Abstract
In this paper, we introduce, analyze, and obtain some features of a new type of Bernstein–Chlodowsky operators using a different technique that is utilized as the classical Chlodowsky operators. These operators preserve the functions \(\exp \left( \mu t\right) \) and \(\exp \left( 2\mu t\right) \), \(\mu >0 \). As a first result, the rate of convergence of the operator using an appropriately weighted modulus of continuity is obtained. Later, Quantitative-Voronovskaya type and Grüss–Voronovskaya type theorems for the new operators are presented. Then, we prove that the first derivative of the Bernstein–Chlodowsky operators applied to a function converges to the function itself. Finally, the variation detracting property of the operators is presented. It is proved that the variation seminorm property is preserved. Also, it is shown that the operators converge to \(f/\exp _{\mu }\) in variation seminorm is valid if and only if the function is absolutely continuous.
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1 Introduction
Recall that the classical Bernstein–Chlodowsky operator \(C_{n}\) defined from \(C\left[ 0,\infty \right) \rightarrow C\left[ 0,\infty \right) \) is given by
where f is a function defined on \(\left[ 0,\infty \right) \) and bounded on every finite interval \(\left[ 0,b_{n}\right] \subset \left[ 0,\infty \right) \) with a certain rate, and \(b_{n}\) is a monotone increasing, positive and real sequence such that \(\lim \limits _{n\rightarrow \infty }b_{n}=\infty \) and \(\lim \limits _{n\rightarrow \infty }\frac{b_{n}}{n}=0\).
The classical Bernstein–Chlodowsky polynomials were introduced by I. Chlodovsky in 1937 as a generalization of the Bernstein polynomials. Note that the case \(b_{n}=1\), \(n\in \mathbb {N} \), in Eq. 1.1, defines an approximation to the function f on the interval \(\left[ 0,1\right] \) (or, suitably modified on any fixed finite interval \(\left[ -b,b\right] \) ).
For \(b>0\), let \(M\left( b;f\right) :=\sup \limits _{0\le t\le b}\left| f\left( t\right) \right| \). It is shown by Chlodowsky that when \(f\in \) \(C\left[ 0,\infty \right) \) and \(\lim \limits _{n\rightarrow \infty }M\left( b;f\right) \exp \left( -\frac{\sigma n}{b_{n}}\right) =0\) for each \(\sigma >0 \), then the classical Bernstein–Chlodowsky operator converges to \(f\left( x\right) \) at each point where f is continuous. Chlodovsky also showed that the simultaneous convergence of the derivative \(\left( C_{n}f\right) ^{^{\prime }}\left( x\right) \) to \(f^{^{\prime }}\left( x\right) \) at points x, where the derivative of f(x) exists, a result taken up by Butzer [4, 5]. Due to these two former results, the classical Bernstein–Chlodowsky operators and their generalizations have been an increasing interest in the field of approximation theory.
During the paper, \(\mu >0\) is a fixed real parameter and \(\exp _{\mu }\) represents the exponential function defined by \(\exp _{\mu }\left( t\right) =e^{\mu t}\).
Herein, we consider a generalization of Bernstein–Chlodowsky operators of the form
with the property that
Then, the operator \(\mathcal {C}_{n}\) is more explicitly given by
with
Note that the connection of this operator with the classical Bernstein–Chlodowsky operator can be expressed as
Namely,
Also note that the Bernstein–Chlodovsky operators \(\mathcal {C}_{n}\), based on functions defined on \(\left[ 0,\infty \right) \), are bounded on every \(\left[ 0,b_{n}\right] \subset \left[ 0,\infty \right) \) with a certain rate. Thus, they are a very natural polynomial process in approximating unbounded functions on the unbounded infinite interval \(\left[ 0,\infty \right) \); but this approximation process is not so easy to handle.
We know that the classical Bernstein–Chlodowsky operators have the degree of exactness one, that is, they preserve the monomials 1 and x. On the other side, the operator (1.4) does not preserve 1 and x, but it satisfies the exponential moments (1.3) that play an important role in our calculations.
