Research Article | Open Access

# Bivariate Positive Operators in Polynomial Weighted Spaces

**Academic Editor:**Sung Guen Kim

#### Abstract

This paper aims to two-dimensional extension of some univariate positive approximation processes expressed by series. To be easier to use, we also modify this extension into finite sums. With respect to these two new classes designed, we investigate their approximation properties in polynomial weighted spaces. The rate of convergence is established, and special cases of our construction are highlighted.

#### 1. Introduction

The approximation of functions by using linear positive operators is currently under research. Usually, two types of positive approximation processes are used: the discrete, respectively, continuous form. In the first case, they often are designed through a series. Since the construction of such operators requires an estimation of infinite sums, this restricts the operator usefulness from the computational point of view. In this respect, in order to approximate a function, it is interesting to consider partial sums, the number of terms considered in sum depending on the function argument. Roughly speaking, these discrete operators are truncated fading away their “tails.” Thus, they become usable for generating software approximation of functions. Among the pioneers who approached this direction we mention Gróf [1] and Lehnhoff [2]. In the same direction a class of univariate linear positive operators is investigated in [3].

This work focuses on a general bivariate class of discrete positive linear operators expressed by infinite sums. This class acts in polynomial weighted spaces of continuous functions of two variables defined by , where . By using a certain modulus of smoothness we give theorems on the degree of approximation. Further, we replace the infinite sum by a truncated one, and we study the approximation properties of the new defined family of operators. Compared to what has been done so far, the strengths of this paper consist in using general classes of two-dimensional discrete operators, implying an arbitrary network of nodes. Finally we present some particular classes of operators that can be obtained from our family.

#### 2. The Operators

Our linear and positive operators have the role to approximate functions defined on . Therefore, on this domain we define for every a net of form , where and . Set , and stands for the space of all real-valued continuous functions on .

Products of parametric extensions of two univariate operators are appropriate tools to approximate functions of two variables. For this reason, the starting point is given by the following one-dimensional operators: where , are nonnegative functions belonging to , , such that the following identities take place. These conditions mean that the operators and preserve the monomial , , a property often seen at classical linear positive operators.

For each , define the function by , . Also, for each , set representing the th central moment of the specified operators.

For a simplified writing, we will use the common notation , where for all or for all .

For our purposes, relative to the central moments, we require additional conditions. For each , as a function of is bounded by a polynomial of degree at most . Moreover, is bounded with respect to . These requirements can be brought together and redrafted in the following way: for each , a polynomial exists such that

Apparently is a tough condition, but the examples that we give in the last section show that it is carried out by different classes of operators.

In what follows we specify the function spaces in which the operators act.

For univariate operators , we consider the space , fixed, consisting of all real-valued functions continuous on such that is uniformly continuous and bounded on , where the weight is defined as follows:

The space is endowed with the norm , .

For bivariate operators we consider the space associated to the weighted function , . The norm of this space is denoted by and is defined by

These spaces are ordered with respect to inclusion as follows: if , then . Moreover, for any . Examining the weight , it is easy to verify the inequality for any and .

Indeed, based on the first mean value theorem for integration, between and , a point exists such that

Starting from (1), for each we introduce a linear positive operator in polynomial weighted space as follows:

If the function can be decomposed in the following manner , , then one has

For example, we get .

In order to present the rate of convergence for our bivariate operators, we use a modulus of smoothness associated to any function belonging to . It is given by the formula where for and belonging to . Alternative notation is . More information about moduli of smoothness can be found in the monograph [4].

Further, we indicate a truncated variant of operators defined at (10). Let , be strictly increasing sequences of positive numbers such that

Taking in view the net , we partitioned the set into two parts

Similarly, via the network , we introduce and .

For each and any we define the linear positive operators

#### 3. Auxiliary Results

Throughout the paper, by we denote different real constants, in the brackets specifying the parameter(s) that the indicated constant depends.

At first we collect some useful results relative to the one-dimensional operators where or .

Lemma 1. *Let , and let the weight be given by (5). The operator satisfies
**
where and .*

*Proof. *Let and be fixed.

(i) By using (5), (2), (3), and (4) we can write

Since , the previous expression is bounded with respect to , and inequality (17) follows.

(ii) To be more explicit we consider

By using (17) we obtain the first inequality of the relation (18). Further, applying , the second inequality is proved.

Lemma 2. *Let , and let the weight be given by (5). For any the operator satisfies
**
where and . The polynomials , , are introduced by (4).*

*Proof. *Let and be fixed. Using the identity
we can write the following:

During the previous relations we used notation (3) and hypothesis (4). Considering the significance of , relation (21) follows. Taking into consideration relation (1), we apply the Cauchy-Schwarz inequality, and this allows us to write
Relations (17) and (21) imply (22).

We mention that with the help of the same Cauchy-Schwarz inequality, relations (2) and (4) lead us to the following inequality:

Lemma 1 leads to the following result.

Lemma 3. *Let . For any , the operator given by (10) verifies
*

*Proof. *The first statement follows immediately from the definition of the weight and relations (11), (17).

Regarding the second statement, based on (10), we get

Applying and taking into account (27), one obtains (28).

