## Abstract

The aim of this article is to present a general class of positive linear operators of discrete type related to squared fundamental basis functions. If these operators are expressed by a series, we propose to truncate them by a finite sum while still keeping the property of converging towards the identity operator in a weighted space. A relation between the local smoothness of functions and the local approximation is obtained. In the variant in which the operators are described by a finite sum, we establish the limit of their iterates.

## Authors

Octavian **Agratini**

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

## Keywords

Contraction principle; linear positive operator; Lipschitz function; modulus of smoothness; weighted space

## Paper coordinates

O. Agratini, *Properties of positive linear operators connected with squared fundamental functions, *Numer. Funct. Anal. Optimiz., **45** (2024) no. 2, pp. 103–111, https://doi.org/10.1080/01630563.2024.2316579

## About this paper

##### Journal

Numerical Functional Analysis and Optimization

##### Publisher Name

Taylor and Francis Ltd.

##### Print ISSN

15322467

##### Online ISSN

01630563

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## Paper (preprint) in HTML form

# Properties of positive linear operators connected with squared fundamental functions

###### Abstract

The aim of this article is to present a general class of positive linear operators of discrete type related to squared fundamental basis functions. If these operators are expressed by a series, we propose to truncate them by a finite sum while still keeping the property of converging towards the identity operator in a weighted space. A relation between the local smoothness of functions and the local approximation is obtained. In the variant in which the operators are described by a finite sum, we establish the limit of their iterates.

2020 Mathematics Subject Classification: 41A36, 47A10.

Keywords: Linear positive operator, weighted space, modulus of smoothness, contraction principle, Lipschitz function.

## 1 Introduction

This note falls under the field of Approximation Theory, more precisely it aims at the study of linear approximation processes, the main ingredient being their positivity. We will consider discrete type processes that are often designed as follows

$$({L}_{n}f)(x)=\sum _{k=0}^{\mathrm{\infty}}{l}_{n,k}(x)f({x}_{n,k}),x\in I,$$ | (1) |

where $I$ is a certain real interval, the fundamental functions ${l}_{n,k}:I\to {\mathbb{R}}_{+}$ are continuous on the domain and $f:I\to \mathbb{R}$ belongs to a class, say $\mathcal{F}(I)$, which guarantees that the second member in the above formula is well defined. The nodes ${x}_{n,k}$, $k\ge 0$, form a division of the interval $I$. This construction also includes the case ${l}_{n,k}=0$, $k>n$, i.e., the sum in (1) is finite.

Recently, has been an increasing interest in the study of some operators associated with those defined in (1) implying the squared fundamental functions. They have the following form

$$({\stackrel{~}{L}}_{n}f)(x)=\frac{1}{{\displaystyle \sum _{k=0}^{\mathrm{\infty}}}{l}_{n,k}^{2}(x)}\sum _{k=0}^{\mathrm{\infty}}{l}_{n,k}^{2}(x)f({x}_{n,k})$$ | (2) |

under the additional hypothesis $\sum _{k=0}^{\mathrm{\infty}}}{l}_{n,k}^{2}(x)>0$ for all $x\in I$. Clearly, both ${L}_{n}$ and ${\stackrel{~}{L}}_{n}$, $n\ge 1$, are linear positive operators. We present two examples of ${\stackrel{~}{L}}_{n}$, $n\ge 1$, type operators which are expressed by a finite sum, respectively by a series.

###### Example 1.

Choosing

$$I=[0,1],{l}_{n,k}(x)\equiv {b}_{n,k}(x)=\left(\genfrac{}{}{0pt}{}{n}{k}\right){x}^{k}{(1-x)}^{n-k},{x}_{n,k}=\frac{k}{n},$$ | (3) |

$0\le k\le n$, ${L}_{n}\equiv {B}_{n}$, $n\ge 1$, become Bernstein operators. The rational Bernstein operators ${({\stackrel{~}{L}}_{n})}_{n\ge 1}$ were investigated by Herzog [7]. Among other notable results we mention the papers of Gavrea and Ivan [5], Holhoş [8], Abel and Kushnirevych [2]. Usually, these operators apply to continuous real valued functions on the compact $[0,1]$.

