le \delta ,(t,x)\in [0,a]\times [0,a]\},$$ |

where $\delta \ge 0$. As we can see, $\omega $ gives the maximum oscillation of $f$ in any interval of length not exceeding $\delta $.

Among the properties of $\omega {(f;\cdot )}_{[0,a]}$ we recall the following

$$\omega {(f;|t-x|)}_{[0,a]}\le \left(1+\frac{{(t-x)}^{2}}{{\delta}^{2}}\right)\omega {(f;\delta )}_{[0,a]},\delta >0,$$ | (15) |

and $(t,x)\in [0,a]\times [0,a]$, see, e.g., [8].

Using this modulus we establish the rate of convergence of the sequence ${({D}_{n}f)}_{n\ge 1}$ in a specific case, as shown in the following

Theorem 2. Let ${L}_{n}$, $n\in \mathbb{N}$, be defined by (2) such that conditions (3)-(5) are fulfilled. For any compact $[0,b]$, $b>0$ fixed, the operators ${D}_{n}$, $n\ge 2$, defined by (6) verify the following inequality

$$|({D}_{n}f)(x)-f(x)|\le {M}_{f,b}{\delta}_{n}^{2}(x)+2\omega {(f;{\delta}_{n}(x))}_{[0,b+1]},x\in [0,b],$$ | (16) |

where $f\in {C}_{\rho}({\mathbb{R}}_{+})$, ${M}_{f,b}$ is a constant depending on $f$ and $b$, and

$${\delta}_{n}(x)=\sqrt{c(n)({x}^{2}+x+1)}.$$ | (17) |

Proof. Let $x\in [0,b]$ arbitrarily fixed. Let $t$ belong to ${\mathbb{R}}_{+}$.

If $t\le b+1$, then $|t-x|\le b+1$. With the help of (15) we can write

$|f(t)-f(x)|$ | $\le \underset{|u-v|\le |t-x|}{sup}|f(u)-f(v)|=\omega {(f;|t-x|)}_{[0,b+1]}$ | |||

$\le \left(1+{\displaystyle \frac{{(t-x)}^{2}}{{\delta}^{2}}}\right)\omega {(f;\delta )}_{[0,b+1]},\delta >0.$ | (18) |

If $t>b+1$, then $t-x>1$. Since $f\in {C}_{\rho}({\mathbb{R}}_{+})$, we get

$|f(t)-f(x)|$ | $\le {M}_{f}(2+{x}^{2}+{t}^{2})$ | |||

$={M}_{f}((2+2{x}^{2})+2x(t-x)+{(t-x)}^{2})$ | ||||

$\le {M}_{f}((2+2{x}^{2}){(t-x)}^{2}+2x{(t-x)}^{2}+{(t-x)}^{2})$ | ||||

$\le {M}_{f}\left(\underset{x\in [0,b]}{sup}(3+2{x}^{2}+2x)\right){(t-x)}^{2}$ | ||||

$={M}_{f,b}{(t-x)}^{2},$ | (19) |

where ${M}_{f,b}$ is a constant depending only on $f$ and $b$.

Combining (3) and (3), we obtain

$$|f(t)-f(x)|\le {M}_{f,b}{(t-x)}^{2}+\left(1+\frac{{(t-x)}^{2}}{{\delta}^{2}}\right)\omega {(f,\delta )}_{[0,b+1]},\delta >0.$$ | (20) |

Since ${D}_{n}$ is linear positive operator, it is monotone. Identity ${D}_{n}{e}_{0}={e}_{0}$ and inequalities (20), (13) allow us to write successively

$|({D}_{n}f)(x)-f(x)|$ | $=|{D}_{n}(f-f(x);x)|\le {D}_{n}(|f-f(x)|;x)$ | ||

$\le {D}_{n}({M}_{f,b}{\phi}_{x}^{2}+\left({e}_{0}+{\displaystyle \frac{{\phi}_{x}^{2}}{{\delta}^{2}}}\right)\omega {(f;\delta )}_{[0,b+1]};x)$ | |||

$\le {M}_{f,b}c(n)({x}^{2}+x+1)(1+{\displaystyle \frac{c(n)({x}^{2}+x+1)}{{\delta}^{2}}})\omega {(f;\delta )}_{[0,b+1]}.$ |

Choosing $\delta ={\delta}_{n}(x)$, see (17), we arrive at (16) and the proof is ended. $\mathrm{\square}$

In the following we will denote by $C$ a positive constant which occurs different values in different relations, its main characteristic being that it does not depend on $n$.

