Abstract
In the present paper, we study a new kind of Kantorovich–Stancu type operators. For this modified form, we discuss a uniform convergence estimate. Some Voronovskaja-type theorems are given.
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1 Introduction
Let \(0 \le \alpha \le \beta\) and \(m \in N\). In [15], D.D. Stancu introduced the linear positive operators
defined by
where
are the fundamental Bernstein polynomials [3].
When \(\alpha = \beta = 0\),
is the classical Bernstein operator.
L.V. Kantorovich [8] introduced the linear positive operators
defined for any nonnegative integer m by
By combining (1.1) and (1.2), D. Bărbosu [2] introduced
defined for any \(m \in N\) by
\(K_{m}^{ ( \alpha,\beta )}\) are linear positive operators called Kantorovich–Stancu operators.
In recent years, Bernstein–Kantorovich–Stancu operators have been modified and studied by many mathematicians. For instance, in [4] Cai et al. defined a new type λ-Bernstein operators, and a Kantorovich variant of the modified Bernstein operators was introduced and studied in [7]. In the last three years, Mursaleen et al. investigated several approximation properties for a Kantorovich type generalization of q-Bernstein–Stancu operators in [14], applied (\(p,q\))-calculus in approximation theory, and constructed the (\(p,q\))-analogue of Bernstein operators [12], (\(p,q\))-Bernstein–Kantorovich operators [13], and a Kantorovich variant of (\(p,q\))-Szász–Mirakjan operators [11]. Also, in [1] Ansari and Karaisa introduced and studied Chlodowsky variant of (\(p,q\))-Bernstein operators.
H. Khosravian-Arab, M. Delghan, and M.R. Eslahchi introduced in [9] the following operators:
where
and
Here, \(a_{0} ( m )\) and \(a_{1} ( m )\) are two unknown sequences which are determined in an appropriate way. Note that, for \(a_{0} ( m ) = 1\) and \(a_{1} ( m ) = - 1\), (1.5) becomes the well-known identity for the fundamental Bernstein polynomials
From (1.5), the operators (1.4) become
We try to extend some results to the Kantorovich–Stancu operators considering the operators denoted by
2 Auxiliary results
Lemma 2.1
For \(p \in N^{ *}\), we have
Proof
(i)
(ii) We have that
□
Corollary 2.2
For any \(p \in N^{ *}\), there exists a constant \(C(p)\), independent of m and x, such that
for every \(x \in [0,1]\).
Proof
First we have
where \(M = \max \{ \alpha,\beta - \alpha \}\) for \(x \in [0,1 ]\).
The following inequality
where \(c ( i )\) is a constant independent of m, can be found in [16] for \(mX \ge 1\), \(X = x ( 1 - x )\) and in [5] for \(mX < 1\).
Taking \(c ( p ) = \max_{i = \overline{0,p}}c ( i )\) in (2.3), by (2.2) it follows
From (2.4) and Lemma 2.1, we obtain estimate (2.1). □
The first four central moments for \(K_{m}^{ ( \alpha,\beta )}\) are as follows:
Remark 2.3
Using the results obtained by Gavrea and Ivan ([5], Theorem 14, Theorem 15, Remark 16), it is straightforward to give the following estimates:
-
(i)
For any \(p \ge 4\) and \(x \in ( 0,1 )\), there exists a constant \(A ( p )\) independent of m and x such that
$$ \frac{K_{m}^{ ( \alpha,\beta )} ( \vert t - x \vert ^{p + 1};x )}{K_{m}^{ ( \alpha,\beta )} ( \vert t - x \vert ^{p};x )} \le \frac{A ( p )}{\sqrt{m}},\quad m \ge 5. $$(2.5) -
(ii)
For any \(p \ge 1\) and \(x \in ( 0,1 )\), there exists a positive constant \(B ( p )\) independent of m and x such that
$$ \biggl\Vert \frac{K_{m}^{ ( \alpha,\beta )} ( \vert t - x \vert ^{p + 1};x )}{K_{m}^{ ( \alpha,\beta )} ( \vert t - x \vert ^{p};x )} \biggr\Vert \ge \frac{B ( p )}{\sqrt{m}}, $$(2.6)where \(\Vert \cdot \Vert \) is the uniform norm on \([0,1]\).
-
(iii)
From (i) and (ii) it follows
$$ \biggl\Vert \frac{K_{m}^{ ( \alpha,\beta )} ( \vert t - x \vert ^{p + 1};x )}{K_{m}^{ ( \alpha,\beta )} ( \vert t - x \vert ^{p};x )} \biggr\Vert = O \biggl( \frac{1}{\sqrt{m}} \biggr). $$(2.7)
Remark 2.4
From Mamedov’s theorem [10] it follows that:
If \(p \in N^{ *} \) is even and \(f \in C^{p} ( [0,1] )\), for any \(x \in [0,1]\), we have that
3 Modified Kantorovich–Stancu operators
Now, we modify the Kantorovich–Stancu operator as follows:
Lemma 3.1
The moments \(\overline{K}_{m}^{ ( \alpha,\beta )} ( t^{i};x )\), \(i = 0,1,2\), are given by
Lemma 3.2
The central moments of the operators \(\overline{K}_{m}^{ ( \alpha,\beta )}\), \(\overline{K}_{m}^{ ( \alpha,\beta )} ( ( t - x )^{i};x )\), \(i = 1,2,3,4\), are given by
We will study the uniform convergence of the sequence \(( \overline{K}_{m}^{ ( \alpha,\beta )}f )_{m \in N}\) for the case
We observe that (3.2) implies \(\overline{K}_{m}^{ ( \alpha,\beta )} ( 1;x ) = 1\).
