## Abstract

Sequences of binomial operators introduced by using umbral calculus are investigated from the point of view of statistical convergence. This approach is based on a detailed presentation of delta operators and their associated basic polynomials. Bernstein–Sheffer linear positive operators are analyzed, and some particular cases are highlighted: Cheney–Sharma operators, Stancu operators, Lupaş operators.

## Authors

O. **Agratini**

(Babes-Bolyai University, Cluj-Napoca

Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)

## Keywords

### References

See the expanding block below.

## Paper coordinates

O. Agratini,* From uniform to statistical convergence of binomial-type operators*, In: Advances In Summability And Approximation Theory, 169 – 179, (Eds. S. A. Mohiuddine, T. Acar), Springer, Singapore, 2018. ISBN: 978-981-13-3076-6, DOI https://doi.org/10.1007/978-981-13-3077-3_10

## About this paper

##### Print ISSN

?

##### Online ISSN

aici se va trece ISSN-ul online

##### Google Scholar Profile

soon

## Paper (preprint) in HTML form

# From uniform to statistical convergence of binomial type operators

###### Abstract.

Sequences of binomial operators introduced by using umbral calculus are investigated from the point of view of statistical convergence. This approach is based on a detailed presentation of delta operators and their associated basic polynomials. Bernstein-Sheffer linear positive operators are analyzed and some particular cases are highlighted: Cheney-Sharma operators, Stancu operators, Lupaş operators.

Keywords and phrases: Statistical convergence, binomial sequence, linear positive operator, umbral calculus, Bernstein-Sheffer operator, Pincherle derivative.

2010 Mathematics Subject Classification: 05A40, 41A36, 47A58.

## 1. Introduction

Let ${({L}_{n})}_{n\ge 1}$ be a sequence of linear positive operators acting on the space $C([a,b])$ of all real-valued and continuous functions defined on the interval $[a,b]$, equipped with the norm $\parallel \cdot \parallel $ of the uniform convergence, namely $\Vert h\Vert =\underset{a\le t\le b}{sup}|h(t)|$. Bohman-Korovkin’s theorem asserts: if the operators ${L}_{n}$, $n\in \mathbb{N}$, map $C([a,b])$ into itself such that

$$\underset{n\to \mathrm{\infty}}{lim}\Vert {L}_{n}{e}_{j}-{e}_{j}\Vert =0\text{for}j\in \{0,1,2\},$$ | (1.1) |

then, one has

$$\underset{n\to \mathrm{\infty}}{lim}\Vert {L}_{n}f-f\Vert =0\text{for every}f\in C([a,b]).$$ | (1.2) |

In the above ${e}_{j}$ represents the monomial of $j$-th degree, ${e}_{0}(x)=1$ and ${e}_{j}(x)={x}^{j}$, $j\ge 1$.

A current subject in Approximation Theory is the approximation of continuous functions by using the statistical convergence, the first research of this topic being done by Gadjiev and Orhan [5]. This approach models and improves the technique of signals approximation in different function spaces.

On the other hand, sequences of polynomials of binomial type have been the subject of many mathematical studies, drawing to light their role in Approximation Theory. Practically, the theory of the approximation operators of binomial type is based on the technique of the umbral calculus. In its modern form, this is a strong tool for calculations with polynomials representing a successful combination between the finite differences calculus and certain chapters of Probability Theory. The topic discussed in this chapter is at the confluence of the two concepts mentioned above, statistical convergence and binomial type operators, from the point of view of the approximation of some function classes. The material is structured in three sections.

First of all we recall the variant of Bohman-Korovkin theorem via statistical convergence and we present elementary facts about polynomial sequences of binomial type. Further on, we deal with delta operators and their basic polynomials. In the last section we will analyze the approximation properties of some binomial operators in terms of the statistical convergence.

We mention that at the first sight this work seems disproportionate, dominated by a lot of notions introduced and results already achieved. The goal was to be self contained paper. It will be seen that for a clear understanding of the last paragraph, it was necessary to structure the article in this way.