The aim of the present paper is to investigate the operators \(\mathcal {C}_{n} \), \(n\in \mathbb {N}\) in deeper to reveal, in addition to elementary properties, their advanced properties. Moreover, the development of the some theoretical results of the generalized operator is within the aim of the paper. After Voronovskaya type theorems for the generalized operator is stated , it is compared to the classical Bernstein–Chlodovsky operators in terms of effectiveness. For this purpose, the convergence of the derivative \(\left( \mathcal {C}_{n}f\right) ^{^{\prime }}\left( x\right) \) to \(f^{^{\prime }}\left( x\right) \) is also considered. Finally, in the last section, the variation detracting property of the operators and variation seminorm property is stated. Moreover, it is proved that the operators converge to \(f/\exp _{\mu }\) in variation seminorm is valid if and only if the function is absolutely continuous.
2 Preliminary Results
For the operator \(\mathcal {C}_{n}\), \(n\in \mathbb {N}\), we give here some of their properties and results. At first, we calculate all the moments of operator (1.4).
Lemma 1
For each \(n\in \mathbb {N}\) and \(x\in \left[ 0,b_{n}\right] \), the following identities hold:
Using Mathematica, we give two limits, which play an important role in both the uniform approximation of operator to functions and Voronoskaya type result.
For each \(x\in \left( 0,\infty \right) \), we shall consider the function \(\exp _{\mu ,x}\), defined for \(t\in \left( 0,\infty \right) \) by
Using Lemma 1 and (1.3) , one easily finds that
and
Lemma 2
For each \(x\in \left[ 0,\infty \right) ,\) the following identities hold:
and
3 Quantitative Results
All concepts mentioned below can be found in [7] more generally. We denote by \(C_{\mu }\left[ 0,\infty \right) \) the space of continous functions \(f\in C\left[ 0,\infty \right) \) with the property that exists \(M>0\) such that \(\left| f\left( x\right) \right| \le Me^{\mu x}\), for every \(x\in \left[ 0,b_{n}\right] .\) This space endowed with norm
Also,
For \(f\in C_{\mu }^{k}\left[ 0,\infty \right) \) we use the following modulus of continuity:
In [7], the authors proved the most general form of the following lemmas.
In the following, we give the main properties of the modulus of continuity.
Lemma 3
([7]) If \(f\in C_{\mu }\left[ 0,\infty \right) \) and \(\lambda >0\), then
holds for every \(\delta >0.\)
Lemma 4
([7]) For \(\delta >0,\) \(f\in C_{\mu }\left[ 0,\infty \right) \) and \(x,t\in \left[ 0,b_{n}\right] ,\) the inequality
holds.
Lemma 5
([7]) For any \(f\in C_{\mu }^{k}\left[ 0,\infty \right) ,\) we have
Quantitative approximation theorems for sequences of linear positive operators play an important role not only in approximating functions, but also in estimating the error of the approximation. One of the most important convergence results in approximation theory is the Voronovskaya theorem. Roughly speaking, it is obtained to describe the rate of pointwise convergence.
Moreover, the other results presented in this paper are a quantitative-Voronovskaya type and a Grüss–Voronovskaya type theorems for the new operators. For more details, see [1]. Recently, Gal and Gonska obtained a Voronovskaya type theorem with the aid of Grüss inequality for Bernstein operators in [8] and called it Grüss–Voronovskaya type theorem. In this paper, we extend some of these results for our operators \(\mathcal {C}_{n}\).
First, in the following theorem, we give quantitative type theorem for our operator \(\mathcal {C}_{n}\):
Theorem 1
For \(f\in C_{\mu }^{k}\left[ 0,\infty \right) \) and \(x\in \left[ 0,b_{n} \right] \), we have
Proof
Suppose that \(\delta <1.\) Using Lemma 3, 4 and (2.2) , we have
\(\square \)
We have that our operator has a different approach charecteristics
Remark 1
If in the previous theorem, we assume
then the estimate reads as
Hence, velocity of convergence of \(\mathcal {C}_{n}f\left( x\right) \) to \( f\left( x\right) \) is managed by the velocity of convergence of \(\mathcal {C} _{n}e_{0}\left( x\right) \) to \(e_{0}\left( x\right) =1\), or equivalently, the one of \(\lambda _{n}\left( x\right) \) to 0, and this is given by the undermentioned limit, that can be easily computed by elementary calculus.