Due to (5), all sequences from relation (12), ${({b}_{i,j}(n))}_{n\ge 1}$, $0\le i\le j\le 2$, are bounded with respect to $n$. Consequently, $0\le {L}_{n}\rho \le C\rho $. This relation and the definition of ${D}_{n}$ operators imply $0\le {D}_{n}\rho \le C\rho $. Taking into account the result established in [3, Eq. (4)], the previous inequality represents the necessary and sufficient condition which allows us to draw the following

Remark 2. Each operator ${D}_{n}$ maps ${C}_{\rho}({\mathbb{R}}_{+})$ into ${B}_{\rho}({\mathbb{R}}_{+})$ and

$${\Vert {D}_{n}\Vert}_{{C}_{\rho}({\mathbb{R}}_{+})\to {B}_{\rho}({\mathbb{R}}_{+})}={\Vert {D}_{n}\rho \Vert}_{\rho}.$$ |

Theorem 3. Let ${L}_{n}$, $n\in \mathbb{N}$, be defined by (2) such that conditions (3)-(5) are fulfilled. The operators ${D}_{n}$, $n\ge 2$, defined by (6) satisfy the following property

$$\underset{n\to \mathrm{\infty}}{lim}{\Vert {D}_{n}f-f\Vert}_{\rho}=0$$ | (21) |

for every function $f\in {C}_{\rho}^{\ast}({\mathbb{R}}_{+})$.

Proof. Considering the statement drawn in Remark 2, we will apply a result set by Gadjiev [3, Theorem 2]. According to this result, in order to obtain (21), it is sufficient to prove the following relations

$$\underset{n\to \mathrm{\infty}}{lim}{\Vert {D}_{n}{e}_{1}^{k}-{e}_{1}^{k}\Vert}_{\rho}=0,k=0,1,2.$$ | (22) |

We were based on the fact that in our case $\rho ={e}_{0}+{e}_{1}^{2}$.

Using (8) and (11), we can write

${\Vert {D}_{n}{e}_{1}-{e}_{1}\Vert}_{\rho}$ | $=\underset{x\ge 0}{sup}{\displaystyle \frac{|({L}_{n}{e}_{1})(x)-x|}{1+{x}^{2}}}=\underset{x\ge 0}{sup}{\displaystyle \frac{|{\alpha}_{0,1}(n)+({\alpha}_{1,1}(n)-1)x|}{1+{x}^{2}}}$ | ||

$\le |{\alpha}_{0,1}(n)|+|{\alpha}_{1,1}(n)-1|.$ |

Then, by calling (9) and (12), we have

$${\Vert {D}_{n}{e}_{2}-{e}_{2}\Vert}_{\rho}=\underset{x\ge 0}{sup}\frac{\left|\frac{n}{n-1}({\alpha}_{0,2}(n)+{\alpha}_{1,2}(n)x+{\alpha}_{2,2}(n){x}^{2})+\frac{n{\lambda}_{n}}{n-1}({\alpha}_{0,1}(n)+{\alpha}_{1,1}(n)x)-{x}^{2}\right|}{1+{x}^{2}}$$ |

$$\le \left|\frac{n}{n-1}{\alpha}_{2,2}(n)-1\right|+\frac{n}{n-1}|{\alpha}_{1,2}(n)+{\lambda}_{n}{\alpha}_{1,1}(n)|+\frac{n}{n-1}|{\alpha}_{0,2}(n)+{\lambda}_{n}{\alpha}_{1,1}(n)|.$$ |

Relations (1) and (5) guarantee the veracity of the identity (22) in the case $k=2$. Thus, the limit from (21) is completely proved. $\mathrm{\square}$

Since ${(1+{x}^{2})}^{1+\gamma}\ge \rho (x)$, $(x,\gamma )\in {\mathbb{R}}_{+}\times {\mathbb{R}}_{+}^{\ast}$, from Theorem 3 we enunciate

Remark 3. For any $f\in {C}_{\rho}^{\ast}({\mathbb{R}}_{+})$ and $\gamma >0$,

$$\underset{n\to \mathrm{\infty}}{lim}\underset{x\ge 0}{sup}\frac{|({D}_{n}f)(x)-f(x)|}{{(1+{x}^{2})}^{1+\gamma}}=0$$ |

takes place. In the particular case of Baskakov-Durrmeyer-Stancu type operators, the above relation was included in [7, Theorem 3.4].