## 2. Preliminaries

The concept of statistical convergence was first defined by Steinhaus [13] and Fast [4]. It is based on the notion of the asymptotic density of subsets of $\mathbb{N}$. The density of $S\subseteq \mathbb{N}$ denoted by $\delta (S)$ is given by

$$\delta (S)=\underset{n\to \mathrm{\infty}}{lim}\frac{1}{n}\sum _{j=1}^{n}{\chi}_{S}(j),$$ |

where ${\chi}_{S}$ stands for the characteristic function of the set $S$. Clearly, $0\le \delta (S)\le \delta (\mathbb{N})=1$. A sequence ${({x}_{n})}_{n\ge 1}$ of real numbers is said to be statistically convergent to a real number $l$, if, for every $\epsilon >0$,

$$\delta (\{n\in \mathbb{N}:|{x}_{n}-l|\ge \epsilon \})=0,$$ |

the limit being denoted by $st-\underset{n\to \mathrm{\infty}}{lim}{x}_{n}=l$. It is known that any convergent sequence is statistically convergent but the converse of this statement is not true. Even though this notion was introduced in 1951, its application to the study of sequences of positive linear operators was attempted only in 2002. We refer to the A.D. Gadjiev and C. Orhan [5] result, which reads as follows.

###### Theorem 2.1.

If the sequence of positive linear operators ${L}_{n}:C([a,b])\to B([a,b])$ satisfies the condition

$$st-\underset{n\to \mathrm{\infty}}{lim}\Vert {L}_{n}{e}_{j}-{e}_{j}\Vert =0,j\in \{0,1,2\},$$ | (2.1) |

then, one has

$$st-\underset{n\to \mathrm{\infty}}{lim}\Vert {L}_{n}f-f\Vert =0\text{for every function}f\in C([a,b]).$$ | (2.2) |

As usual, $B([a,b])$ stands for the space of all real-valued bounded functions defined on $[a,b]$, endowed with the sup-norm. The identities (2.1) and (2.2) generalize respectively relations (1.1), (1.2). From this moment, the statistical convergence of positive linear operators represented a new direction in the study of so-called KAT-Korovkin-type Approximation Theory.

Set ${\mathbb{N}}_{0}:=\mathbb{N}\cup \{0\}$. For any $n\in {\mathbb{N}}_{0}$ we denote by ${\mathrm{\Pi}}_{n}$ the linear space of polynomials of degree no greater than $n$ and by ${\mathrm{\Pi}}_{n}^{\ast}$ the set of all polynomials of degree $n$. We also set

$$\mathrm{\Pi}:=\bigcup _{n\ge 0}{\mathrm{\Pi}}_{n},$$ |

representing the commutative algebra of polynomials with coefficients in $\mathbb{K}$, this symbol standing either for the field $\mathbb{R}$ or for the field $\u2102$.

A sequence $p={({p}_{n})}_{n\ge 0}$ such that ${p}_{n}\in {\mathrm{\Pi}}_{n}^{\ast}$ for every $n\in {\mathbb{N}}_{0}$ is called a polynomial sequence.

###### Definition 2.2.

A polynomial sequence $b={({b}_{n})}_{n\ge 0}$ is called of binomial type if for any $(x,y)\in \mathbb{K}\times \mathbb{K}$ the following identities hold

$${b}_{n}(x+y)=\sum _{k=0}^{n}\left(\genfrac{}{}{0pt}{}{n}{k}\right){b}_{k}(x){b}_{n-k}(y),n\in {\mathbb{N}}_{0}.$$ | (2.3) |

###### Remark 2.3.

Knowing that $\mathrm{deg}({b}_{0})=0$, we get ${b}_{0}(x)=1$ for any $x\in \mathbb{K}$ and by induction we easily obtain ${b}_{n}(0)=0$ for any $n\in \mathbb{N}$.

