It is widely acknowledged that the most studied linear positive operators are Bernstein operators which have known innumerable generalization over time.

Bernstein operators $B_n:C([0,1])\to C([0,1])$ are defined by

where $b_{n,k}(x)=\text{binom}nk{x}^k{(1-x)}^{n-k}$, $0\le k\le n$, represent the $n+1$ Bernstein basis polynomials of degree $n$. As usual, $C([0,1])$ denotes the real Banach space of all continuous functions $f:[0,1]\to ℝ$ endowed with the sup-norm $\parallel \cdot \parallel$, $\Vert f\Vert =\underset{x\in [0,1]}{sup}|f(x)|$. With the same norm we endow $B([0,1])$, the space of bounded real valued functions defined on $[0,1]$.

Based on the generalized Lototsky matrix, an extension of these operators was given by King [1]. We present it in the following. For each $j\in \mathbb{N}$, let $h_j:[0,1]\to [0,1]$ be a continuous function. Further, for each $n\in \mathbb{N}$ is defined a system of functions ${(a_{n,k})}_{k=\overline{0,n}}$ for $x\in [0,1]$ by the relation

From the above identity we immediately obtain the coefficient of $y^k$, $k=\overline{0,n}$, that is

where ${\mathbb{N}}_n=\{1,2,\mathrm{\dots },n\}$ and $\overline{K}={\mathbb{N}}_n\setminus K$. For each real valued function $f$ defined on $[0,1]$, the $n$-th Lototsky-Bernstein operator is defined as follows

see [1, Eq. 4]. $L_n$ operators are linear. Since $h_j([0,1])\subset [0,1]$ for all $j\in \mathbb{N}$, they are also positive. It is clear that in the special case $h_j(x)=x$, $j\in \mathbb{N}$, the functions ${(a_{n,k})}_{0\le k\le n}$, $n\in \mathbb{N}$, become Bernstein bases ${(p_{n,k})}_{0\le k\le n}$, $n\in \mathbb{N}$, consequently $L_n$ operator turns into $B_n$ operator.

King has established the sufficient condition on ${(h_j)}_{j\in \mathbb{N}}$ sequence to ensure that ${(L_n)}_{n\ge 1}$ is an approximation process on $C([0,1])$, his result can be written as follows. If

then

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

In recent years the study of these operators has been deepened, see, for example, the papers of Ron Goldman, Xiao-Wei Xu, Xiao-Ming Zeng [2], [3].

The purpose of this note is to define and to establish approximation properties for a Durmmeyer-type extension of $L_n$, $n\in \mathbb{N}$, operators. We mention that, using elements of probability theory, a Kantorovich-type extension was achieved in 2020 by Popa [4]. A second goal of this note is to extend the univariate operators for vector functions with real values.

Set ${\mathbb{N}}_0=\{0\}\cup \mathbb{N}$ and $e_j$, $j\in {\mathbb{N}}_0$, monomials of degree $j$, $e_0(x)=1$, $e_j(x)=x^j$, $j\in \mathbb{N}$. The statement (5) is motivated by Bohman-Korovkin criterion which says: If a sequence of linear positive operators ${({\mathrm{\Lambda }}_n)}_{n\ge 1}$ defined on $C([a,b])$ has the property that ${({\mathrm{\Lambda }}_n{e}_k)}_{n\ge 1}$ converges to $e_k$ uniformly on $[a,b]$, $k\in \{0,1,2\}$, then ${({\mathrm{\Lambda }}_{n}f)}_{n\ge 1}$ converges to $f$ uniformly on $[a,b]$ for each $f$ belonging to $C([a,b])$. In relation (1) let us denote by $P_n(x;y)$ the polynomial of degree $n$ in $y$ and with the parameter $x\in [0,1]$. Following [5] we get

Taking in view the mentioned criterion, the proof of (5) is completed.

Moreover, for any $k\ge 1$,

see [5, Eq. (2.6)]. In the above $s(k,j)$ denotes a Stirling number of the second kind. For $1\le j\le k$, its closed form is given as follows

see, e.g., [6, p. 824]. For $k=1$ and $k=2$, from (9) we reobtain the identities (7) and (2).

Usually, in the papers that approached Lototsky operators, in order to obtain significant results, the authors imposed additional conditions on the functions $h_j$, $j\in \mathbb{N}$. For similar reason, we define a particular Durrmeyer type construction that involves both Lototsky-Bernstein and genuine Bernstein bases. $L_1([0,1])$ stands for the Banach space of all real valued integrable functions on $[0,1]$ endowed with the norm $\parallel \cdot {\parallel }_1$, ${\Vert f\Vert }_1={\displaystyle {\int }_0^1}|f(x)|𝑑x.$ Define $D_n:L_1([0,1])\to C([0,1])$ by formula

(a) The operators keep the properties of linearity and positivity.

(b) By using a bivariate kernel we can write $D_{n}f$ in a more compact form as follows

where

(c) If $f\in C([0,1])$, then $\Vert D_{n}f\Vert \le \Vert f\Vert$, $n\in \mathbb{N}$, consequently the operators are non expansive in the space $C([0,1])$.

(d) For the particular case $h_j=e_1$, $j\in \mathbb{N}$, $D_n$ operators turn into the genuine Durrmeyer operators [7].

Let $D_n$, $n\in \mathbb{N}$, be the operators defined by (10). The following identities

take place for each $n\in \mathbb{N}$.

At this point we introduce the $j$-th central moment of $D_n$ operators, $j\in \mathbb{N}$, i.e. $D_n{\phi }_x^j$, where

The second order central moments of the $D_n$, $n\in \mathbb{N}$, operators satisfy the following relations