###### Remark 1.

The construction indicated at (2) can be correlated with the theory of probability. To any discrete probability distribution

$$P={({p}_{\lambda})}_{\lambda \ge 0}$$ |

we can associate the index of coincidence [6],

$$IC(P)=\sum _{\lambda \ge 0}{p}_{\lambda}^{2}$$ |

which is used in the definition of entropy. For instance, if $P$ is the binomial distribution with parameters $n\in \mathbb{N}$ and $x\in [0,1]$, its index of coincidence is given by $\sum _{k=0}^{n}}{b}_{n,k}^{2}(x)$, see (3).

###### Example 2.

Set ${\mathbb{N}}_{0}=\{0\}\cup \mathbb{N}$. Choosing

$$I=[0,\mathrm{\infty}),{l}_{n,k}(x)\equiv {s}_{n,k}(x)={e}^{-nx}\frac{{(nx)}^{k}}{k!},{x}_{n,k}=\frac{k}{n},k\in {\mathbb{N}}_{0},$$ |

${L}_{n}\equiv {S}_{n}$, $n\ge 1$, represent the classical Favard-Szász-Mirakjan operators.

In this case $f$ is commonly considered from the Banach lattice ${E}_{1}$. We recall, in general the polynomial weighted spaces ${E}_{m}$ are defined as follows

$${E}_{m}={E}_{m}({\mathbb{R}}_{+})=\{f\in C({\mathbb{R}}_{+}):\underset{x\to \mathrm{\infty}}{lim}{w}_{m}(x)f(x)\text{is finite}\},m\in \mathbb{N},$$ | (4) |

${w}_{m}(x)={(1+{x}^{2m})}^{-1}$ and they are endowed with the norm $\parallel \cdot {\parallel}_{m}$,

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

These spaces are nested as follows ${C}_{B}({\mathbb{R}}_{+})\subset {E}_{m}\subset {E}_{m+1}\subset C({\mathbb{R}}_{+})$, $m\in \mathbb{N}$, where ${C}_{B}({\mathbb{R}}_{+})$ indicates the space of continuous and bounded real valued functions defined on ${\mathbb{R}}_{+}=[0,\mathrm{\infty})$.

Results related to squared fundamental functions ${s}_{n,k}$, $k\ge 0$, can be found in [1, Section 5.1]. Moreover, in Abel’s paper are obtained Voronovskaja theorems for the general class defined by (2).

The purpose of this note is to highlight some properties of ${\stackrel{~}{L}}_{n}$, $n\in \mathbb{N}$, operators. Due to the fact that their expression is very general, we admit that the results cannot be spectacular. This is the reason why we consider additional assumptions on the operators.

The first approach assumed to modify the series into finite sums and to investigate their approximation properties in weighted spaces of continuous functions defined on ${\mathbb{R}}_{+}$. More exactly, the operators are truncated fading away their tails and sufficient conditions are provided to ensure their convergence to the identity operator. A study for some bivariate classes of discrete operators of type (1) defined following the truncated process is developed, e.g., in [4]. We also establish a relation between the local smoothness of functions and the local approximation.

The second approach involves the operators acting on the space $C([0,1])$ and we determine the convergence of their iterates.

## 2 Results

It should be noted that a property of ${\stackrel{~}{L}}_{n}$, $n\ge 1$, operators is to reproduce the constants, more precisely ${\stackrel{~}{L}}_{n}{e}_{0}={e}_{0}$, where ${e}_{0}(x)=1$, $x\in I$.

Such operators are called Markov type operators. The study of the operators defined by (2) requires an estimation of infinite sum which in a certain sense restricts the operators usefulness from computational point of view. In this respect, in order to approximate a function $f$, it is interesting and useful to consider partial sums of ${\stackrel{~}{L}}_{n}f$ which have only finite numbers of terms depending upon $n$ and $x$. The classes of operators defined by (2) have a very general form. In order to obtain tangible results we will particularize these classes. The first specifications concern the interval $I$ and the network of nodes ${({x}_{n,k})}_{k\ge 0}$. We consider

