Let $L_n$ be Bernstein operator $B_n$, see (1). It is known that $B_n$ is a Markov type operator interpolating the functions at the extremities of the domain $[0,1]$. Moreover, ${B_n|}_{X_{\alpha ,\beta }}$ is a contraction for all $\alpha ,\beta \in ℝ$, where Lipschitz constant is ${\lambda }_n=1-2^{1-n}$.

Consequently, applying (11) we deduce

where all components of the function ${𝐟}_0$ are given by $p_{\alpha ,\beta }^{\ast }=f(0)e_0+(f(1)-f(0))e_1$.

For $p=1$ we reobtain the identity (2) which is the classical result of Kelisky and Rivlin [11, Eq. (2.4)].

By using this concept, Kelisky-Rivlin result can be reformulated as follows: for each $n\in \mathbb{N}$, $B_n$ is WPO and $(B_n^{\mathrm{\infty }}f)=f(0)e_0+(f(1)-f(0))e_1$.

Recently, following the same route of WPO, in [4], the limit of iterates of modified Bernstein operators in Durrmeyer sense has been approached.

In our attention is a generalization of Bernstein operators, the basis of its construction being a combinatorial identity of Jensen [10]

The inception of its motivation is Lagrange’s formula

and proceeds by setting $u_1(z)=e^{xz}$, $u_2(z)=e^{\gamma z}$. Choosing $y=1-x$ in (15), Cheney and Sharma [5] have investigated the operators

where $\gamma$ is a non-negative parameter and

Obviously, the Bernstein polynomials represent a particular case of (16) obtained by setting $\gamma =0$.

The authors proved that each $Q_n$ preserves the constant functions. In [19] was shown that $Q_n$ also reproduces the monomial $e_1$. It is easy to see that for all $(\alpha ,\beta )\in ℝ\times ℝ$, $X_{\alpha ,\beta }$ defined by (3) with $a=0$, $b=1$ is an invariant set of $Q_n$. Moreover, ${Q_n|}_{X_{\alpha ,\beta }}$ is a contraction. Indeed, if $f_1$ and $f_2$ belong to $X_{\alpha ,\beta }$, then we get

Examining (4), we can choose and all requirements for one-dimensional operators $L_n\equiv Q_n$ are met. Thus, we can form the multidimensional operators $𝐋\equiv 𝐐$ as in (9) and the relation (11) takes place ($a=0$, $b=1$).

Here we have in mind operators constructed with the help of binomial polynomials. This involves a foray into umbral calculus, consequently at the beginning we will point out the basics. The first rigorous version of this calculus belongs to Gian-Carlo Rota and his collaborators, see, e.g., [16].

For any $n\in {\mathbb{N}}_0=\{0\}\cup \mathbb{N}$, 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 $n$.

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.

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

hold. We get $b_0(x)=1$ and by induction we obtain $b_n(0)=0$ for any $n\in \mathbb{N}$. Set

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

An operator $T\in ℒ$ which switches with all shift operators, that is $T{E}^a=E^{a}T$ for every $a\in ℝ$, is called a shift-invariant operator and the set of these operators are denoted by ${ℒ}_s$.

An operator $Q$ is called a delta operator if $Q\in {ℒ}_s$ and $Q{e}_1$ is a nonzero constant. Let ${ℒ}_{\delta }$ denote the set of all delta operators. More generally, according to [16, Proposition 2] for every $Q\in {ℒ}_{\delta }$ we have

A polynomial sequence $p={(p_n)}_{n\ge 0}$ is called the sequence of basic polynomials associated to the delta operator $Q$ if, for any $n\in \mathbb{N}$ and $x\in ℝ$, we get

(i) $p_0(x)=1$,

(ii) $p_n(0)=0$,

(iii) $(Q{p}_n)(x)=n{p}_{n-1}(x)$.

It was proved [16, Proposition 3] that every delta operator has a unique sequence of basic polynomials.