The most common example of binomial sequence is $e={({e}_{n})}_{n\ge 0}$ (the monomials). Some nontrivial examples are given below.

a) The generalized factorial power with the step $a$: $p={({p}_{n})}_{n\ge 0}$,

$${p}_{0}(x)={x}^{[0,a]}:=1\text{and}{p}_{n}(x)={x}^{[n,a]}:=x(x-a)\mathrm{\dots}(x-(n-1)a),n\in \mathbb{N}.$$ |

The Vandermonde formula, i.e.

$${(x+y)}^{[n,a]}=\sum _{k=0}^{n}\left(\genfrac{}{}{0pt}{}{n}{k}\right){x}^{[k,a]}{y}^{[n-k,a]},$$ |

guarantees that this is a binomial type sequence. There are two particular cases: for $a=1$ we obtain the lower-factorials which, usually, are denoted by ${\u27e8x\u27e9}_{n}$; for $a=-1$ we obtain the upper-factorials denoted by Pochhammer’s symbol ${(x)}_{n}$. By convention we consider

$${x}^{[-n,a]}:=1/{(x+na)}^{[n,a]}.$$ |

b) Abel polynomials: $\stackrel{~}{a}={({a}_{n}^{\u27e8a\u27e9})}_{n\ge 0}$,

$${a}_{0}^{\u27e8a\u27e9}=1,{a}_{n}^{\u27e8a\u27e9}(x)=x{(x-na)}^{n-1},n\in \mathbb{N},a\ne 0.$$ |

Rewriting the identity (2.3) for these polynomials we obtain the Abel-Jensen (1902) combinatorial formula

$$(x+y){(x+y+na)}^{n-1}=\sum _{k=0}^{n}\left(\genfrac{}{}{0pt}{}{n}{k}\right)xy{(x+ka)}^{k-1}{(y+(n-k)a)}^{n-1-k},n\in \mathbb{N}.$$ |

c) Gould polynomials: $g={({g}_{n}^{\u27e8a,b\u27e9})}_{n\ge 0}$,

$${g}_{0}^{\u27e8a,b\u27e9}=1,{g}_{n}^{\u27e8a,b\u27e9}(x)=\frac{x}{x-an}{\u27e8\frac{x-an}{b}\u27e9}_{n},n\in \mathbb{N},ab\ne 0.$$ |

The space of all linear operators $T:\mathrm{\Pi}\to \mathrm{\Pi}$ will be denoted by $\mathcal{L}$. Among these operators an important role will be played by the shift operator, named ${E}^{a}$. For every $a\in \mathbb{K}$, ${E}^{a}$ is defined by

$$({E}^{a}p)(x)=p(x+a),\text{where}p\in \mathrm{\Pi}.$$ |

An operator $T\in \mathcal{L}$ which switches with all shift operators, that is

$$T{E}^{a}={E}^{a}T\text{for every}a\in \mathbb{K},$$ |

is called a shift-invariant operator and the set of these operators is denoted by ${\mathcal{L}}_{s}$.

## 3. On delta operators

###### Definition 3.1.

An operator $Q:\mathrm{\Pi}\to \mathrm{\Pi}$ is called delta operator if $Q\in {\mathcal{L}}_{s}$ and $Q{e}_{1}$ is a non zero constant.

Let ${\mathcal{L}}_{\delta}$ denote the set of all delta operators. For a better understanding we present some examples of delta operators. In the following the symbol $I$ stands for the identity operator on the space $\mathrm{\Pi}$.

a) The derivative operator, denoted by $D$.

b) The operators used in calculus of divided differences. Let $h$ be a fixed number belonging to the field $\mathbb{K}$. We set

${\mathrm{\Delta}}_{h}$ | $:={E}^{h}-I,\text{the}\text{forward difference operator},$ | ||

${\nabla}_{h}$ | $:=I-{E}^{-h},\text{the}\text{backward difference operator},$ | ||

${\delta}_{h}$ | $:={E}^{h/2}-{E}^{-h/2},\text{the}\text{central difference operator}.$ |

It is evident that ${\nabla}_{h}={\mathrm{\Delta}}_{h}{E}^{-h}$, ${\delta}_{n}={\mathrm{\Delta}}_{h}{E}^{-h/2}={\nabla}_{h}{E}^{h/2}$. The properties of these operators as well as their usefulness can be found in [6].

c) Abel operator, ${A}_{a}:=D{E}^{a}$. For any $p\in \mathrm{\Pi}$, $({A}_{a}p)(x)={\displaystyle \frac{dp}{dx}}(x+a)$.