In our investigation we appeal to the Steklov function. This can be used to approximate continuous functions by smoother functions. The Steklov function associated with is given as follows: where and . By using (13) we deduce

In the next lemma we have gathered some known properties of Steklov function , where . These properties establish connections between and the modulus indicated at (12). For the sake of completeness we present the proofs of these inequalities.

Lemma 4. *Let belong to , and let be defined by (30). The following relations take place:
**
where , and is defined by (12).*

*Proof. *Let and be arbitrarily fixed.

(i) For and we deduce
On the other hand,
Keeping in mind (31), we consequently obtain
and (32) is completed.

(ii) We justify only the first inequality, and the second inequality can be proven in the same manner.

Occurs
(see (13)). Further, we can write

Since , it is clear that , and we obtain

In the same manner we show . Returning at (38), the proof is ended.

In the following we denote by the space of all functions having the first order partial derivatives such that the functions , and belong to .

Lemma 5. *Let . If , then for any the operator given by (10) verifies
**
where
**
the polynomials , , being indicated at (4).*

*Proof. *Let and be arbitrarily fixed. Since , for any we can write

Since is linear monotone and reproduces the constants, from the previous identity we obtain

Further, one has
see (8). Applying and by using successively (11), (26), (17), and (22) we have
where is a suitable constant. Following the same pathway, we find

Considering the increases established for and returning to the relation (43), the inequality (40) is completely proven.

#### 4. Main Results

The rate of convergence for operator will be read as follows.

Theorem 6. *Let . For any , the operator given by (10) satisfies
**, where is given by (41) and is a suitable constant.*

*Proof. *Setting
we can write

We establish upper bounds for these three quantities. Relations (28) and (32) imply that

For we use Lemma 5 choosing ; see definition (31).

One has
(see also (33)). Finally, inequality (32) implies

Setting and coming back to (49), we can affirm that a certain constant exists such that (47) holds.

Knowing that the modulus enjoys the property , from Theorem 6 we deduce the following result.

Theorem 7. *Let , and let the operators , , be defined by (10).**For any the pointwise convergence takes place
**If are compact intervals included in , then (53) holds uniformly on the domain .*

Theorem 8. *Let , and let the operators , , be defined by (16). For any the pointwise convergence takes place
**If are compact intervals included in , then (54) holds uniformly on the domain .*

*Proof. *Let be arbitrarily fixed. Taking in view the partitions of (see (15)), we use the following decomposition:

Setting
we can write

If we show , , then, based on (53), our statement (54) follows, and the proof is ended.

Further, we prove the previous limit only for , other two following similar routes.

Since , a constant exists such that

Based on the classical inequality , , , , we deduce

Consequently, (58) implies that
where .

Using this relation we have

We establish an upper bound for . Since , clearly
and we can write
(see (26) and (4)). Arguing similarly we get

Considering (14), relation (61) leads to the claimed result.

#### 5. Particular Cases

In presenting these cases, we are looking for one-dimensional linear operators that verify conditions (2) and (4).

(1) Baskakov operators [5] are given by the formula where , hence (2) fulfilled. Since reproduces the monomial , , its first central moment is null.

For , the th central moment is given as follows [6, Lemma 4]: where if is odd, if is even, and are positive coefficients bounded with respect to . In particular, is a polynomial of degree without a constant term. These properties ensure that condition (4) is achieved.

The study of these operators in polynomial weighted spaces was carried out in [6]. Choosing in (10) and we obtain the Baskakov operator for functions of two variables. The net is . Our results indicated at (47) and (53) are identified with the results established by Gurdek et al. [7, Equations , ].

The univariate truncated operators has been approached in [8]. The truncated version specified in (16) coincides with the operators studied by Walczak [9, Equation ]. In this case from (15) becomes . Here indicates the largest integer not exceeding .

(2) Szász operators [10] are of the form: One has and . The central moments have the following form: where are constants; see, for example, [11, Equations -]. For this class, conditions (2) and (4) are achieved.

The research of , , operators in polynomial weighted spaces has appeared in [6]. The truncated univariate Szász operators and another extension to functions of two variables in weighted spaces have been considered in [2] and [12], respectively. In the latter paper instead was used the weight , .

Our theorems of the previous section lead us to two-dimensional versions of genuine Szász operators and of their truncated form. In this case the net is .

The next example comes from the world of Quantum Calculus which, in the past two decades, has gained popularity in the construction of linear approximation processes. We choose a -analogue of Szász-Mirakjan operators recently introduced and studied by Mahmudov [13].

(3) -Szász-Mirakjan-Mahmudov operators. Let and . For one defines the operator where . We recall the standard notations in -calculus. For consider

Also, and . In [13] it was proved that and is a linear positive operator from to for any . Withal, for all moments , , explicit formulas were given as follows: where represent -analogue of Stirling numbers; see [13, Lemma 2.6]. One can see that is a polynomial in of degree without a constant term, and it is bounded with respect to . These properties are transferred to the central moments , and consequently (4) takes place, provided to replace with . To construct two-dimensional operators of the form (10) and (16) we choose , , and the network will be

In time were carried out -analogues of these operators not only for but for the case ; see, for example, [14, 15]. Extensions of these classes of operators by our method also work there.

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#### Copyright

Copyright © 2013 Octavian Agratini. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.