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Abstract
In this paper, we introduce a generalization of the Kantorovich-type Bernstein operators based on q-integers and get a Bohman–Korovkin-type approximation theorem of these operators. We also compute the rate of convergence using the first modulus of smoothness.
Public Interest Statement
The approximation of functions by positive linear operators is a significant research area in mathematical analysis with key relevance to studies of computer aided, geometric design, numerical analysis, solution of differential equations, etc. In this work, we introduce a generalization of the Kantorovich-type Bernstein operators based on q-integers and prove the basic convergence of the introduced operators and also obtain the rate of convergence in terms of modulus of continuity. Further, we study the local approximation property using the modulus of continuity.
1. Introduction
During the last two decades, the applications of q-calculus emerged as a new area in the field of approximation theory. The rapid development of q-calculus has led to the discovery of various generalizations of Bernstein polynomials involving q-integers. Lupaş (Citation1987) introduced the first q-analogue of Bernstein operators (Bernstein, Bernstein) and investigated its approximating and shape preserving properties. Another q-generalization of the classical Bernstein polynomials is due to Phillips (Citation1997). Several generalizations of well-known positive linear operators based on q-integers were introduced and their approximation properties have been studied by several authors.
For each positive integer n, Philips (Citation1997) defined q-Bernstein polynomials as(1.1)
(1.1)
When is the classical Bernstein polynomial
(1.2)
(1.2)
Kantorovich (Lorentz, Citation1953) modified the Bernstein operators and defined the linear positive operators defined for any
by
(1.3)
(1.3)
where These operators are known as Kantorovich operators in literature. Radu (Citation2008) has obtained the statistical convergence of q-Bernstein–Kantorovich polynomials. Also, the Kantorovich-type generalizations of the linear positive operators based on q-integers were studied by some authors (see e.g. Mahmudov, Mahmudov2010; Mishra, Khatri, & Mishra, Citation2012; Mishra, Khatri, Mishra, & Deepmala, Citation2013; Mishra, Sharma, Kiliçman, & Jain, Citationin press; Mishra, Sharma, & Mishra, Citation2015; Mursaleen, Khan, Srivastava, & Nisar, Citation2013; Srivastava, Citation2011; Srivastava & Choi, Citation2012). Gairola, Deepmala, and Mishra (Citationin press), Wafi, Rao, and Deepmala (Citation2016) studied rate of approximation and some approximation properties of linear positive operators using quantum calculus approach. Recently, Agrawal, Finta, and Kumar (Citation2015a) introduced a new Kantorovich-type generalization of the q-Bernstein–Schurer operators,
In 2007, Dalmanoglu (Citation2007) defined Kantorovich-type q-Bernstein Operator as follows:(1.4)
(1.4)
Before proceeding further, we recall certain notations of q-calculus as follows. Such notations can be found in Ernst (Citation2000), Kac and Cheung (Citation2002). We consider q as a real number satisfying .
For
and
Then for and integers
, we have
We observe that
Also, for any real number , we have
In special case, when is a whole number, this definition coincides with the above definition.
The q-Jackson integral and q-improper integral in the interval [0, a] defined as
and
provided sum converges absolutely.
Let and
. Following Marinković, Rajković, and Stanković (Citation2008), we consider the Riemann-type q-integral defined as follows
This Riemann-type q-integral is appropriate to derive the q-analogues of some well-known integral inequalities. Then the Riemann-type q-integral for a bivariate function is given by(1.5)
(1.5)
where and
. Also, f is a
-integrable function, so the series in (1.5) converges.
The bivariate case for the operators are first introduced by Stancu (Citation1969). He studied the bivariate Bernstein polynomials and estimated the order of approximation for these operators. The aim of this paper was to construct bivariate q-Bernstein–Kantorovich operators, and investigate Korovkin-type approximation properties and estimate the order of approximation in terms of a modulus of continuity.
2. The construction of the bivariate operators of Kantorovich type
The aim of this part was to construct the bivariate extension of the operator (1.1). Let , where
and
. For
, let
denote the space of all real-valued continuous functions on
endowed with the norm
. Analogously, for
, we denote by
the sup-norm on
.
