On approximation properties of a generalization of Bernstein polynomials in symmetric range

In the present paper, in order to make the convergence faster to a function being approximated we identify a new generalization of Bernstein operators depending on symmetric range. The rate of convergence of these operators are given by using the modulus of continuity. Furthermore,we establish Korovkin-type approximation theorem and Voronovskaja type asymptotic theorem. Finally, we show that using graphics in Maple this new generalization of Bernstein operators converge faster than Bernstein operators on symmetric range for certain functions.


Introduction
In 1912, Bernstein [1] introduced the classical Bernstein polynomials B n ( f ; x) for f ∈ C[0, 1] as below Bernstein polynomials can uniformly approximate any continuous function over a closed interval.
In the papers [4,5,6], various generalization of Bernstein polynomials are investigated. Also, approximating continuous functions by classical Bernstein polynomials have been studied for two dimensional Bernstein polynomials in [7]- [9]. The idea of constructing linear and nonlinear positive operators have been studied intensively in approximation theory (see [11], [12]). A generalization of Bernstein polynomials in symmetric range are defined in [2] C n ( f ; x) = n ∑ k=0 n k for f ∈ C[−1, 1] and n ∈ N. These operators given in (2) are linear positive in symmetric range and provide the Korovkin theorem's conditions. Also, the operators (2) are smooth convergence on the range of [−1, 1].
The purpose of this paper is to introduce a new generalization of Bernstein polynomials in symmetric range and their certain elementary properties. One of our main important results is uniform convergence to a continuous function. Also, in that paper we calculate rate of convergence of this new generalization by using modulus of continuity and give the Voronovskaja type asymptotic theorem.
Let I = − m+a m+b , m+a m+b , a, b ∈ N and f ∈ C [I]. Then we define the linear positive operators E m ( f ; x) in the following way It is clear that let take a = b = 0 for the operators given (3) then it reduce to Bernstein polynomials in symmetric range given in [2].
In the following section, we obtain the properties of approximation of these operators.

Main results
Lemma 1. For ∀x ∈ I and m ∈ N, symmetric Bernstein operators (3) is satisfied the following equalities: and Hence, we get lim Accordingly the Bohman-Korovkin theorem [3], we get the desired result.
Lemma 2. Let obtain the following results for the central moments of the E m ( f ; x) operators defined by Then we get C m,0 (x) = 1, C m,1 (x) = 0, Also, the equality Proof. It is obvious that C m,0 (x) = 1, C m,1 (x) = 0, hold from Lemma 1. From (6), we obtain Theorem 2. If f ∈ C(I), then the below inequality holds Proof. Let f ∈ C(I), we get Since 1 2 m ≥ 0 and p m, j,a,b (x) ≥ 0 and using e Cauchy-Schwartz inequality, we obtain From the properties of modulus of continuity, we can write Now, replacement in the inequality (7), we obtain

Let take
Applying Cauchy-Schwarz inequality, we get Also, we obtain max m+a m+b Hence, choosing δ = 1 √ m , we get the desired result as below Theorem 3. Voronovskaja Type Theorem. For f ∈ C 2 (I), we obtain the following equality Proof. Taylor expansion of a function f at point x is as follows

Competing interests
The authors declare that they have no competing interests.