$$I={\mathbb{R}}_{+},{x}_{n,k}=\frac{k}{{a}_{n}}\le k,k\in {\mathbb{N}}_{0},\text{with}\underset{n\to \mathrm{\infty}}{lim}{a}_{n}=\mathrm{\infty},$$ | (5) |

where ${({a}_{n})}_{n\ge 1}$ is a positive strictly increasing sequence. Set

$${\psi}_{n}(x)=\sum _{k=0}^{\mathrm{\infty}}{l}_{n,k}^{2}(x),x\ge 0.$$ | (6) |

Also, ${\tau}_{n,r}$, $r\in {\mathbb{N}}_{0}$, stands for the $r$-th central moment of ${\stackrel{~}{L}}_{n}$, i.e.,

$${\tau}_{n,r}(x)={\stackrel{~}{L}}_{n}({(\cdot -x)}^{r},x).$$ |

Starting from (2), under the additional assumptions (5), we define

$$({\stackrel{~}{L}}_{n,\delta}f)(x)=\frac{1}{{\psi}_{n}(x)}\sum _{k=0}^{[{a}_{n}(x+\delta (n))]}{l}_{n,k}^{2}(x)f\left(\frac{k}{{a}_{n}}\right),x\ge 0,f\in \mathcal{F}({\mathbb{R}}_{+}),$$ | (7) |

where $\delta ={(\delta (n))}_{n\ge 1}$ is a certain sequence of positive numbers and ${\psi}_{n}$ was defined at (6). Here $[\alpha ]$ indicates the largest integer not exceeding $\alpha $. The study of these operators will be developed in ${E}_{m}$ spaces defined by (4).

###### Remark 2.

In what follows we set ${e}_{j}$, $j\in \mathbb{N}$, for the $j$-th monomial, ${e}_{j}(t)={t}^{j}$, $t\ge 0$. The first result represents the answer to the question whether ${({\stackrel{~}{L}}_{n,\delta})}_{n\ge 1}$ can inherit the property of the approximation process of the original sequence ${({\stackrel{~}{L}}_{n})}_{n\ge 1}$. We mention that the conditions required by us ensure only sufficiency.

###### Theorem 1.

Let ${\stackrel{~}{L}}_{n}$, $n\in \mathbb{N}$, be defined by (2) subject to conditions (5). Let ${\stackrel{~}{L}}_{n,\delta}$, $n\in \mathbb{N}$, be defined by (7). We suppose that $\underset{n\to \mathrm{\infty}}{lim}\sqrt{{a}_{n}}\delta (n)=\mathrm{\infty}$ and the central moment ${\tau}_{n,2m}$ satisfies

$${\tau}_{n,2m}(x)\le C{a}_{n}^{-m},$$ | (8) |

where $C$ is a constant depending only on $m$. Let $f\in {E}_{m}\cap \mathcal{F}({\mathbb{R}}_{+})$. If ${({\stackrel{~}{L}}_{n}f)}_{n\ge 1}$ converges pointwise to $f$, then ${({\stackrel{~}{L}}_{n,\delta}f)}_{n\ge 1}$ enjoys the same property.

###### Proof.

Since $\underset{\lambda \ge 0}{\mathrm{min}}({\lambda}^{2m}+{(1-\lambda )}^{2m})={2}^{1-2m}$, $m\in \mathbb{N}$, the following elementary inequality

$${t}^{2m}\le {2}^{2m-1}({x}^{2m}+{(t-x)}^{2m}),t\ge 0,x\ge 0,m\in \mathbb{N},$$ | (9) |

holds. On the other hand, for every $f\in {E}_{m}$, the positive constants ${a}_{f}$, ${b}_{f}$ exist such that $|f|\le {a}_{f}+{b}_{f}{e}_{2m}$ and, consequently, by using (9) we get

$$|f(t)|\le {g}_{m}(x)+{2}^{2m-1}{b}_{f}{(t-x)}^{2m},$$ |

where ${g}_{m}={a}_{f}+{2}^{2m-1}{b}_{f}{e}_{2m}$. This implies

$$\left|f\left(\frac{k}{{a}_{n}}\right)\right|\le {g}_{m}(x)+{2}^{2m-1}{b}_{f}{\left(\frac{k}{{a}_{n}}-x\right)}^{2m},k\in {\mathbb{N}}_{0},x\ge 0.$$ | (10) |