For a good understanding, we collect below some results established in [16] on this topic.

(a) If $p={(p_n)}_{n\ge 0}$ is a basic sequence for some delta operator $Q$, then it is a sequence of binomial type. Reciprocally, if $p$ is a sequence of binomial type, then it is a basic sequence for some delta operator.

(b) Let $T\in {ℒ}_s$ and $Q\in {ℒ}_{\delta }$ with the basic sequence $p={(p_n)}_{n\ge 0}$. One has

(c) An isomorphism $\mathrm{\Psi }$ exists from the ring $(ℱ,+,\cdot )$ of the formal power series in the variable $t$ over $ℝ$ field, onto $({ℒ}_s,+,\cdot )$ such that

(d) An operator $P\in {ℒ}_s$ is a delta operator if and only if it corresponds under the isomorphism defined by (18), to a formal series $\varphi (t)$ such that $\varphi (0)=0$ and ${\varphi }^{\prime }(0)\ne 0$.(e) Let $Q\in {ℒ}_{\delta }$ with $p={(p_n)}_{n\ge 0}$ its sequence of basic polynomials. Led $\varphi (D)=Q$ and $\tau (t)$ be the inverse formal series of $\varphi (t)$, where $D$ indicates the derivative operator. Then one has

where $\tau (t)$ has the form $c_{1}t+c_2{t}^2+\mathrm{\dots }$ $(c_1\ne 0)$.

At this moment we are ready to present a new class of operators. Let $Q$ be a delta operator and $p={(p_n)}_{n\ge 0}$ be its sequence of basic polynomials under the additional assumption $p_n(1)\ne 0$ for every $n\in \mathbb{N}$. We define $L_n^Q:C([0,1])\to C([0,1])$ as follows

P. Sablonniere [18] called them Bernstein-Sheffer operators, but as D.D. Stancu and M.R. Occorsio motivated in [20] these operators can be namely Popoviciu operators. These operators check all the requirements in order to be able to create the multidimensional operators ${𝐋}^Q$, see (9).

The operators $L_n^Q$, $n\in \mathbb{N}$, are linear and reproduce the constants. Indeed, choosing in (17) $y:=1-x$, from (20) 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 $\tau (t)$ defined at (19). Tiberiu Popoviciu [14] and later P. Sablonniere [18, Theorem 1] have established the following result.

$L_n^Q$ defined by (20) is a positive operator on $C([0,1])$ for every $n\in \mathbb{N}$ if and only if

In [18, Theorem 2(i)] it was shown that if (21) takes place, then $L_n^Q{e}_1=e_1$.

Further, from the definition of basic polynomials we get $p_k(0)={\delta }_{k,0}$, consequently $(L_n^{Q}f)(x_0)=f(x_0)$ for $x_0=0$ and $x_0=1$. It remains to check if ${L_n^Q|}_{X_{\alpha ,\beta }}$ is a contraction. Let $f_1$ and $f_2$ belong to $X_{\alpha ,\beta }$, where $\alpha \in ℝ$, $\beta \in ℝ$ are arbitrarily fixed. Using (20) we can write

Due to positivity of $L_n^Q$ operators, $n\in \mathbb{N}$, see relation (21) corroborated with (19), we have . Thus, we deduce that ${L_n^Q|}_{X_{\alpha ,\beta }}$ is a contraction and

Considering that the additional condition (21) occurs, we can apply our result for multidimensional operators. Consequently, we state that (11) is valid for $𝐋\equiv {𝐋}^Q$.

We end this application by indicating Crăciun’s paper [6]. Here a more complex class of linear operators is built that mixes two sequences $p={(p_n)}_{n\ge 0}$, $s={(s_n)}_{n\ge 0}$, the first containing basic polynomials associated to a delta operator $Q$ and the second is defined by the identity $s_n=S^{-1}{p}_n$, where $S$ is an invertible shift invariant operator.