On Convergence Properties of Szasz-Mirakyan-Bernstein Operators of two Variables

In this study, we have constructed a sequence of new positive linear operators with two variable by using Szasz-Mirakyan and Bernstein Operators, and investigated its approximation properties.

For f ∈ C[0, ∞), the Szasz-Mirakyan operators are defined by where q n,k (x) := e −nx (nx) k k! , Some approximation properties of S n f can be found in works [3,16,8] and references therein. Now, taking into account the Bernstein polynomials and the Szasz-Mirakyan operators, we introduce some positive linear operators for functions of two variables.
where q n,m and p ν,k are defined in (4) and (2), respectively, and ν := ν(m, n) is a natural double sequence which is tends to infinity when m, n → ∞ and ν(0, 1) := 1. It could be seen easily that the operators L n are linear and positive. They are called as Szasz-Mirakyan-Bernstein operators. For y ∈ [0, ∞) and f ∈ C(D), let us define the function f y ∈ C[0, 1] by f y (x) := f (x, y). With this notation, the positive linear operators L n given by (5) can be written in the form The function L n f defined by (6) is became the ν(0, n)th Bernstein polynomial for the function f 0 , on the set Some positive linear operators for the functions of two variables are introduced and investigated their approximation properties by the authors in [13,5,11,15,14,12,10,4].
In this study we investigate some approximation properties of the sequence of positivie linear operators L n defined by (5) in the space of functions which are continuous on compact subsets of D, and the order of approximation by means modulus of continuity. For any nonnegative number R, let

Some Notations and Auxiliary
Let us denote the space of real valued continuous functions of two variables on D R equipped with the uniform norm: where the maximum is taken for all r 1 , (8) and with respect to y is defined by We shall need some well known properties of full and partial modulus of continuity: for any λ ≥ 0. lim δ→0 ω(f ; δ) = 0, when f is uniformly continuous.
For M > 0 and 0 < α ≤ 1, the class of the functions f ∈ C(D) satisfying the relation is called a Lipschitz class and denoted by Lip M (α).

Let x be fixed point in R. Let us define the moment functions E i by
The following lemma follows by the definition of Szasz-Mirakyan Operators (3),

Approximation properties of L n on C(D R )
Let L n be the positive linear operators defined by (5) with the condition ν = O(m), m → ∞, that is, there exists natural sequences α n and β n such that α n ≤ ν/m ≤ β n . In this section we give some classical approximation properties of the operators L n . Let e j , j = 0, 1, 2, 3 be the test functions defined by By simple calculations, we get the following lemma.
The following theorem gives the Baskakov-type theorem (see [2]) to get uniform approximation to the functions in C(D R ) satisfying some additional conditions by the sequence of the positive linear operators L n .
Proof. Let f ∈ C ρ (D) with ρ(x, y) = 2 + y 2 and (x, y) ∈ D R . By the continuity of f at the point (x, y), for any positive number ϵ, there exists a number δ > 0 such that for all (s, where M 1 > 0 is a constant depending on f and R. Therefore, we obtain that for (s, Applying the operators L n to the last inequality, we get ≤ ϵL n e 0 (x, y) + M 1 δ 2 (L n e 3 (x, y) − 2xL n e 1 (x, y) −2yL n (e 2 )(x, y) + (x 2 + y 2 )L n e 0 (x, y)) + ∥f ∥ C(DR) |L n e 0 (x, y) − e 0 (x, y)| .
By using the assumptions (12), the desired assertion (13) is proved. The rates of convergence of the sequence {L n f } to f by means of full and partial modulus of continuity are given in the following theorem. where Proof. For (x, y) ∈ D R , y) . (16) First, using the inequality which is obtained from (10), then applying the Cauchy-Schwartz inequality, finally, using Lemma 3.1, the inequality (16) gives the inequality (14). By similar arguments with the inequality we get the inequality (15).

Voronovskaya-Type Theorem
From now on we make the assumption: ν(m, n) = (m + 1)n, for all m + 1, n ∈ N. Let (x, y) be a fixed point in R 2 . Let us define the moment functions E i,j by Using Lemma 2.1 and Lemma 2.2, we get the following lemma.

By simple calculations, it can be obtained that
where 1 ≤ c s ≤ (s + 1)!, and hence for fixed x ∈ (0, ∞), We shall denote by f ′ x , f ′′ xx the partial derivatives of f . Let us denote the space of functions have continuous partial derivatives up to order 2 on D, by C 2 (D).

Remark 3.10
The method of the proof of the Theorem 3.9 does not work to prove of convergence of partial derivative of L n f with respect to x.