#### Abstract

The present paper deals with approximation properties of *q*-Szász-Mirakyan-Kantorovich operators. We construct new bivariate generalization by -integral and these operators' approximation properties in polynomial weighted spaces are investigated. Also, we obtain Voronovskaya-type theorem for the proposed operators in polynomial weighted spaces of functions of two variables.

#### 1. Introduction

In the past two decades, -calculus has gained popularity in the construction of linear approximation processes. Lupaş [1] and Phillips [2] defined generalizations of the Bernstein operators called -Bernstein operators. Then, as Phillips has done for Bernstein operators, the authors introduced modifications of the other important operators based on the -integers, for example, -Meyer-König operators [3, 4], -Bleimann, Butzer, and Hahn operators [5, 6], -Szász-Mirakyan operators [7–9], -Baskakov operators [10, 11].

On the other hand, Stancu [12] first introduced new linear positive operators in two- and several dimensional variables. Recently, Barbosu [13] introduced a Stancu-type generalization of two-dimensional Bernstein operators based on -integers and called them bivariate -Bernstein operators. Doru and Gupta [14] constructed a bivariate generalization of the Meyer-König and Zeller operators based on the -integers. Agratini [15] presented two-dimensional extension of some univariate positive approximation processes expressed by series.

All the above mentioned new operators motivate us for current work. In this paper, we firstly extend the -Szász-Mirakyan-Kantorovich operators to the case of bivariate functions. Then these operators' approximation properties in polynomial weighted spaces are investigated. Also we obtain Voronovskaya-type theorem for the proposed operators in polynomial weighted spaces of functions of two variables.

Now we recall some definitions about -integers. For any nonnegative integer , the -integer of the number is defined by where is a positive real number. The -factorial is defined as Two -analogues of the exponential function are given as The following relation between -exponential functions and holds: The -derivative of a function , denoted by , is defined by Also, it is known that .

The -integral of the function over the interval is defined by If is integrable over , then Generally accepted definition for -integral over an interval is In order to generalize and spread the existing inequalities, Marinković et al. considered new type of the -integral. So, the problems which ensue from the general definition of -integral were overcome. The Riemann-type -integral [16] in the interval was defined as This definition includes only point inside the interval of the integration.

Details of -integers can be found in [17].

#### 2. Construction of the Bivariate Operators

For and , we now define new operators that we call the -Szász-Mirakyan-Kantorovich operators of functions of two variables as follows: where and is a -integrable function, so the series in (11) converges. It is clear that the operators given in (10) are linear and positive. For the operator , if is a -integrable function and , , then Now, in order to obtain approximation properties of proposed operators, we give some auxiliary results. For a fixed , by the -Taylor theorem [18], we write where Choosing and taking into account we get for that Similarly, choosing and taking into account we obtain for that Also, using we have for that

Lemma 1. *Let and . One has
*

*Proof. *Using and from , we can write
is obtained from the above identity (16).

Now, taking and since , we get from the linearity of that
From this, applying (16) and (18), we have
Similarly, we write that
Now, taking and from
we have
From this, using we get
Replacing by and by in the abovementioned, we obtain
Equation (20) implies
Similarly, we write that
So, the proof is completed.

Similarly, given by the proof of Lemma 1, we calculate and , shortly. Since we write Using , we have Then, we rewrite Finally, we have Since we obtain

#### 3. Approximation Properties in Polynomial Weighted Spaces

For bivariate operators, the space is considered as follows: associated with the weighted function , . The weight is defined as , . The norm of this space is denoted by and is defined by Now, we give some useful results given by Agratini [15].

For each , define the function by , , . For the one-dimensional operator , , and for each , a polynomial exists such that

Theorem 2 (see [15]). *Consider . For any , given by
**
the operator verifies
*

Theorem 3 (see [15]). *Let . For any , the operator given by (42) satisfies**, where is given by
**
with the polynomials , , being indicated at (41), and is a suitable constant.*

Theorem 4 (see [15]). *Let . Let the operator , be defined by (42). For any the pointwise convergence takes place
**
If , are compact intervals included in , then (46) holds uniformly on the domain .*

In the latter paper, we use the weight function instead of and instead of the space , the space associated with the weighted function is used. We denote the norm of this space by .

Lemma 5. *The operator , , , given by (10) verifies
*

*Proof. *Since and by Lemma 1, we have
So, inequality (47) is proved. Since the operator is linear and positive and by using (47)
we obtain inequality (48).

The Steklov function associated with is given as follows: where .

The modulus of smoothness function associated with any function is given by One can see that The following inequalities verify where . In order to justify these inequalities, one can see that

Theorem 6. *For any and , the operator given by (10) satisfies
**
where , .*

* Proof. *For any , we can write
Inequalities (48) and (53) imply that
Let be the class of all functions in which partial derivatives belong to . Since and given by (10) is linear and monotone, we get
Then, by definition of norm and the first mean value theorem for integration, we have
Following the same way, one finds
Using inequalities (59), (61), and (60), since is linear and monotone, we get
From Lemmas 1 and 5 and (36), we can write the following:
where
is a polynomial of degree . Then, by inequalities (54), we have
Finally, we write from (53)
If we go back to (57) and take , , then the proof is completed.

We replace and in (10) by sequences , so that So, as . Knowing that modulus of smoothness function satisfies the property , from Theorem 6, we deduce the following result.

Theorem 7. *Let and let , be sequences in the interval satisfying (67). Let the operator given by (10) and -integrable function. For any the pointwise convergence takes place
**
If , are compact intervals included in , then (68) holds uniformly on the domain .*

#### 4. Voronovskaya-Type Theorem

We will prove the Voronovskaya-type theorem.

Theorem 8. *Let , be sequences in the interval satisfying (67). Suppose that is the class of all functions in which the second partial derivatives belong to and -integrable function. Then, for every , one has
*

*Proof. *Let and -integrable function and let be fixed point. Then, by the Taylor formula, we can write
where , belongs to , and for . From the linearity of , we have