Because $x$, $\delta (n)$, ${a}_{n}$ are positive quantities, if $k\ge [{a}_{n}(x+\delta (n))]+1$, then

$$\{k\in {\mathbb{N}}_{0}:k\ge [{a}_{n}(x+\delta (n))]+1\}\subset \{k\in {\mathbb{N}}_{0}:\left|\frac{k}{{a}_{n}}-x\right|>\delta (n)\}:={I}_{n,x,\delta}.$$ | (11) |

Setting ${R}_{n}:={\stackrel{~}{L}}_{n}-{\stackrel{~}{L}}_{n,\delta}$, we get

$$|({\stackrel{~}{L}}_{n,\delta}f)(x)-f(x)|\le |({R}_{n}f)(x)|+|({\stackrel{~}{L}}_{n}f)(x)-f(x)|.$$ | (12) |

Taking into account both (11) and (10), we can write

$|({R}_{n}f)(x)|$ | $\le {\displaystyle \sum _{k\in {I}_{n,x,\delta}}}{\displaystyle \frac{{l}_{n,k}^{2}(x)}{{\psi}_{n}(x)}}f\left({\displaystyle \frac{k}{{a}_{n}}}\right)$ | |||

$\le {\displaystyle \sum _{k\in {I}_{n,x,\delta}}}{\displaystyle \frac{{l}_{n,k}^{2}(x)}{{\psi}_{n}(x)}}{g}_{m}(x)+{2}^{2m-1}{b}_{f}{\displaystyle \sum _{k\in {I}_{n,k,\delta}}}{\displaystyle \frac{{l}_{n,k}^{2}(x)}{{\psi}_{n}(x)}}{\left({\displaystyle \frac{k}{{a}_{n}}}-x\right)}^{2m}$ | ||||

$:={\displaystyle {\sum}_{1}}+{\displaystyle {\sum}_{2}}.$ | (13) |

Concerning the sums $\sum}_{1$ and $\sum}_{2$ we have

$\sum}_{1$ | $\le {g}_{m}(x){\delta}^{-2m}(n){\displaystyle \sum _{k=0}^{\mathrm{\infty}}}{\displaystyle \frac{{l}_{n,k}^{2}(x)}{{\psi}_{n}(x)}}{\left({\displaystyle \frac{k}{{a}_{n}}}-x\right)}^{2m}$ | ||

$={g}_{m}(x){\delta}^{-2m}(n){\tau}_{n,2m}(x),$ |

respectively

$${\sum}_{2}\le {2}^{2m-1}{b}_{f}{\tau}_{n,2m}(x),x\ge 0.$$ |

Returning to (2) and using both (8) and our hypotheses

$$\underset{n\to \mathrm{\infty}}{lim}\sqrt{{a}_{n}}\delta (n)=\mathrm{\infty},$$ |

we obtain

$$|({R}_{n}f)(x)|\le ({g}_{m}(x){\delta}^{-2m}(n)+{2}^{2m-1}{b}_{f})\frac{C}{{a}_{n}^{m}}=o(1)(n\to \mathrm{\infty}).$$ |

Using (12), the above relation along with the hypothesis of the theorem completes the proof of the statement. ∎

Next, our result concerns the operators defined by (2) in the version that the sum is finite. This time a compact interval $I$ will be chosen. Since any compact $[a,b]$ is isomorphic to $[0,1]$, in what follows we use only the compact $[0,1]$. For this sequence is of interest to investigate the sequences of iterates ${({\stackrel{~}{L}}_{n}^{j})}_{n\ge 1}$, where ${\stackrel{~}{L}}_{n}^{0}=I$, the identity operator of the space $C([0,1])$ and

$${\stackrel{~}{L}}_{n}^{1}={\stackrel{~}{L}}_{n},{\stackrel{~}{L}}_{n}^{j}={\stackrel{~}{L}}_{n}({\stackrel{~}{L}}_{n}^{j-1}),j>1,$$ |

assuming that $j$ does not depend on $n$. In our study we maintain the general net ${\mathrm{\Delta}}_{n}$ on $[0,1]$,

$$ |

Our operators have therefore the following form

$$({\stackrel{~}{L}}_{n}f)(x)=\sum _{k=0}^{n}\frac{{l}_{n,k}^{2}(x)}{{\displaystyle \sum _{k=0}^{n}}{l}_{n,k}^{2}(x)}f({x}_{n,k}),$$ | (14) |