Now, we state quantitative-Voronovskaya type theorem for \(\mathcal {C}_{n}\):
Theorem 2
If \(\ f\in C_{\mu }^{k}\left[ 0,\infty \right) \) and \(x\in \left( 0,b_{n}\right) \), then we get
Proof
By Taylor’s theorem, we have
where
with \(\xi \) a number between \(\ x\) and t. Applying the operator \(\mathcal {C }_{n}\) to both side of above inequality, we get
Using Lemma 4 and the fact that \(\left| e^{\mu \xi }-e^{\mu x}\right| \le \left| e^{\mu t}-e^{\mu x}\right| ,\) then we can write
Suppose that \(\delta <1.\) Thus, we can write
Multiplying this relation with \(\exp _{\mu ,x}^{2}\) and applying the operator \(\mathcal {C}_{n},\) we get
We know that, since
and
we have
Also since
and
we get
Therefore, since
and
we can write
Choosing \(\delta =\sqrt{\frac{\mathcal {C}_{n}\left( \exp _{\mu ,x}^{4};x\right) }{\mathcal {C}_{n}\left( \exp _{\mu ,x}^{2};x\right) }},\) we have desired result. \(\square \)
Later, we express quantitative-Grüss–Voronovskaya type theorem for \( \mathcal {C}_{n}\):
Theorem 3
If f,\(g\in C_{\mu }^{k}\left[ 0,\infty \right) \) , then for all \(x\in \left[ 0,b_{n}\right] \) and \(n\in \mathbb {N}\) we have
where
and
Also, \(\mathcal {G}_{n}\left( \mathcal {C}_{n},g;x\right) \), \(\mathcal {G} _{n}\left( \mathcal {C}_{n},\left( fg\right) ;x\right) \), and \(I_{n}\left( g\right) \) are the analogous one.
Proof
For \(x\in \left[ 0,\infty \right) \) and \(n\in \mathbb {N}\), it is easily seen that we can write
So, we get
By Theorem 2, we have the estimates
and
On the other hand, since \(f\in C_{\mu }^{k}\left[ 0,\infty \right) \) we write
and so we get
where \(\xi \) is a number between t and x. If \(t<\xi <x\), then \(e^{\mu \xi }\le e^{\mu x}\). In this case, we have
or if \(x<\xi <t\), then \(e^{\mu \xi }\le e^{\mu t}\). In this case, with the help of Hölder’s inequality, we get
Hence, we gain for two cases of \(\xi \) that
A similar reasoning yields \(\left| \mathcal {C}_{n}\left( g;x\right) -g\left( x\right) \right| \le I_{n}\left( g\right) \). Therefore we get
as desired. \(\square \)
Theorem 4
For each \(n\in \mathbb {N}\) and \(x\in \left[ 0,\infty \right) \), we have
Proof
Using (1.5), we obtain
First, we take into account the case \(x=0\).
From (3.2), we have
For \(x=0\), because of \(a_{n}\left( x\right) =0\), we get
If the limit of both sides is taken above equality, then we obtain
Now, let’s \(x>0.\)
We consider the following function:
In that case, \(\lim \limits _{t\rightarrow x}\lambda _{x}\left( t\right) =0\). We get
If \(\frac{kb_{n}}{n}\) is changed instead of t, then we have
If this equality is written in (3.2), then we attain
We know
We can write
Because
and
we have
Now, we use Hölder inequality:
From Korovkin theorem, we know
As
and
we obtain desired result. \(\square \)
4 Variation Detracting Property of Bernstein–Chlodowsky Operators
The first study about the variation detracting property and the convergence in variation of a sequence of linear positive operators was come out by Lorentz (1953). He proved that \(B_{n}\) have
and it is called the variation detracting property.
The main purpose of this section is to confirm the variation detracting property and convergence in the variation seminorm for the Bernstein–Chlodowsky operators. We firstly give the definitions related to variation detracting property.
Definition 1
([11]) The least upper bound of the set of all possible sums V is called the total variation of the function \(f\left( x\right) \) on \(\left[ a,b \right] \) and is designated by \(V_{\left[ a,b\right] }\left[ f\right] \).