Writing (symbolically) the Taylor’s series in the following manner

$${E}^{h}=\sum _{\nu =0}^{\mathrm{\infty}}\frac{{h}^{\nu}{D}^{\nu}}{\nu !}={e}^{hD},$$ | (3.1) |

we can also get ${A}_{a}=D({e}^{aD})$.

d) Gould operator, ${G}_{a,b}:={\mathrm{\Delta}}_{b}{E}^{a}={E}^{a+b}-{E}^{a}$, $ab\ne 0$.

###### Definition 3.2.

Let $Q$ be a delta operator. A polynomial sequence $p={({p}_{n})}_{n\ge 0}$ is called the sequence of basic polynomials associated to $Q$ if

(i) ${p}_{0}(x)=1$ for any $x\in \mathbb{K}$,

(ii) ${p}_{n}(0)=0$ for any $n\in \mathbb{N}$,

(iii) $(Q{p}_{n})(x)=n{p}_{n-1}(x)$ for any $n\in \mathbb{N}$ and $x\in \mathbb{K}$.

###### Remark 3.3.

If $p={({p}_{n})}_{n\ge 0}$ is a sequence of basic polynomials associated to $Q$, then $\{{p}_{0},{p}_{1},\mathrm{\dots},{p}_{n-1},{e}_{n}\}$ is a basis of the linear space ${\mathrm{\Pi}}_{n}$. Taking this fact into account, by induction it can be proved that every delta operator has a unique sequence of basic polynomials, see [9, Proposition 3].

Here are some examples. The basic polynomials associated to the operators $Q=D$, $Q={\mathrm{\Delta}}_{h}$ and $Q={\nabla}_{h}$, are respectively ${({e}_{n})}_{n\ge 0}$, ${({x}^{[n,h]})}_{n\ge 0}$ and ${({(x+(n-1)h)}^{[n,h]})}_{n\ge 0}$. Also, we can easily prove that $\stackrel{~}{a}={({a}_{n}^{\u27e8a\u27e9})}_{n\ge 0}$ respectively $g={({g}_{n}^{\u27e8a,b\u27e9})}_{n\ge 0}$ is the sequence of basic polynomials associated to Abel operator ${A}_{a}$, respectively Gould operator ${G}_{a,b}$.

The connection between delta operator and the binomial type sequences is given by the following result [9, Theorem 1].

###### Theorem 3.4.

Let $p={({p}_{n})}_{n\ge 0}$ be a sequence of polynomials. It is a sequence of binomial type if and only if it is a basic sequence for some delta operator.

The following statement generalizes the Taylor expansion theorem to delta operators and their basic polynomials.

###### Theorem 3.5.

Let $T$ be a shift-invariant operator and let $Q$ be a delta operator with its basic sequence ${({p}_{n})}_{n\ge 0}$. Then the following identity holds

$$T=\sum _{k\ge 0}\frac{(T{p}_{k})(0)}{k!}{Q}^{k}.$$ | (3.2) |

Let $Q$ be a delta operator and let $(\mathcal{F},+,\cdot )$ be the ring of the formal power series in the variable $t$ over the same field. Here the product means the Cauchy product between two series. Further, let $({\mathcal{L}}_{s},+,\cdot )$ be the ring of shift-invariant operators, the product being defined as usually: for any ${P}_{1},{P}_{2}\in {\mathcal{L}}_{s}$ we have ${P}_{1}{P}_{2}:\mathrm{\Pi}\to \mathrm{\Pi}$, $({P}_{1}{P}_{2})(q)={P}_{1}({P}_{2}(q))$, $q\in \mathrm{\Pi}$. Then there exists an isomorphism $\psi $ from $\mathcal{F}$ onto ${\mathcal{L}}_{s}$ such that

$$\psi (f(t))=T,\text{where}f(t)=\sum _{k\ge 0}\frac{{a}_{k}}{k!}{t}^{k}\text{and}T=\sum _{k\ge 0}\frac{{a}_{k}}{k!}{Q}^{k}.$$ | (3.3) |

This isomorphism allows us to conclude: a shift-invariant operator $T$ is invertible if and only if $T{e}_{0}\ne 0$. Since for every $Q\in {\mathcal{L}}_{\delta}$ we have $Q{e}_{0}=0$ we deduce that any delta operator is not invertible. Also, we can write $T=\varphi (D)$, where $T\in {\mathcal{L}}_{s}$ and $\varphi (t)$ is a formal power series, to indicate that the operator $T$ corresponds to the series $\varphi (t)$ under the isomorphism defined by (3.3).