We are interested in the following cases:
Case 1:
Case 2:
Combining (3.2) and (3.3), we obtain \(a_{0} ( m ) \in [0,1]\) and \(a_{1} ( m ) \in [-1,1]\), which implies that the sequences \(a_{0} ( m )\) and \(a_{1} ( m )\) are bounded. The operators \(\overline{K}_{m}^{ ( \alpha,\beta )}\) are bounded and positive.
Combining (3.2) and (3.4), we obtain that \(a_{0} ( m ) + a_{1} ( m ) > 1\) if \(a_{1} ( m ) < 0\) and \(a_{0} ( m ) > 1\) if \(a_{0} ( m ) + a_{1} ( m ) < 0\). In these cases, the operators \(\overline{K}_{m}^{ ( \alpha,\beta )}\) are not positive.
Remark that, for \(\alpha = \beta = 0\) and \(a_{0} ( m ) = \frac{3}{2}\), \(a_{1} ( m ) = - 2\), we obtain the modified operators introduced and studied in [7].
In order to prove the uniform convergence of the operators \(\overline{K}_{m}^{ ( \alpha,\beta )}\), we give the Korovkin theorem:
Theorem 3.3
([9], Theorem 10)
Let \(0 < h \in C ( [ a,b ] )\) be a function and suppose that \(( L_{n} )_{n \ge 1}\) is a sequence of positive linear operators such that \(\lim_{n \to \infty} L_{n} ( e_{i} ) = he_{i}\), \(i = 0,1,2\), uniformly on \([ a,b ]\). Then, for a given function \(f \in C ( [ a,b ] )\), we have \(\lim_{n \to \infty} L_{n} ( f ) = hf\) uniformly on \([ a,b ]\).
For the first case, we obtain the following result:
Theorem 3.4
Given two sequences \(a_{0} ( m )\) and \(a_{1} ( m )\) that satisfy conditions (3.2) and (3.3), the sequence \(( \overline{K}_{m}^{ ( \alpha,\beta )}f )_{m \in N}\) converges to f, uniformly on \([0,1]\), for any function \(f \in C ( [0,1] )\).
Proof
The operator \(\overline{K}_{m}^{ ( \alpha,\beta )}f\) is a linear convex combination of positive operators \(K_{m - 1}^{ ( \alpha,\beta + 1 )}f\) and \(K_{m - 1}^{ ( \alpha + 1,\beta + 1 )}f\). Consequently, the result follows from Theorem 3.3. □
In the second case, we have the following:
Theorem 3.5
For any function \(f \in C ( [0,1] )\) and all bounded sequences \(a_{0} ( m ), a_{1} ( m )\) that satisfy conditions (3.2) and (3.4), the sequence \(( \overline{K}_{m}^{ ( \alpha,\beta )}f )_{m \in N}\) converges to f, uniformly on \([0,1]\).
Proof
Taking
and
we have
Using the remarks for case 2, it follows that the operators \(\overline{K}_{m,1}^{ ( \alpha,\beta )}\) and \(\overline{K}_{m,2}^{ ( \alpha,\beta )}\) are positive. According to Theorems 3.3 and 3.4, we obtain that
where \(l_{i} = \lim_{m \to \infty} a_{i} ( m )\), \(i = 0,1\). □
The following theorems are Voronovskaja-type results for the operators \(\overline{K}_{m}^{ ( \alpha,\beta )}\).
Theorem 3.6
Let \(a_{0} ( m )\), \(a_{1} ( m )\) be two convergent sequences that verify conditions (3.2) and (3.3) and \(l_{i} = \lim_{m \to \infty} a_{i} ( m )\), \(i = 0,1\). If \(f \in C^{2} ( [0,1] )\), then
uniformly on \([0,1]\).
Proof
Applying Taylor’s formula to the operators \(\overline{K}_{m}^{ ( \alpha,\beta )}\), we have
where \(\rho \in C ( [0,1] )\) and \(\lim_{t \to x}\rho ( t;x ) = 0\).
It is sufficient to prove that \(\lim_{m \to \infty} m\overline{K}_{m}^{ ( \alpha,\beta )} ( \rho ( t;x ) ( t - x )^{2};x ) = 0\) uniformly on \([0,1]\).
Using the Cauchy–Schwarz theorem, we obtain that
Since \(\rho ( x,x ) = 0, \rho^{2} ( \cdot;x ) \in C ( [0,1] )\), by Theorem 3.4, we have
and by Lemma 3.2, we get
uniformly on \([0,1]\). Hence, we obtain the above limit.