$x\in [0,1]$, $f\in C([0,1])$. In addition to the fulfilled condition ${\stackrel{~}{L}}_{n}{e}_{0}={e}_{0}$, we add the following working hypothesis

$$({\stackrel{~}{L}}_{n}{e}_{1})(x)=x,x\in [0,1].$$ | (15) |

Consequently, the degree of exactness of ${\stackrel{~}{L}}_{n}$, $n\ge 1$, has the value 1, which means that they preserve the affine functions. In other words, for any real constant $c$, ${e}_{0}+c{e}_{1}$ is a fixed point for ${\stackrel{~}{L}}_{n}$, $n\ge 1$.

###### Remark 3.

By virtue of the classical result obtained by Shisha and Mond [10], regarding the error of approximation one has

$$|({\stackrel{~}{L}}_{n}f)(x)-f(x)|\le 2{\omega}_{1}(f,{\delta}_{n}(x)),x\in [0,1],$$ | (16) |

where

$${\delta}_{n}(x)={\left(\frac{{\displaystyle \sum _{k=0}^{n}}{l}_{n,k}^{2}(x){x}_{n,k}^{2}}{{\displaystyle \sum _{k=0}^{n}}{l}_{n,k}^{2}(x)}-{x}^{2}\right)}^{1/2}$$ | (17) |

and ${\omega}_{1}(f,\cdot )$ represents the modulus of continuity of $f$ defined by

$${\omega}_{1}(f,\delta )=\underset{\begin{array}{c}{x}_{1},{x}_{2}\in [0,1]\\ |{x}_{1}-{x}_{2}|\le \delta \end{array}}{sup}|f({x}_{1})-f({x}_{2})|,\delta \ge 0.$$ |

For any function $h\in C([0,1])$, ${\omega}_{1}$ enjoys the property $\underset{\delta \to {0}^{+}}{lim}\omega (f,\delta )=0$. Consequently, if we have $\underset{n\to \mathrm{\infty}}{lim}{\delta}_{n}(x)=0$, see (17), then

$$\underset{n\to \mathrm{\infty}}{lim}{\stackrel{~}{L}}_{n}f=f\text{uniformly on}[0,1],$$ |

for any $f\in C([0,1])$.

Considering that ${\stackrel{~}{L}}_{n}$, $n\ge 1$, are interpolating operators at the ends of the interval $[0,1]$, we define

$${\mathrm{\Gamma}}_{\alpha ,\beta}=\{f\in C([0,1]):f(0)=\alpha ,f(1)=\beta \},(\alpha ,\beta )\in \mathbb{R}\times \mathbb{R}.$$ |

The system ${({\mathrm{\Gamma}}_{\alpha ,\beta})}_{(\alpha ,\beta )\in \mathbb{R}\times \mathbb{R}}$ makes up a partition of the space $C([0,1])$ and the interpolation operators of our operators guarantees that ${\mathrm{\Gamma}}_{\alpha ,\beta}$ is an invariant subset of every operator ${\stackrel{~}{L}}_{n}$. Assume that

$${\lambda}_{n}=\underset{x\in [0,1]}{\mathrm{min}}({l}_{n,0}^{2}(x)+{l}_{n,n}^{2}(x))>0.$$ | (18) |

According to [3, Theorem 4] one has ${{\stackrel{~}{L}}_{n}|}_{{\mathrm{\Gamma}}_{\alpha ,\beta}}:{\mathrm{\Gamma}}_{\alpha ,\beta}\to {\mathrm{\Gamma}}_{\alpha ,\beta}$ is a contraction for every $(\alpha ,\beta )\in \mathbb{R}\times \mathbb{R}$ and $n\ge 1$. In addition

$$\Vert {\stackrel{~}{L}}_{n}f-{\stackrel{~}{L}}_{n}g\Vert \le (1-{\lambda}_{n})\Vert f-g\Vert ,f,g\in {\mathrm{\Gamma}}_{\alpha ,\beta},$$ |

where $\parallel \cdot \parallel $ stands for the uniform norm assigns to the space $C([0,1])$.

