The classical Bernstein operators associate the polynomial $B_{n}f$ to any function $f\in C[0,1]$ and the association is defined by

where

In fact, it defines a linear approximation process in the space of all continuous functions on $\left[0,1\right].$ To obtain an approximation process for Lebesgue integrable functions, the Kantorovich version of (1) was defined in Kantorovich by replacing the sample values $f(k/n)$ with the mean values of $f$ in the interval $\left[\frac{k}{n},\frac{k+1}{n}\right]$, that is

where $f:\left[0,1\right]\rightarrow\mathbb{R}$ is a locally integrable function. We note that $B_{n}$ and $K_{n}$ are connected by the relation

where $D$ is the differentiation operator (i.e. $D\left(f\right)=f^{{}^{\prime}}$, $f\in C^{1}\left[0,1\right]$) and $I$ is the antiderivative operator (i.e. $I\left(f;x\right)=\int_{0}^{x}f\left(t\right)dt,$ $f\in C\left[0,1\right]$) and $x\in\left[0,1\right]$. These operators allow us to reconstruct a Lebesgue integrable function by means of its mean values on the sets $\left[\frac{k}{n},\frac{k+1}{n}\right]$. It is well known that Bernstein operators reproduce $e_{i}\left(x\right)=x^{i},$ $i=0,1$ whereas Bernstein-Kantorovich operators reproduce only $e_{0}\left(x\right)=1.$

Recently, Aral et al. Ar-Card-Gar studied a sequence of linear positive operators which generalize the classical Bernstein operators and perform better than $B_{n}$ under sufficient conditions. These operators reproduce the exponential functions $\exp(\mu t)$ and $\exp(2\mu t)$, $\mu>0,$ and are defined by

where

They have close connection with the Bernstein operators that is given by

where for a fixed real parameter $\mu>0$, the exponential function is defined as $\exp_{\mu}\left(x\right)=e^{\mu x}$. We note that the generalization is a special case of the modification introduced by Morigi and Neamtu in Morigi-Nematu .

In order to give the generalization of the operators $K_{n}\left(f\right)$, in Radu2 Păltănea considered the operators $D_{\mu}:C^{1}\left[0,1\right]\rightarrow C\left[0,1\right]$ and $I_{\mu}:C\left[0,1\right]\rightarrow C^{1}\left[0,1\right]$, and defined the operators $K_{n}^{\mu}$ as

where

Using the technique given in Radu2 with the operators in (5), we aim to construct a generalization of the operator $G_{n}$ and call it as generalized Bernstein Kantorovich operator. The construction is described in Section 2. Moreover, certain elementary properties including exponential moments are given in the same section. As we will see in Lemma 2 below, these operators reproduce only the function $\exp\left(2\mu t\right)$, $\mu>0.$ In the same section, we also show that the new operator can be defined with the help of classical Bernstein operator and Bernstein-Kantorovich operator. In Section 3, it is shown that these operators form an approximation process in $C\left[0,1\right]$ by obtaining uniform convergence of them and an estimate in terms of modulus of continuity. In Section 4, a quantitative Voronovskaya type result is presented. In Section 5, we show that the new operators converge to the function in $L_{p,\mu}$-norm. Furthermore, the approximation error of the operators is expressed in terms of an appropriate integral modulus of continuity.

The obtained results show that our technique is more convenient in order to introduce a Kantorovich version of $G_{n}.$

We mention that Kantorovich type operators have been object of the investigation by several mathematicians. Altomare et al. in Altomare-Va introduced a unitary approach to the study of the approximation properties for a large class of Kantorovich type operators including the classical Bernstein-Kantorovich operators and their various generalizations. Also, more results related to the mentioned operators can be found in Altomare2 and Altomare 3 . Various results on Kantorovich-type operators were given in Zhou and Ivanov . It is worth mentioning that Kantorovich type operators in classical sense were considered in the aforementioned more general framework, as well as in the context of the $q$-operators and $\left(p,q\right)$-operators, a very active area of research (see Aral-Gupta , tuncer- aral and the references therein).

Throughout this section, we introduce a new family of operators and present some elementary properties.

For a given $f\in L_{1}\left[0,1\right],$ we define $F_{\mu}\in C\left[0,1\right]$ as

Then, we have

Using the notation

the operators $\widetilde{K}_{n}$ can be represented as

where $B_{n}$  and $K_{n}$ denote Bernstein and Bernstein-Kantorovich operators, respectively. Note that $\widetilde{K}_{n}f(x)$ is an exponential polynomial, based on the representation of Bernstein polynomials.

When we compare with the definition of the operators in Radu2 , the above relationships show that the approach which will be used below is more convenient to apply to the generalized Bernstein operators $G_{n},$ which preserve exponential functions. As we can see in Radu2 , the operator $K_{n}^{\mu}$ preserves good properties of Kantorovich operator $K_{n}$ . So we can expect that $\widetilde{K}_{n}$ also can preserve good properties of $K_{n}.$

Our process depends on the following lemma from Radu2 .

For this family of operators, we give here some of their properties and results.

Using the above limits, we get

and

We shall denote by $C[0,1]$ the space of all real valued continuous functions on the interval $[0,1]$ endowed with the sup-norm $\left\|\cdot\right\|_{\infty}.$

Considering the calculations in the proof of Lemma 4, we get $\left\|\widetilde{K}_{n}e_{0}\right\|_{\infty}=\alpha_{n,\mu}+1$, where $\alpha_{n,\mu}$ is defined as in that proof. So we can say that the linear operator $\widetilde{K}_{n}$ maps $C\left[0,1\right]$ into itself and it is continuous with respect to the sup-norm. Since the sequence $\alpha_{n,\mu}$ is convergent, we get $\left\|\widetilde{K}_{n}\right\|\leq d,$ $d$ is a positive constant.

In the following theorem, we establish the convergence of the operators $\widetilde{K}_{n}$ towards the identity operator.