Abstract

We introduce positive linear operators which are combined with the Chlodowsky and Szász type operators and study some approximation properties of these operators in the space of continuous functions of two variables on a compact set. The convergence rate of these operators are obtained by means of the modulus of continuity. And we also obtain weighted approximation properties for these positive linear operators in a weighted space of functions of two variables and find the convergence rate for these operators by using the weighted modulus of continuity.

1. Introduction and Preliminaries

Let and . The well-known Chlodowsky polynomial of degree , denoted by , is where is a sequence of positive numbers with the properties

Some generalization of these polynomials in the one-dimensional case may be found in [1, 2]. Recently, in [3, 4] some approximation theorems for different positive linear operators in the space of continuous functions for one variable case are done.

Let and . The modified Szász-Mirakjan operators denoted by are where and  are given increasing and unbounded sequences of positive numbers such that In [5], Walczak introduced the modified Szász-Mirakjan operators in the polynomial weighted spaces of functions of one and two variables. He investigated approximation properties of modified Szász-Mirakjan operators in the weighted space of continuous functions of two variables for which is uniformly continuous and bounded on , where is a polynomial weight function. In [6], İspir and Atakut studied the theorems on convergence of , defined by (1) to , in the weighted spaces of continuous and obtain the convergence rate of the operators by using the weighted modulus of continuity on all positive semiaxis. They also study the modified Szász-Mirakjan operators in the polynomial weighted spaces of functions of two variables.

In this study, inspired by the operators (1) and (3), we consider certain linear positive operators of functions of two variables. To this end, let , where and define the linear positive operators in the following way: with and .

If we take and in (5), then the operators reduce to the Chlodowsky and Szász type operators, respectively. Approximation of functions of one or two variables by some positive linear operators in weighted spaces may be found in [1, 2, 5–8].

In this paper, firstly we study some approximation properties of the sequence of linear positive operators given by (5) in the space of continuous functions on compact set and find the order of this approximation using full and partial modulus of continuity. We finally investigate the convergence of the sequence of linear positive operators , defined on a weighted space of functions of two variables, and find the rate of this convergence by means of weighted modulus of continuity.

Now we give some basic definitions which we will use in our theorems.

Let and let be the set of all functions defined on the real axis satisfying the condition , where is a constant depending only on . is a normed space with the norm Let with being the subspace of all continuous functions belonging to . Then the weighted modulus of continuity of defined by Let . The full modulus of continuity of is defined as follows: Partial modulus of continuity with respect to and is defined by respectively. It is known that the full modulus of continuity and the partial modulus of continuity satisfy the following properties:

2. Approximation Properties on

In this section we give some classical approximation properties of the operators on the compact set .

Let . Then by simple calculations, one can obtain the following lemmas.

Lemma 1. Let be defined by (5). Then one has for all ,

From Lemma 1, we obtain following lemmas.

Lemma 2. If the operator is defined by (5), then for all and ,

Lemma 3. If the operator is defined by (5), then for all and sufficiently large ,

The approximation theorem for functions of two variables is as follows.

Theorem 4 (see [8]). If is a sequence of linear positive operators satisfying the conditions then for any function , which is bounded in , where is a compact set.

In the following theorem we show that the linear positive operator which we define by (5) converges to uniformly with the help of Theorem 4 given by Volkov in [8].

Theorem 5. Let , then the operators defined by (5) converge uniformly to on as .

Proof. From (13)–(17) and conditions (2) and (4), we have Applying Theorem 4, we obtain the desired result.

The following theorem gives the convergence rate of the sequence of linear positive operators to on , by means of partial and full modulus of continuity.

Theorem 6. Let ; then the following inequalities hold: where , ,; , , and are given by (9), (10), and (11), respectively.

Proof. From (5) and (13) and using the property of the partial modulus of continuity, we can write applying the Cauchy-Schwartz inequality, we have and using equalities (16) and (17) and choosing , , we obtain inequality (20). If we use inequality we can easily obtain inequality (21). Thus, the proof of theorem is completed.

Below in Example 7, we will try to see the agreement of our linear positive operator with using different values of and , whereas in Example 8, we also compare with for another function .

Example 7. For , , and , ; the convergence of to is illustrated in Figure 1.

Figure 1 

The convergence of to for and .

Example 8. For and , , , the convergences of and to will be illustrated in Figure 2.

Figure 2 

The convergence of and to for .

3. Weighted Approximation Properties of

In this section, we investigate the convergence properties of the operators given by (5) in the weighted spaces of continuous functions on positive semiaxis by using weighted Korovkin type theorem. The Korovkin type theorem in weighted spaces for linear positive operators , acting from to , has been proved by Gadjiev [9], Gadjiev and Hacısalihoğlu [10].

Theorem 9 (see [10]). There exists a sequence of positive linear operators , acting from to , satisfying the conditions Then there exists a function for which

Theorem 10 (see [9]). Let be a sequence of positive linear operators acting from to and let be a continuous function for which The conditions (31)–(34) imply for all .

Now, we give the following results in [9, 10] which we use in the proofs of our main theorems.

Theorem 11. Let be the sequence of linear positive operators defined by (5). Then for all one has where and is the continuous function, satisfying the condition (36).

Proof. Firstly let us show that is acting from to . Using (13), (16), and (17), we write where . Since as , there is a positive constant such that for all natural numbers and . Hence we have From [10], we have . If we prove that the conditions (31)–(34) are satisfied then the proof is completed by Theorem 10. By using (13), (14), and (15), we get (31)–(33). Finally, using (16) and (17), we have where , and as . Thus we obtain the desired result.