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.

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 [

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

Our linear and positive operators have the role to approximate functions defined 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:

For each

For a simplified writing, we will use the common notation

For our purposes, relative to the central moments, we require additional conditions. For each

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

The space is endowed with the norm

For bivariate operators we consider the space

These spaces are ordered with respect to inclusion as follows: if

Indeed, based on the first mean value theorem for integration, between

Starting from (

If the function

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

Further, we indicate a truncated variant of operators defined at (

Taking in view the net

Similarly, via the network

For each

Throughout the paper, by

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

Let

Let

(i) By using (

Since

(ii) To be more explicit we consider

By using (

Let

Let

Using the identity

During the previous relations we used notation (

Taking into consideration relation (

We mention that with the help of the same Cauchy-Schwarz inequality, relations (

Lemma

Let

The first statement follows immediately from the definition of the weight

Regarding the second statement, based on (

Applying

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

In the next lemma we have gathered some known properties of Steklov function

Let

Let

(i) For

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

Occurs

Since

In the same manner we show

In the following we denote by

Let

Let

Since

Further, one has

Considering the increases established for

The rate of convergence for

Let

Setting

We establish upper bounds for these three quantities. Relations (

For

One has

Setting

Knowing that the modulus

Let

For any

If

Let

If

Let

Setting

If we show

Further, we prove the previous limit only for

Since

Based on the classical inequality

Consequently, (

Using this relation we have

We establish an upper bound for

Considering (

In presenting these cases, we are looking for one-dimensional linear operators that verify conditions (

(1) Baskakov operators [

For

The study of these operators in polynomial weighted spaces was carried out in [

The univariate truncated operators has been approached in [

(2) Szász operators [

The research of

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

(3)

Also,

In time were carried out