Definition 2
([2]) The class of all functions of bounded variation on I is called BV space and denoted by \(BV\left( I\right) \). This space can be endowed both with seminorm \(\left| .\right| _{BV\left( I\right) }\) and with a norm, \(\left\| .\right\| _{BV\left( I\right) }\), where
\(f\in BV\left( I\right) \), a being any fixed point of I.
Definition 3
([3])
Let \(I\subseteq \mathbb {R} \) be a fixed integral, and \(V_{I}\left[ f\right] \) the total variation of the function \(f:I\rightarrow \mathbb {R} \). The class of all bounded functions of bounded variation on I endowed with the seminorm
is called TV space and is denoted by \(TV\left( I\right) \).
Definition 4
([11]) Let \(f\left( x\right) \) be a finite function defined on the closed interval \(\left[ a,b\right] \). Suppose that for every \(\epsilon >0\), there exists a \(\delta >0\) such that
for all numbers \(a_{1},b_{1},\ldots ,a_{n},b_{n}\) such that \(a_{1}<b_{1}\le a_{2}<b_{2}\le \cdots \le a_{n}<b_{n}\) and
Then the function \(f\left( x\right) \) is said to be absolutely continuous. The class of all absolutely continuous function on \(\left[ a,b\right] \) is denoted by \(AC\left[ a,b\right] \).
Now, we give the variation detracting property of the Bernstein–Chlodowsky operators:
Theorem 5
If \(\ f\in TV\left[ 0,b_{n}\right] \), then \(V_{\left[ 0,b_{n}\right] }\left[ \frac{\mathcal {C}_{n}f}{\exp _{\mu }}\right] \le \) \(V_{\left[ 0,b_{n}\right] }\left[ \frac{f}{\exp _{\mu }}\right] \).
Proof
As \(\frac{\mathcal {C}_{n}f}{\exp _{\mu }}\) polynomials are differentiable and their derivatives are integrable, by [9, 10], the equality
is implemented. From (1.5), we can write
By Theorem 3.13 in [6], we get
If \(\frac{a_{n}\left( x\right) }{b_{n}}=y\) is changed, then we have
Now, let’s consider the integral on the left side of the inequality. From definition of Beta function, we obtain
\(\square \)
Theorem 6
Let \(f\in TV\left[ 0,b_{n}\right] \). There holds
Proof
Since \(\frac{f}{\exp _{\mu }}\) and \(\frac{\mathcal {C}_{n}f}{\exp _{\mu }}\in AC\left[ 0,b_{n}\right] \), then \(\frac{\mathcal {C}_{n}f}{\exp _{\mu }}-\frac{ f}{\exp _{\mu }}\in AC\left[ 0,b_{n}\right] \). By Theorem 3.13 and Remark 3.20 in [6], it is written
From Theorem 4, it can be seen easily that \(\left( \frac{\mathcal {C}_{n}f}{ \exp _{\mu }}\right) ^{\prime }\left( x\right) \longrightarrow \left( \frac{f }{\exp _{\mu }}\right) ^{\prime }\left( x\right) \) as \(n\rightarrow \infty \). Therefore,
Conversely, let \(\lim \limits _{n\rightarrow \infty }\left\| \frac{ \mathcal {C}_{n}f}{\exp _{\mu }}-\frac{f}{\exp _{\mu }}\right\| _{TV\left[ 0,\infty \right) }=0\). This means that \(\frac{\mathcal {C}_{n}f}{\exp _{\mu }} \longrightarrow \frac{f}{\exp _{\mu }}\) in TV space. Therefore \(\frac{f}{ \exp _{\mu }}\) is in AC because of AC is closed. \(\square \)
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Acknowledgements
The first author of this paper would like to thank TUBITAK (The Scientific and Technological Research Council of Turkey) for their financial supports during his Ph.D. studies.
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Ozsarac, F., Aral, A., Karsli, H. (2020). On Bernstein–Chlodowsky Type Operators Preserving Exponential Functions. In: Deo, N., Gupta, V., Acu, A., Agrawal, P. (eds) Mathematical Analysis I: Approximation Theory . ICRAPAM 2018. Springer Proceedings in Mathematics & Statistics, vol 306. Springer, Singapore. https://doi.org/10.1007/978-981-15-1153-0_11
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