If and
, let us define the bivariate generalization of operator (1.4) as follows:
(2.1)
(2.1)
where and
(2.2)
(2.2)
In 2003, Erkuş and Duman (DU) proved the statistical Korovkin-type approximation theorem for the bivariate linear positive operators to the functions in space In 2009, Ersan and Doğru (Ersan2009) obtained the statistical Korovkin-type theorem and lemma for the bivariate linear positive operators defined in the space
as follows
Theorem 1
(Ersan & Doğru, Ersan2009) Let be the sequence of linear positive operator acting from
into
where
Then, for any
,
Lemma 1
The bivariate operators defined in Ersan (Ersan2007) satisfy the following :
(i) |
| ||||
(ii) |
| ||||
(iii) |
| ||||
(iv) | |||||
(v) |
In order to obtain the convergence properties of bivariate operator (2.1), we need the following lemma.
Lemma 2
Let ,
with
be the two-dimensional test functions. Then, we have
(i) |
| ||||
(ii) |
| ||||
(iii) |
| ||||
(iv) | |||||
(v) |
Proof
The proof can be obtained similar to the proof of bivariate operator in Agrawal, Finta, and Kumar (Sathish2015Bivariate). So, we shall omit this proof.
Lemma 3
For the operator (2.1), we have
(i) | |||||
(ii) | |||||
(iii) | |||||
(iv) |
3. Rates of convergence of bivariate operators
Let Then the sup norm on
is given by
We consider the modulus of continuity , where
for bivariate case given by
(3.1)
(3.1)
It is clear that a necessary and sufficient condition for a function is
and satisfy the following condition:
(3.2)
(3.2)
for each . Then observe that any function in
is continuous and bounded on K. The details of the modulus of continuity for bivariate case can be found in Anastassiou and Gal (Citation2000).
Now, the rate of statistical convergence of bivariate operator (2.1) by means of modulus of continuity in will be given in the following theorem.
Theorem 2
Let such that
and
. So, we have
(3.3)
(3.3)
where defined as in Lemma (3).
Proof
By using the condition in (3.2), for and
, we get
If the Cauchy–Schwarz inequality is applied, we have
So, if it is substituted in the above equation, the proof is completed.
The next theorem represents the rate of statistical convergence of bivariate operator (2.1) by means of Lipschitz functions for the bivariate case, where
and
and
,
, then let us define
as
We have the following theorem.
Theorem 3
Let and
be sequence satisfying
and
, and let
and
. Then
(3.4)
(3.4)
where and
are defined in Theorem (2).
Proof
Since are linear positive operators and
and
we can write
If we take applying Hölder’s inequality, we obtain
which is the required result.
In what follows we shall use the following notations:
and
respectively. We have the next result.
Theorem 4
Let and
such that
and
. Then, we have
(3.5)
(3.5)
where and
are defined in Lemma 3.
Proof
Let be fixed point. Then, we can write for
that
Now, applying the operator defined by (2.1) on both sides and Lemma 2, (i), we obtain
Since and
, we have
Now, applying the Cauchy–Schwarz inequality
This completes the proof of the theorem.
Acknowledgements
The authors would like to express their deep gratitude to the anonymous learned referee(s) and the editor for their valuable suggestions and constructive comments, which resulted in the subsequent improvement of this research article.
Additional information
Funding
Notes on contributors
Preeti Sharma
Preeti Sharma is an assistant professor at Banasthali University and pursuing PhD in Mathematics from SVNIT, Surat under the supervision of Dr Vishnu Narayan Mishra. She received the MSc degree in Mathematics & Computing from Indian Institute of Technology Guwahati, in 2012. Her research interests are in the areas of Approximation Theory and Operator Theory.
Vishnu Narayan Mishra
Vishnu Narayan Mishra received the PhD in Mathematics from Indian Institute of Technology, Roorkee. His research interests are in the areas of pure and applied mathematics. He has published more than 100 research articles in reputed international journals of mathematical and engineering sciences. He is a referee and an editor of several international journals in frame of Mathematics. He guided many postgraduate and PhD students. Citations of his research contributions can be found in many books and monographs, PhD thesis and scientific journal articles.
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