Finally we present an estimate of error of approximation by using a more general weight $\rho $ and a suitable modulus. Keeping the same notation, from this moment, we consider the weight $\rho $ given by

$$\rho (x)=1+ax+b{x}^{2},x\ge 0,$$ | (23) |

where $a\ge 1$ and $b>0$. Since $\rho (0)=1$ and $\underset{x\ge 0}{inf}{\rho}^{\prime}(x)\ge 1$, in accordance with [4], we can use the weighted modulus ${\mathrm{\Omega}}_{\rho}(f;\cdot )$ defined by

$${\mathrm{\Omega}}_{\rho}(f;\delta )=\underset{\begin{array}{c}x,t\in {\mathbb{R}}_{+}\\ |\rho (t)-\rho (x)|\le \delta \end{array}}{sup}\frac{|f(t)-f(x)|}{(|\rho (t)-\rho (x)|+1)\rho (x)},\delta >0,$$ | (24) |

for each $f\in {C}_{\rho}({\mathbb{R}}_{+})$.

Among the properties of this modulus proved by Gadjiev and Aral [4, Lemmas 4,5] we recall

$$\underset{\delta \to 0}{lim}{\mathrm{\Omega}}_{\rho}(f;\delta )=0,$$ | (25) |

$$|f(t)-f(x)|\le 2\rho (x){(1+\delta )}^{2}\left(1+\frac{{(\rho (t)-\rho (x))}^{2}}{{\delta}^{2}}\right){\mathrm{\Omega}}_{\rho}(f;\delta ),$$ | (26) |

for each $f\in {C}_{\rho}^{\ast}({\mathbb{R}}_{+})$ and $\delta >0$.

Regarding the weight $\rho $, from (23) we deduce

$$\frac{{e}_{k}}{\rho}\le 1,k\in \{1,2\},\text{and}\frac{{e}_{k}}{{\rho}^{2}}\le 1,k\in \{1,2,3,4\}.$$ | (27) |

${D}_{n}{e}_{j}$, $j\le 4$, are polynomials with coefficients expressed in terms of ${({\alpha}_{i,j})}_{0\le i\le j}$, $j\le 4$, subject to conditions (5). We assume that ${\rho}^{2}\in \mathcal{F}({\mathbb{R}}_{+})$. These facts combined with (27) allow us to establish the relations

$${\Vert {D}_{n}\rho -\rho \Vert}_{\rho}\le {\mu}_{1}(n),{\Vert {D}_{n}{\rho}^{2}-{\rho}^{2}\Vert}_{{\rho}^{2}}\le {\mu}_{2}(n),$$ | (28) |

for $n\ge 4$, where

$$\underset{n\to \mathrm{\infty}}{lim}{\mu}_{k}(n)=0,k\in \{1,2\}.$$ | (29) |

Theorem 4. Let ${L}_{n}$, $n\in \mathbb{N}$, be defined by (2) such that conditions (3)-(5) are fulfilled. The operators ${D}_{n}$, $n\ge 4$, defined by (6) satisfy the following property

$${\Vert {D}_{n}f-f\Vert}_{{\rho}^{3}}\le 4{(1+{\delta}_{n}^{\ast})}^{2}{\mathrm{\Omega}}_{\rho}(f;{\delta}_{n}^{\ast}),f\in {C}_{\rho}^{\ast}({\mathbb{R}}_{+}),$$ |

where $\rho $ is given at (23), ${\mathrm{\Omega}}_{\rho}$ is defined at (24) and

$${\delta}_{n}^{\ast}=\sqrt{{\mu}_{2}(n)+2{\mu}_{1}(n)},$$ | (30) |

in accordance with (28).

Proof. Like how we reasoned Remark 2, we deduce that ${D}_{n}$ is a mapping from ${C}_{\rho}({\mathbb{R}}_{+})$ into ${B}_{\rho}({\mathbb{R}}_{+})$ and also a mapping from ${C}_{{\rho}^{2}}({\mathbb{R}}_{+})$ into ${B}_{{\rho}^{2}}({\mathbb{R}}_{+})$. By using relations (26) and (7) we obtain the following estimate

$$\frac{|({D}_{n}f)(x)-f(x)|}{\rho (x)}\le 4{(1+\delta )}^{2}{\delta}^{-2}{D}_{n}({(\rho -\rho (x))}^{2};x){\mathrm{\Omega}}_{\rho}(f;\delta ),$$ | (31) |

where $f\in {C}_{\rho}^{\ast}({\mathbb{R}}_{+})$. Further, we can write

$${D}_{n}({(\rho -\rho (x))}^{2};x)\le |({D}_{n}{\rho}^{2})(x)-{\rho}^{2}(x)|+2\rho (x)|({D}_{n}\rho )(x)-\rho (x)|.$$ |

Using the inequalities from (28), for $n\ge 4$ and $x\ge 0$, we have

$${D}_{n}({(\rho -\rho (x))}^{2};x)\le ({\mu}_{2}(n)+2{\mu}_{1}(n)){\rho}^{2}(x).$$ |

Returning at (31) and choosing $\delta :={\delta}_{n}^{\ast}$, see (30), the conclusion of the theorem is fully motivated. $\mathrm{\square}$

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