THE KANTOROVICH FORM OF SOME EXTENSIONS FOR SZ´ASZ-MIRAKJAN OPERATORS

. Recently, C. Mortici deﬁned a class of linear and positive operators depending on a certain function ϕ . These operators generalize the well known Sz´asz-Mirakjan operators. A convergence theorem for the deﬁned sequence by the mentioned operators was given. Other interesting approximation properties of these generalized Sz´asz-Mirakjan operators and also their bivariate form were obtained by D. B˘arbosu, O. T. Pop and D. Micl˘au¸s. In the present paper we are dealing with the Kantorovich form of the generalized Sz´asz-Mirakjan operators. We construct the Kantorovich associated operators and then we establish a convergence theorem for the deﬁned operators. The degree of approximation is expressed in terms of the modulus of continuity. Next, we construct the bivariate and respectively the GBS corresponding operators and we establish some of their approximation properties.

Remark 1.1.Similar generalization of this type are the operators defined and studied by Jakimovski and Leviatan [15] or the operators defined by Baskakov in 1957 (see, e.g., the book [2], subsection 5.3.11, p. 344, where they are attributed to Mastroianni).The main goal of the present paper is to construct the Kantorovich type operators, associated to the ϕ-Szász-Mirakjan operators (1.1).
Using the method of parametric extensions [7], [12], the bivariate ϕ 1 ϕ 2 -Szász-Mirakjan-Katorovich operators are constructed and some of their approximation properties are established.The last section is devoted to the construction of the associated GBS ϕ 1 ϕ 2 -Szász-Mirakjan-Kantorovich operators and to study some of their approximation properties.
In order to obtain the convergence of the sequence (ϕK n ) n∈N we need the following: Lemma 2.2.Let e j (x) = x j , j = 0, 1, 2 be the test functions.The ϕ-Szász-Mirakjan-Kantorovich operators satisfy the following relations: and next, by differentiation For the test functions e 0 , e 1 , e 2 , the following identities Recall that, the images of test functions by the operators ϕ-Szász-Mirakjan [19] are Next, from the well known Bohman-Korovkin Theorem one arrives to the desired result.
In order to obtain the degree of approximation of f ∈ C 2 ([0, +∞[), by means of the ϕ-Szász-Mirakjan-Kantorovich operators, let us to recalling some known results, concerning the modulus of continuity.
Let I ⊂ R be an interval, C(I) be the set of real-valued functions continuous on I, B(I) be the set of real-valued functions bounded on I and C B (I) be the set of real-valued functions continuous, bounded on I.
is called the modulus of continuity (the first modulus of smoothness) of the function f .
Remark 2.5.Its properties can be found in the monograph [1].
In 1968, O. Shisha and B. Mond [21] established the following: Theorem 2.6.[1] Let L : C(I) → B(I) be a linear positive operator and let the function ϕ be defined by If f ∈ C B (I), then for any x ∈ I and δ > 0 the following For obtaining the degree of approximation of f ∈ C 2 ([0, +∞[), on any compact interval [a, b] ⊂ [0, +∞[, by means of the ϕ-Szász-Mirakjan-Kantorovich operators we need the following: Lemma 2.7.Let the function ϕ x be defined by (2.6) Proof.Because the operators ϕ-Szász-Mirakjan-Kantorovich are linear, then taking the definition of ϕ x into account, we get Next, one applies Lemma 2.2.
In the following, we suppose that the analytic function ϕ : R →]0, +∞[ satisfy the conditions (2.3) and taking these conditions into account, it results lim n→∞ ϕ (2) (nx) Then we suppose that, there exists 0 < γ ≤ 1, so that for any x ∈ [0, +∞[ and where β 2 is a function, and any δ > 0, the ϕ-Szász-Mirakjan-Kantorovich operators verify the inequality where Proof.The relation (2.8) yields from (2.5), if we choose δ = b−a √ n γ and if we take the definition of limit and relation (2.7) into account.
It is immediately the result contained in the following: for any x, y ∈ [0, +∞[ and m, n ∈ N.
Lemma 3.2.Let e ij (x, y) = x i y j , i, j ∈ N 0 , i + j ≤ 2 be the test functions.
The operators (3.3) verify the following identities: The operators (3.3) satisfy Proof.Taking the linearity of operator (3.3) and the definitions of the functions ϕ x , ϕ y into account, one obtains Next, applying Lemma 3.2.one arrives to (3.4) and (3.5).
The function is called modulus of continuity of the bivariate function f .Its properties are similar with the properties of the modulus of continuity for univariate functions [3], [7].
It is known from [7], [24] the following analogous of Shisha-Mond Theorem for the bivariate case: holds.
Then, for every function  In this section we shall construct the ϕ 1 ϕ 2 -Szász-Mirakjan-Kantorovich operator associated to a B-continuous function.
A function f : We shall use the function sets: is called the GBS (Generalized Boolean Sum) operator associated to L.
Next we recall the Korovkin type theorem for B-continuous functions due to C. Badea, I. Badea and H. H. Gonska in [5].

Theorem 4 . 5 . [ 4 ]
Let L : C b (I × J) → B(I × J) be an linear and positive operator reproducing constants and let U : C b (I × J) → B(I × J) be the GBS associated operator.