The above results imply the following statement.

###### Theorem 2.

Let ${\stackrel{~}{L}}_{n}$ be defined by (14) such that (15) and (18) hold. For each $n\in \mathbb{N}$, the iterates sequence ${({\stackrel{~}{L}}_{n}^{j})}_{j\ge 1}$ satisfies

$$\underset{j\to \mathrm{\infty}}{lim}({\stackrel{~}{L}}_{n}^{j}f)(x)=f(0)+(f(1)-f(0))x\text{uniformly on}[0,1]$$ | (19) |

for every $f\in C([0,1])$.

Notice that the affine function ${f}^{\ast}=f(0)+(f(1)-f(0)){e}_{1}$ is a fixed point in ${\mathrm{\Gamma}}_{f(0),f(1)}$ of any operator ${\stackrel{~}{L}}_{n}$ and thus (19) is nothing else then the conclusion of Banach’s fixed point theorem applied in the mentioned space ${\mathrm{\Gamma}}_{f(0),f(1)}$ regarding the limit of the sequence of successive approximations of the contraction ${{\stackrel{~}{L}}_{n}|}_{{X}_{f(0),f(1)}}$.

Knowing that $(C([0,1]),d)$ is a complete metric space,

$$d(f,g)=\Vert f-g\Vert ,$$ |

we establish upper limits of approximation error between ${\stackrel{~}{L}}_{n}^{j}f$ and ${f}^{\ast}$.

###### Theorem 3.

Let ${\stackrel{~}{L}}_{n}$ be defined by (14) such that (15) and (18) take place. For each $n\in \mathbb{N}$, the iterates of ${\stackrel{~}{L}}_{n}$ satisfy the following relation

$$\Vert {\stackrel{~}{L}}_{n}^{j}f-{f}^{\ast}\Vert \le \frac{2{(1-{\lambda}_{n})}^{j}}{{\lambda}_{n}}\underset{x\in [0,1]}{sup}{\omega}_{1}(f,{\delta}_{n}(x)),f\in C([0,1]),j\ge 1,$$ |

where ${\delta}_{n}(x)$ is given by (17).

###### Proof.

Let $n\in \mathbb{N}$ and $f\in C([0,1])$ be arbitrarily chosen. We use the known inequality describing the speed of convergence

$$d({x}_{j},{x}^{\ast})\le \frac{{q}^{j}}{1-q}d({x}_{1},{x}_{0}),j\ge 1,$$ | (20) |

valid in any complete metric space, where ${x}_{j}=\mathrm{\Lambda}({x}_{j-1})$, $j\ge 1$, $\mathrm{\Lambda}$ is a contraction and $q\in (0,1)$ is a Lipschitz constant of $\mathrm{\Lambda}$. In our case we assign the following values

$${x}_{0}=f,{x}_{j}={\stackrel{~}{L}}_{n}^{j}f,{x}^{\ast}={f}^{\ast}.$$ |

Then (20) reads as follows

$$\Vert {\stackrel{~}{L}}_{n}^{j}f-{f}^{\ast}\Vert \le \frac{{(1-{\lambda}_{n})}^{j}}{{\lambda}_{n}}\Vert {\stackrel{~}{L}}_{n}f-f\Vert .$$ |

By using (16) one gets

$$\Vert {\stackrel{~}{L}}_{n}f-f\Vert \le 2\underset{x\in [0,1]}{sup}{\omega}_{1}(f,{\delta}_{n}(x)).$$ |

Combining the previous two inequalities, the statement of the theorem is proved. ∎

