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

then, one has

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.

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

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 $\epsilon >0$,

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.

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

representing the commutative algebra of polynomials with coefficients in $𝕂$, this symbol standing either for the field $ℝ$ or for the field $ℂ$.

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.

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

Let ${ℒ}_{\delta }$ 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 $\mathrm{\Pi }$.

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 $\tilde{a}={(a_n^{⟨a⟩})}_{n\ge 0}$ respectively $g={(g_n^{⟨a,b⟩})}_{n\ge 0}$ 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].

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

Let $Q$ be a delta operator and let $(ℱ,+,\cdot )$ 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 $({ℒ}_s,+,\cdot )$ be the ring of shift-invariant operators, the product being defined as usually: for any $P_1,P_2\in {ℒ}_s$ 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 $ℱ$ onto ${ℒ}_s$ such that

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 $Q\in {ℒ}_{\delta }$ we have $Q{e}_0=0$ we deduce that any delta operator is not invertible. Also, we can write $T=\varphi (D)$, where $T\in {ℒ}_s$ 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).

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

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

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

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

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

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

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.

Moreover, if $L_n^Q$ satisfies the above conditions, then one has

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

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

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

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 $Q\in {ℒ}_{\delta }$, $p={(p_n)}_{n\ge 0}$, $Q=\varphi (D)$, ${\varphi }^{-1}=\phi$, 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

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 $U\in ℒ$ is defined by the formula

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