ON SZ´ASZ-MIRAKYAN TYPE OPERATORS PRESERVING POLYNOMIALS

. In this paper, a modiﬁcation of Sz´asz-Mirakyan operators is studied [1] which generalizes the Sz´asz-Mirakyan operators with the property that the linear combination e 2 + αe 1 of the Korovkin’s test functions e 1 and e 2 are reproduced for α ≥ 0 . After providing some computational results, shape preserving properties of mentioned operators are obtained. Moreover, some estimations for the rate of convergence of these operators by using diﬀerent type modulus of continuity are shown. Furthermore, a Voronovskaya-type formula and an approximation result for derivative of operators are calculated.


INTRODUCTION
Approximation theory is based on finding the best approximation of a function by polynomials or other type of simple functions. For many years, there have been lots of improvements about the approximation theory. In 1853, Russian mathematician Chebyshev focused on this matter. However, the big step was in 1885 when Karl Weierstrass [13] presented the theorem on approximation.
In approximation theory, positive linear operators play an essential role. The study of approximation sequences of linear positive operators was started at the beginning of the 1950s. One of the most important positive linear operators is Bernstein polynomials. Bernstein polynomials on the space C [0, 1] are defined by Up to now, there have been lots of extensions and modifications of Bernstein polynomials. It was King [9] who constructed Bernstein type linear positive operator defined on C [0, 1] having an approximation order better then the classical operators such that they reproduce the test function e 0 and e 2 . This operator has an approximation order better than the classical operators on 0, 1 3 .
They found the shape preserving properties for B n,α and worked on the comparison with Bernstein polynomials. Furthermore, they also showed that the sequences B n,α for α ≥ 0 are an approximation process. Besides, for different Bernstein Durrmeyer type operators, similar results were given in [5]. Szász-Mirakyan operators are the generalizations of Bernstein polynomials on the interval [0, ∞) which are defined by Notice that, for all functions f : C [0, ∞) → R the series at the right hand side convergences absolutely. There are many papers about different type of generalizations of Szász-Mirakyan operators where the basic properties of approximation are analyzed. In the recent years, the number of the articles related to this fact has increased (see [2], [7], [3] and [10]).
In order to furnish better error estimation in a certain sense than classical Szász-Mirakyan operators, in [7], authors defined the following operators, Note that, both of the results of [4] and [9] were obtained in finite intervals. On the infinite interval, using similar technique, authors introduced Szász-Mirakyan operators King type by reproducing e 1 and e 2 [3].
In this paper, as in [4] for Bernstein polynomials, we consider a similar modification of the Szász-Mirakyan modified operators given in [1] using the function r n,α which is defined by {S n,α : Noting the fact that when n → ∞, r n,α → x, S n,α f reduces to the classical Szász-Mirakyan operator. That is, classical Százs-Mirakyan operators turn out to be a limit element of S n,α f and also if we take α = 0, the sequence D * n f of operators appears which is introduced in [7].
In [1], authors showed the approximation properties of Szász-Mirakyan modified operator. In the lights of the definition of the operator of (2), different kinds of results which are related to the mentioned operator are obtained.
The organization of the paper is as follows: In section 2, shape preserving properties of the Szász-Mirakyan modified operators are investigated. Using the convexity and generalized convexity, relations between the given functions, S n,α f and S n f operators are obtained. Then, the results of Voronovskaya-type theorem are given. Moreover, the rate of convergence properties of this operator for two different modulus of continuities are studied and a theorem which is satisfied by derivative of S n,α f is given.
Throughout the paper, we use following definition and notations.

Definition 1. [5]
A function f ∈ C k [0, ∞) (the space of k times continuously differentiable functions) is said to be τ convex of order k ∈ N whenever The classical convexity is obtained for τ = e 1 and k = 2.
The function space C 2 [0, ∞) is defined by, where k f is a constant depending on f and For α ≥ 0 and n > 0, using the results for the Szász-Mirakyan operator (2), it is found that In view of the definition of r n,α yields To obtain the shape preserving properties, we need to find the first and second order derivatives of S n,α f. For Szász-Mirakyan operators similar results were first established in [12].
Calculating the first and the second order derivatives of r n,α , it is directly seen that By applying first and second order derivatives of the operators, it leads to the following theorem: Theorem 4. If f is convex and decreasing, we have Proof. We know that a function f is convex if and only if all second order divided differences of f are nonnegative, (see [11, p. 259]). Thus, using Lemma holds true. Considering f is decreasing function, we have the desired result.
As an immediate consequence of the above result, one can stated the following theorem.
Proof. From the remark of [14,Remark,p.426], we know that because of the f is convex with respect to τ = e 2 +αe 1 1+α , α > 0. It is known that the Szász-Mirakyan operators, n ∈ N, of a convex function f , satisfy for all n ∈ N and α ∈ [0, ∞) (see [14,Remark,p. 438]). Thus, since τ is convex we have S n (τ ) ≥ τ. Since (S n (τ )) −1 is increasing, so we get Applying the operator S n f on both sides of the above inequality (S n is monotone operator), then we obtain which completes the proof.
Proof. It is easily checked that This completes the proof.