Finally, Lemma 3.2 gives us (3.6). □
Theorem 3.7
Let \(a_{0} ( m )\), \(a_{1} ( m )\) be two bounded convergent sequences that verify conditions (3.2) and (3.4) and \(l_{i} = \lim_{m \to \infty} a_{i} ( m )\), \(i = 0,1\). If \(f \in C^{2} ( [0,1] )\), then
uniformly on \([0,1]\).
Proof
From (3.5), we have
where
and
Applying Theorem 3.6 to the operators \(\overline{K}_{m,2}^{ ( \alpha,\beta )}\) and \(\overline{K}_{m,1}^{ ( \alpha,\beta )}\), we obtain
and
uniformly on \([0,1]\).
Combining these two results, the proof is finished. □
In what follows, we will denote by \(\omega ( f; \cdot )\) the first order modulus of continuity of the function f
Theorem 3.8
Let \(a_{0} ( m ), a_{1} ( m )\) be two bounded sequences that verify (3.2). If \(f ( x )\) is bounded for \(x \in [0,1]\), then
where \(\Vert \cdot \Vert \) is the uniform norm on \([0,1]\).
Proof
By (3.1), we have that
We need an upper bound for \(a ( x;m )\) and \(a ( 1 - x;m )\). Note that this is the same upper bound for both. From (3.2), it follows that
and (3.9) becomes
By ([2], Theorem 2.6), we have
and
where
So,
By using the properties of the first order modulus of continuity together with the above forms of \(\delta_{m - 1,1}^{ ( \alpha,\beta + 1 )}\)and \(\delta_{m - 1,1}^{ ( \alpha + 1,\beta + 1 )}\) in (3.11), we obtain (3.8). □
Assume that \(\beta = 2\alpha\), \(\overline{K}_{m}^{ ( \alpha,\beta )} ( 1;x ) = 1\) and \(\overline{K}_{m}^{ ( \alpha,\beta )} ( t;x ) = x\).
Consequently, we get
which implies that
and from (3.4), it follows that the operators \(\overline{K}_{m}^{ ( \alpha,\beta )}\) are not positive.
Now, we can formulate a new quantitative Voronovskaja-type result:
Theorem 3.9
For \(g \in C^{2} ( [0,1] )\), \(x \in [0,1]\) fixed, we have the following estimate:
where C is a positive constant independent of m and x.
Proof
Under the above assumptions, by applying Taylor’s formula to the operators \(\overline{K}_{m}^{ ( \alpha,\beta )}\), we have
where
From the mean value theorem, it follows that there exists \(\xi \in ( \min ( x,t ),\max ( x,t ) )\) such that
So,
When \(x \in [0,1]\), an upper bound for \(a ( x;m )\) and \(a ( 1 - x;m )\) is
Using (3.14), it follows that
Applying Corollary 2.2, it follows that there exists a constant \(C'\) independent of m and x such that (3.17) becomes
Thus,
and the proof is completed. □
Corollary 3.10
For \(g \in C^{2} ( [0,1] )\), \(x \in [0,1]\) fixed, we have
Proof
By Theorem 3.9 and Lemma 3.2(ii), we obtain (3.20). □
Corollary 3.11
For \(g \in C^{2} ( [0,1] )\), the following estimate holds:
where \(\Vert \cdot \Vert \) is the uniform norm on \([0,1]\).
Proof
Since \(\omega ( g'';\delta ) \le 2 \Vert g'' \Vert \), by Lemma 3.2(ii) and Theorem 3.9, we obtain (3.21). □
We can reformulate Theorem 3.9 in terms of second order moduli of continuity.
Theorem 3.12
Assuming \(\beta = 2\alpha\), for \(a_{0} ( m ) = \frac{2\alpha + 3}{2}\), \(a_{1} ( m ) = - 2 ( \alpha + 1 )\), and \(g \in C ( [0,1] )\), we have
Proof
The operators \(\overline{K}_{m}^{ ( \alpha,\beta )}\)are bounded, and by (3.1), we have
It is well known that the second order modulus of continuity is equivalent to the K-functional
From Gonska ([6], Corollary 2.7),
Combining the above inequalities and taking the infimum over all \(h \in C^{2} ( [0,1] )\) in the following inequality
leads to the desired result. □
4 Conclusions
In this paper, we introduce and study a modified form of the Kantorovich–Stancu operators.
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Acknowledgements
The author is grateful to the PhD coordinator, Prof. Ioan Gavrea, Department of Mathematics, Technical University of Cluj-Napoca, Romania. Also, the author would like to thank the anonymous reviewers for their careful reading of the manuscript and their recommendations which improved the quality of the paper.
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Opriş, AA. Approximation by modified Kantorovich–Stancu operators. J Inequal Appl 2018, 346 (2018). https://doi.org/10.1186/s13660-018-1939-9
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DOI: https://doi.org/10.1186/s13660-018-1939-9