###### Remark 3.6.

In relation (3.1) we choose $T={E}^{x}$ and expand ${E}^{x}$ in terms of $Q$. Due to the identity $({E}^{x}{p}_{k})(0)={p}_{k}(x)$ and the relation (3.2), one obtains

$${e}^{xD}=\sum _{k\ge 0}\frac{{p}_{k}(x)}{k!}{\varphi}^{k}(D).$$ |

Substituting $D$ by $u$, the series terms lead us to the following result [9, Corollary 3].

###### Theorem 3.7.

Let $Q$ be a delta operator with $p={({p}_{n})}_{n\ge 0}$ its sequence of basic polynomials. Let $\varphi (D)=Q$ and $\phi (t)$ be the inverse formal power series of $\varphi (u)$. Then

$${e}^{x\phi (t)}=\sum _{n\ge 0}\frac{{p}_{n}(x)}{n!}{t}^{n},$$ | (3.4) |

where $\phi (t)$ has the form ${c}_{1}t+{c}_{2}{t}^{2}+\mathrm{\dots}$ $({c}_{1}\ne 0)$.

Another characterization of delta operators was included in [9] without proof. For this reason we prove the following statement.

###### Theorem 3.8.

$Q\in {\mathcal{L}}_{s}$ is a delta operator if and only if $Q=DP$ for some shift-invariant operator $P$, where the inverse operator ${P}^{-1}$ exists.

Proof. If in (3.3) we substitute $T$ by a delta operator $Q$, then we get ${a}_{0}=Q({e}_{0})=0$ and ${a}_{1}=Q({e}_{1})=c\ne 0$. Consequently, we can write

$$Q=\sum _{k\ge 1}\frac{{a}_{k}}{k!}{D}^{k}.$$ | (3.5) |

Denoting $\sum _{k\ge 1}}{\displaystyle \frac{{a}_{k}}{k!}}{D}^{k-1$ by $P$ we have $P\in {\mathcal{L}}_{s}$ and $P({e}_{0})={a}_{1}\ne 0$, thus $P$ is invertible, see the conclusion that emerges from (3.3). So, $Q$ can be written as $DP$.

Reciprocally, for every $P\in {\mathcal{L}}_{s}$ such that $P$ is invertible, $DP$ is a shift-invariant operator, ${E}^{a}(DP)=(DP){E}^{a}$, and

$$(DP)({e}_{1})=P(D({e}_{1}))=P({e}_{0})=c\ne 0$$ |

thus $DP\in {\mathcal{L}}_{\delta}$. $\mathrm{\square}$

Now we are ready to analyze some binomial operators investigating their statistical convergence to the identity operator.

## 4. Classes of binomial operators

We consider a delta operator $Q$ and its sequence of basic polynomials $p={({p}_{n})}_{n\ge 0}$, under the assumption that ${p}_{n}(1)\ne 0$ for every $n\in \mathbb{N}$. Also, according to Theorem 3.7 we shall keep the same meaning of the functions $\varphi $ and $\phi $. For every $n\ge 1$ we consider ${L}_{n}^{Q}:C([0,1])\to C([0,1])$ defined as follows

$$({L}_{n}^{Q}f)(x)=\frac{1}{{p}_{n}(1)}\sum _{k=0}^{n}\left(\genfrac{}{}{0pt}{}{n}{k}\right){p}_{k}(x){p}_{n-k}(1-x)f\left(\frac{k}{n}\right).$$ | (4.1) |

They are called by P. Sablonnière [10] Bernstein-Sheffer operators. As D.D. Stancu and M.R. Occorsio motivated in [12], these operators can be named Popoviciu operators. T. Popoviciu [8] indicated the construction (4.1) in front of the sum appearing the factor ${d}_{n}^{-1}$ from the identities

$${(1+{d}_{1}t+{d}_{2}{t}^{2}+\mathrm{\dots})}^{x}={e}^{x\phi (t)}=\sum _{n=0}^{\mathrm{\infty}}{p}_{n}(x)\frac{{t}^{n}}{n!},$$ |

see (3.4). If we choose $x=1$ it becomes obvious that ${d}_{n}={p}_{n}(1)/n!$.

