NTMSCI 9, No. 1, 52-58 (2021) 52
New Trends in Mathematical Sciences
http://dx.doi.org/10.20852/ntmsci.2021.414
On approximation properties of a generalization of
Bernstein polynomials in symmetric range
Ecem Acar and Aydin Izgi
Department of Mathematics, University of Harran, Sanliurfa, Turkey
Received: 28 December 2020, Accepted: 30 December 2020
Published online: 21 March 2021.
Abstract: 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.
Keywords: Bernstein Operators, Modulus of continuity, Rate of convergence, Voronovskaja type theorem.
1 Introduction
In 1912, Bernstein [1] introduced the classical Bernstein polynomials B
n
( f ; x) for f ∈ C[0, 1] as below
B
n
( f ; x)=
n
∑
k=0
f
k
n
n
k
x
k
(1 − x)
n−k
. (1)
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
1
2
+
x
2
k
1
2
−
x
2
n−k
f
2
k
n
− 1
(2)
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.
© 2021 BISKA Bilisim Technology ∗
Corresponding author e-mail: karakusecem@harran.edu.tr