Bivariate Positive Operators in Polynomial Weighted Spaces

and Applied Analysis 3 For each (m, n) ∈ N × N and any f ∈ C p,q (R2 + ) we define the linear positive operators


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 R + × R + , where R + = [0, ∞).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.
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 (R + ), (, ) ∈ N 0 × N 0 , such that the following identities take place.These conditions mean that the operators   and   preserve the monomial  0 ,  0 () = 1, a property often seen at classical linear positive operators.
For each  ∈ R + , define the function   by   () =  − ,  ∈ R + .Also, for each  ∈ N, set representing the th central moment of the specified operators.
For a simplified writing, we will use the common notation   , where   =   for all  ∈ N or   =   for all  ∈ N.
For our purposes, relative to the central moments, we require additional conditions.For each  ∈ N, M  (  ; ) as a function of  is bounded by a polynomial of degree at most .Moreover,  /2 M  (  ; ⋅) is bounded with respect to .These requirements can be brought together and redrafted in the following way: for each  ∈ N, a polynomial Γ  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   (R + ),  ∈ N 0 fixed, consisting of all real-valued functions  continuous on R + such that    is uniformly continuous and bounded on R + , where the weight   is defined as follows: The space is endowed with the norm ‖ ⋅ ‖  , ‖‖  = sup ≥0   ()|()|.
In order to present the rate of convergence for our bivariate operators, we use a modulus of smoothness associated to any function  belonging to  , (R 2 + ).It is given by the formula where for (, ) and (, V) belonging to  , (R 2 + ).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 (  ) ≥1 , (V  ) ≥1 be strictly increasing sequences of positive numbers such that lim Taking in view the net Δ 1, , we partitioned the set N 0 into two parts Similarly, via the network Δ 2, , we introduce (, V  ) and (, V  ).

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 onedimensional operators   where   =   ( ∈ N) or   =   ( ∈ N).

Lemma 1.
Let  ∈ N 0 , and let the weight   be given by (5).The operator   satisfies where  ≥ 0 and  ∈ N.
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  ∈ (R 2 + ) is given as follows: where ℎ > 0 and  > 0. By using (13) we deduce In the next lemma we have gathered some known properties of Steklov function  ℎ, , where  ∈  , (R 2 + ).These properties establish connections between  ℎ, and the modulus   indicated at (12).For the sake of completeness we present the proofs of these inequalities.Lemma 4. Let  belong to  , (R 2 + ), and let  ℎ, be defined by (30).The following relations take place: and ( 32) is completed.
(ii) We justify only the first inequality, and the second inequality can be proven in the same manner. Occurs |Δ  1 , 2 (, )|, and we obtain In the same manner we show  2 ≤ (1/ℎ)  (ℎ, ).Returning at (38), the proof is ended.
In the following we denote by  1 , (R 2 + ) the space of all functions  : R 2 + → R having the first order partial derivatives such that the functions /, /, and  belong to  , (R 2 + ).
, then for any (, ) ∈ N × N the operator  , given by (10) where (, ) is a suitable constant.Following the same pathway, we find Considering the increases established for  1 ,  2 and returning to the relation (43), the inequality (40) is completely proven.

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

Particular Cases
In presenting these cases, we are looking for one-dimensional linear operators that verify conditions ( 2) and ( 4).
For  ≥ 2, the th central moment is given as follows [6, Lemma 4]: where   = 1 if  is odd,   = 0 if  is even, and  ,, are positive coefficients bounded with respect to .In particular, M  (  ; ) is a polynomial of degree  without a constant term.These properties ensure that condition (4) is achieved.
Here [] indicates the largest integer not exceeding .
The research of   ,  ∈ N, 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 , (, ) = 1 +  2 +  2 .
Our theorems of the previous section lead us to twodimensional versions of genuine Szász operators and of their truncated form.In this case the net is Δ , = (/, /) ,≥0 .
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]. ( where   () = ∏ ∞  = 0 (1 + ( − 1)(/ +1 )).We recall the standard notations in -calculus.For  ∈ N consider (70) Also, [0] = 0 and [0]! = 1.In [13] it was proved that  ,  0 =  0 and  , is a linear positive operator from   (R + ) to   (R + ) for any  ∈ N 0 .Withal, for all moments  ,   ,   () =   , explicit formulas were given as follows: ( ,   ; ) = In time were carried out -analogues of these operators not only for  > 1 but for the case  ∈ (0, 1); see, for example, [14,15].Extensions of these classes of operators by our method also work there.