In the particular case $Q=D$, ${L}_{n}^{D}$ becomes genuine Bernstein operator of degree $n$. An integral generalization of ${L}_{n}^{Q}$ in Kantorovich sense was introduced and studied in [1].

The operators ${L}_{n}^{Q}$, $n\in \mathbb{N}$, are linear and reproduce the constants. Indeed, choosing in (2.3) $y:=1-x$ we obtain ${L}_{n}^{Q}{e}_{0}={e}_{0}$. The positivity of these operators are given by the sign of the coefficients of the series $\phi (t)={c}_{1}+{c}_{2}t+\mathrm{\dots}$ $({c}_{1}\ne 0)$. More precisely, in [8] and in [10] the authors established the followings.

###### Lemma 4.1.

${L}_{n}^{Q}$ is a positive operator on $C([0,1])$ for every $n\ge 1$ if and only if ${c}_{1}>0$ and ${c}_{n}\ge 0$ for all $n\ge 2$.

Moreover, if ${L}_{n}^{Q}$ satisfies the above conditions, then one has

$${L}_{n}^{Q}{e}_{1}={e}_{1},n\in \mathbb{N},\text{and}{L}_{n}^{Q}{e}_{2}={e}_{2}+{a}_{n}({e}_{1}-{e}_{2}),n\ge 2,$$ | (4.2) |

where ${a}_{n}={\displaystyle \frac{1}{n}}\left(1+(n-1){\displaystyle \frac{{r}_{n-2}(1)}{{p}_{n}(1)}}\right)$, the sequence ${({r}_{n}(x))}_{n\ge 0}$ being generated by

$${\phi}^{\prime \prime}(t)\mathrm{exp}(x\phi (t))=\sum _{n\ge 0}{r}_{n}(x)\frac{{t}^{n}}{n!}.$$ |

###### Theorem 4.2.

Let the operators ${L}_{n}^{Q}$, $n\in \mathbb{N}$, be defined by (4.1) such that the hypothesis of Lemma 4.1 takes place.

If $st-\underset{n\to \mathrm{\infty}}{lim}{\displaystyle \frac{{r}_{n-2}(1)}{{p}_{n}(1)}}=0$, then

$$st-\underset{n\to \mathrm{\infty}}{lim}\Vert {L}_{n}^{Q}f-f\Vert =0,f\in C([0,1]).$$ | (4.3) |

Proof. We apply Theorem 2.1. Based on algebraic operations with statistically convergent sequences of real numbers, our hypothesis guarantees the identity (4.3). For a profound documentation of operations with such sequences [2, Theorem 3.1] can be consulted. $\mathrm{\square}$

Further, choosing particular delta operators $Q$ we reobtain some classical linear positive operator of discrete type.

###### Example 4.3.

If $Q={A}_{a}$ with its basic sequence $\stackrel{~}{a}$ and assuming that the parameter $a$ depends on $n$, $a:={t}_{n}$, one obtains the Cheney-Sharma operators [3]. The corresponding operators ${Q}_{n}$, $n\in \mathbb{N}$, are defined by the equation

$$({Q}_{n}f)(x):={(1+n{t}_{n})}^{1-n}\sum _{k=0}^{n}\left(\genfrac{}{}{0pt}{}{n}{k}\right)x{(x+k{t}_{n})}^{k-1}(1-x){[1-x+(n-k){t}_{n}]}^{n-1-k}.$$ |

###### Example 4.4.

If $Q={\displaystyle \frac{1}{\alpha}}{\nabla}_{\alpha}$, $\alpha \ne 0$, with its basic polynomials

$${p}_{n}(x)={(x+(n-1)\alpha )}^{[n,\alpha ]},$$ |

${L}_{n}^{Q}$ becomes Stancu operator [11] denoted by ${P}_{n}^{[\alpha ]}$,

$({P}_{n}^{[\alpha ]}f)(x)={\displaystyle \sum _{k=0}^{n}}{w}_{n,k}(x;\alpha )f\left({\displaystyle \frac{k}{n}}\right),$ | ||