ASYMPTOTIC EXPRESSION
We begin by the following Voronovskaya-type theorem: Proof. Let f, f and f ∈ C * 2 [0, ∞) and x ∈ [0, ∞) be fixed. By the Taylor formula, we can write By the Cauchy-Schwarz inequality, we have . To prove Voronovskaya-type theorem, we must compute S n,α e x i , i = 1, 2, 3, 4.
Moreover, we compute the following via Mathematica Then if we use equalities which are mentioned above, we obtain nS n,α e x 4 (x) =n S n,α e 4 − 4xS n,α e 3 + 6x 2 S n,α e 2 − 4x 3 S n,α e 1 + x 4 S n,α e 0 (x) Finally by letting n → ∞, we get Putting this results in (10), we have the desired result.

RATE OF CONVERGENCE OF S n,α f
In this Section, the rate of convergence of S n,α operators in terms of both the weighted modulus of continuity and classical one is obtained.
Examining relations (4) and using the fact that r n,α → x as n → ∞, then, on the basis of Korovkin's first theorem, we observe that S n,α is an approximation process on compact subsets included in [0, ∞). Now, we want to give sufficient conditions which ensure both uniform convergence of the sequence S n,α to the identity operator on the whole interval [0, ∞) and the rate of convergence. For Bernstein type operators, a similar result was first established in [6]. This problem was further studied by de la Cal and Carcamo in [6].
We have for all x ≥ 0 and α > 0 Therefore, S n,α f converges to f uniformly on [0, ∞) as n → ∞, whenever f * is uniformly continuous.
Proof. Let x > 0 be arbitrary fixed. Since using the the definition of modulus of smoothness w (f * ; ·) , we have Further on, using the property w (f * ; ), we have Using Cauchy-Schwarz inequality, we have From (4) and (5), we deduce According to (13) we have Thus by (12), we have the inequality (11).
Under the hypothesis of our theorem, f * is uniformly continous on [0, ∞), we know that lim δ→0 w (f * ; δ) = 0. Since the inequality (11) valid for all x ∈ [0, ∞) leads us to the conclusion of our theorem. Now, we focus on weighted space C 2 [0, ∞). Using (4) and (8), then we obtain Therefore, we can say that S n,α acts from C 2 [0, ∞) to C 2 [0, ∞). Also, we give an estimation in terms of following weighted modulus of continuity. It is known that, if f is not uniformly continuous on the interval [0, ∞), then the usual first modulus of continuity w(f, δ) does not tend to zero, as δ → 0. Here, we use the following weighted modulus of continuity to gain this property. For f ∈ C 2 [0, ∞) and for every δ > 0, the weighted modulus of continuity considered in [8] is defined as follows: It is known that for every f ∈ C * 2 [0, ∞) , Ω(f, δ), δ > 0, the following properties hold true lim δ→0 Ω(f, δ) = 0 and (16) For f ∈ C 2 [0, ∞), by (15) and (16), we can write where C α is a positive constant depending only on α.

CONVERGENCE OF DERIVATIVE OF S n,α f
Before considering the main results of this section, we state the derivative of the operator (2) in following lemma.
Lemma 10. Let f be a continuously differentiable on [0, ∞). Then, we have Proof. From Lemma 2, we know that By reason of f ∈ Lip β M , we attain Since φ k < 1, we deduce Applying Holder inequality, we have Now by (6) and (14) , we get Combining (19), (20) and (21), we have the the desired result. We have the following estimates for S n f , D * n f and S n,α f in terms of the modulus of continuity w(f, δ), where δ 2 1 (x) = x n and δ 2 2 (x) = (α + 2x)(x − r n,α (x)). In the following theorem, we present analogues theorem for S n,α f to show a better order of approximation.
Theorem 12. For every f ∈ C B [0, ∞), x ≥ 0 and n ∈ N, we have δ 2 (x) ≤ δ 1 (x) and one can get the best approximation using S n,α f.