In the final part, we expand the domain of the definition of the functions on which the operators from (14) act, returning to the interval $I={\mathbb{R}}_{+}$. Our aim is to prove a relation between the local smoothness of functions and the local approximation. We recall that a continuous function $f:{\mathbb{R}}_{+}\to \mathbb{R}$ is locally $Lip\alpha $ on $E$ ($$, $E\subset {\mathbb{R}}_{+}$) if it satisfies the condition

$$|f(x)-f(y)|\le {M}_{f}{|x-y|}^{\alpha},(x,y)\in {\mathbb{R}}_{+}\times E,$$ | (21) |

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

Also, the distance between $x\in {\mathbb{R}}_{+}$ and $E$ denoted by $d(x,E)$ is given by

$$d(x,E)=inf\{|x-y|:y\in E\}.$$ |

###### Theorem 4.

Let ${\stackrel{~}{L}}_{n}$, $n\in \mathbb{N}$, be defined by (14) where $x\in {\mathbb{R}}_{+}$ and $f\in C({\mathbb{R}}_{+})$. If $f$ is locally $Lip\alpha $ on $E\subset {\mathbb{R}}_{+}$, then

$$|({\stackrel{~}{L}}_{n}f)(x)-f(x)|\le {M}_{f}({\tau}_{n,2}^{\alpha /2}(x)+2{d}^{\alpha}(x,E)),x\ge 0$$ |

takes place.

###### Proof.

In the relation $1/r+1/s=1$ ($r>0$, $s>0$) that characterizes Hölder’s inequality, choosing $r=2{\alpha}^{-1}$ and knowing that ${\stackrel{~}{L}}_{n}{e}_{0}={e}_{0}$, we get

$${\stackrel{~}{L}}_{n}({h}^{\alpha};x)\le {({\stackrel{~}{L}}_{n}({h}^{2};x))}^{\alpha /2},x\ge 0,$$ |

where $\alpha \in (0,1]$ and $h\ge 0$. Taking $h=|{e}_{1}-x{e}_{0}|$, we can write

$${\stackrel{~}{L}}_{n}({|{e}_{1}-x{e}_{0}|}^{\alpha};x)\le {\tau}_{n,2}^{\alpha /2}(x),x\ge 0.$$ | (22) |

By using the continuity of $f$, it is obvious that (21) holds for any $x\ge 0$ and $y\in \overline{E}$, $\overline{E}$ being the closure of the set $E$ in ${\mathbb{R}}_{+}$. Let $(x,{x}_{0})\in {\mathbb{R}}_{+}\times \overline{E}$ be such that $d(x,E)=|x-{x}_{0}|$. In the following obvious inequality

$$|f-f(x)|\le |f-f({x}_{0})|+|f({x}_{0})-f(x)|{e}_{0},$$ |

by applying the linear and positive operators ${\stackrel{~}{L}}_{n}$, and using (21) we have

$|({\stackrel{~}{L}}_{n}f)(x)-f(x)|$ | $=|{\stackrel{~}{L}}_{n}(f-f(x){e}_{0};x)|\le {\stackrel{~}{L}}_{n}(|f-f(x){e}_{0}|;x)$ | ||

$\le {\stackrel{~}{L}}_{n}(|f-f({x}_{0})|;x)+{\stackrel{~}{L}}_{n}(|f({x}_{0})-f(x)|{e}_{0};x)$ | |||

$\le {\stackrel{~}{L}}_{n}({M}_{f}{|{e}_{1}-{x}_{0}{e}_{0}|}^{\alpha};x)+{M}_{f}{|x-{x}_{0}|}^{\alpha}$ | |||

$\le {M}_{f}({\stackrel{~}{L}}_{n}({|{e}_{1}-x{e}_{0}|}^{\alpha};x)+{\stackrel{~}{L}}_{n}({|x-{x}_{0}|}^{\alpha}{e}_{0};x)+{|x-{x}_{0}|}^{\alpha}).$ |

We also took into account the inequality ${(A+B)}^{\alpha}\le {A}^{\alpha}+{B}^{\alpha}$ ($A\ge 0$, $B\ge 0$, $$) and the monotonicity of ${\stackrel{~}{L}}_{n}$. Based on (22) the conclusion of our theorem is completely motivated. ∎

Disclosure statement. No potential conflict of interest was reported by the author.

Funding. Not applicable.

ORCID: 0000-0002-2406-4274

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