${w}_{n,k}(x;\alpha )=\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\displaystyle \frac{{x}^{[k,-\alpha ]}{(1-x)}^{[n-k,-\alpha ]}}{{1}^{[n,-\alpha ]}}},$ |

$\alpha $ being a parameter which may depend on a natural number $n$. One has

$${P}_{n}^{[\alpha ]}{e}_{j}={e}_{j},j\in \{0,1\},\text{and}({P}_{n}^{[\alpha ]}{e}_{2})(x)=\frac{1}{1+\alpha}\left(\frac{x(1-x)}{n}+x(x+\alpha )\right),$$ |

in accordance with [11, Lemma 4.1].

If $0\le \alpha :=\alpha (n)$ and $st-\underset{n\to \mathrm{\infty}}{lim}\alpha (n)=0$, then (4.3) takes place.

Indeed, it is enough to prove (2.1) for $j=2$. We get

$|({P}_{n}^{[\alpha ]}{e}_{2})(x)-{x}^{2}|$ | $={\displaystyle \frac{1}{1+\alpha}}\left|{\displaystyle \frac{x(1-x)}{n}}-\alpha {x}^{2}+x\alpha \right|$ | ||

$\le {\displaystyle \frac{1}{1+\alpha}}\left({\displaystyle \frac{1}{4n}}+2\alpha \right)\le {\displaystyle \frac{1}{4n}}+2\alpha $ |

and the conclusion follows.

At this moment we take a break in order to illustrate some further properties of the binomial sequences. Keeping the notations $Q\in {\mathcal{L}}_{\delta}$, $p={({p}_{n})}_{n\ge 0}$, $Q=\varphi (D)$, ${\varphi}^{-1}=\phi $, we assume that the conditions of Lemma 4.1 are fulfilled.

In [7] A. Lupaş proved new inequalities between the terms of the binomial sequences $p$. For any $x>0$ and $n\ge 2$ one has

$$ | (4.4) |

In the above ${Q}^{\prime}$ represents Pincherle derivative of $Q$. The concept is detailed further.

Knowing that the operator $X:\mathrm{\Pi}\to \mathrm{\Pi}$, $(Xp)(x)=xp(x)$ is called multiplication operator, we recall that the Pincherle derivative of an operator $U\in \mathcal{L}$ is defined by the formula

$${U}^{\prime}=UX-XU.$$ |

For example we get ${I}^{\prime}=0$, ${D}^{\prime}=I$, ${({D}^{k})}^{\prime}=k{D}^{k-1}$, $k\ge 1$.

###### Example 4.5.

Following Lupaş we can define the operators ${\stackrel{~}{L}}_{n}^{Q}:C([0,1])\to C([0,1])$, $n\in \mathbb{N}$,

$$({\stackrel{~}{L}}_{n}^{Q}f)(x)=\frac{1}{{p}_{n}(n)}\sum _{k=0}^{n}\left(\genfrac{}{}{0pt}{}{n}{k}\right){p}_{k}(nx){p}_{n-k}(n-nx)f\left(\frac{k}{n}\right).$$ |

It is proved that

$${\stackrel{~}{L}}_{n}^{Q}{e}_{i}={e}_{i},i\in \{0,1\},\text{and}{\stackrel{~}{L}}_{n}^{Q}{e}_{2}={e}_{2}+({e}_{1}-{e}_{2}){\rho}_{n}(Q),$$ |

see (4.4). Consequently, if

$$st-\underset{n\to \mathrm{\infty}}{lim}\frac{n(n-1)}{{p}_{n}(n)}({Q}^{\prime -2}{p}_{n-2})(x)=1,$$ |

then the operators ${\stackrel{~}{L}}_{n}^{Q}$, $n\in \mathbb{N}$, satisfy identity (4.3).

### Concluding remarks.

The paper reintroduces some linear positive operators of discrete type by using umbral calculus. Relative to these operators have been studied approximation properties in Banach space $(C[0,1],\parallel \cdot \parallel )$. The approach was based on Bohman-Korovkin theorem via statistical convergence. The usefulness of this type of convergence can be summarized as follows: the statistical convergence of a sequence is that the majority, in a certain sense, of its elements converges and we are not interested in what happens to the remaining elements. The advantage of replacing the uniform convergence by statistical convergence consists in the fact that the second convergence is efficient in summing divergent sequences which may have unbounded subsequences. In short it is more lax.

## References

- [1] O. Agratini, On a certain class of approximation operators, Pure Mathematics and Applications, 11(2000), No. 2, 119-127.
- [2] H. Albayrak, S. Pehlivan, Statistical convergence and statistical continuity on locally solid Riesz spaces, Topology and its Applications, 159(2012), 1887-1893.
- [3] E.W. Cheney, A. Sharma, On a generalization of Bernstein polynomials, Rivista di Matematica della Università di Parma (2), 5(1964), 77-84.
- [4] H. Fast, Sur le convergence statistique, Colloquium Mathematicum, 2(1951), 241-244.
- [5] A.D. Gadjiev, C. Orhan, Some approximation theorems via statistical convergence, Rocky Mountain Journal of Mathematics, 32(2002), 129-138.
- [6] C. Jordan, Calculus of Finite Differences, Chelsea Publishing Company, New York, 1950.
- [7] A. Lupaş, Approximation operators of binomial type, New Developments in Approximation Theory (Dortmund, 1980), 175-198, International Series of Numerical Mathematics, Vol. 132, Birkhäuser Verlag Basel/Switzerland, 1999.
- [8] T. Popoviciu, Remarques sur les polynômes binomiaux, Bulletin de la Société des Sciences de Cluj, Roumanie, 6(1931), 146-148 (also reproduced in Mathematica (Cluj), 6(1932), 8-10).
- [9] G.-C. Rota, D. Kahaner, A. Odlyzko, On the Foundations of Combinatorial Theory, VIII. Finite Operator Calculus, Journal of Mathematical Analysis and Application, 42(1973), 685-760.
- [10] P. Sablonnière, Positive Bernstein-Sheffer operators, Journal of Approximation Theory, 83(1995), 330-341.
- [11] D.D. Stancu, Approximation of functions by a new class of linear polynomial operators, Revue Roumaine des Mathématique Pures et Appliquées, 13(1968), No. 8, 1173-1194.
- [12] D.D. Stancu, M.R. Occorsion, On approximation by binomial operators of Tiberiu Popoviciu type, Revue d’Analyse Numérique et de Théorie de l’Approximation, 27(1998), No. 1, 167-181.
- [13] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloquium Mathematicum, 2(1951), 73-74.

**11**(2), 119–127 (2000)MathSciNetzbMATHGoogle Scholar

**159**, 1887–1893 (2012)MathSciNetCrossRefGoogle Scholar

**5**(2), 77–84 (1964)MathSciNetzbMATHGoogle Scholar

[4] H. Fast, Sur le convergence statistique. Colloq. Math.

**2**, 241–244 (1951)MathSciNetCrossRefGoogle Scholar

**32**, 129–138 (2002)MathSciNetCrossRefGoogle Scholar

*Calculus of Finite Differences*(Chelsea Publishing Company, New York, 1950)zbMATHGoogle Scholar

*New Developments in Approximation Theory (Dortmund, 1980)*. International Series of Numerical Mathematics, vol. 132 (Birkhäuser Verlag, Basel, 1999), pp. 175–198CrossRefGoogle Scholar

**6**, 146–148 (1931) (also reproduced in Mathematica (Cluj)

**6**, 8–10 (1932))Google Scholar

**42**, 685–760 (1973)MathSciNetCrossRefGoogle Scholar

**83**, 330–341 (1995)MathSciNetCrossRefGoogle Scholar

**13**(8), 1173–1194 (1968)MathSciNetzbMATHGoogle Scholar

**27**(1), 167–181 (1998)MathSciNetzbMATHGoogle Scholar

**2**, 73–74 (1951)CrossRefGoogle Scholar