Abstract
The main goal of this paper is to introduce Durrmeyer modifications for the generalized Szász–Mirakyan operators defined in (Aral et al., in Results Math 65:441–452, 2014). The construction of the new operators is based on a function \(\rho \) which is continuously differentiable \(\infty \) times on \( \left[ 0,\infty \right) ,\) such that \(\rho \left( 0\right) =0\) and \( \inf _{x\in \left[ 0,\infty \right) }\rho ^{\prime }\left( x\right) \ge 1.\) Involving the weighted modulus of continuity constructed using the function \( \rho \), approximation properties of the operators are explored: uniform convergence over unbounded intervals is established and a quantitative Voronovskaya theorem is given. Moreover, we obtain direct approximation properties of the operators in terms of the moduli of smoothness. Our results show that the new operators are sensitive to the rate of convergence to f, depending on the selection of \(\rho .\) For the particular case \(\rho \left( x\right) =x\), the previous results for classical Szász-Durrmeyer operators are captured.
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1 Introduction
Approximation theory has an important role in mathematical research, with a great potential for applications. Since Korovkin’s famous theorem in 1950, the study of the linear methods of approximation given by sequences of positive and linear operators became a firmly entrenched part of approximation theory. Due to this fact, the well-known operators such as Bernstein, Szász, Baskakov etc. and their generalizations have been studied intensively. Recently Cárdenas-Morales et al. [8] introduced Bernstein-type operators defined for \(f\in C\left[ 0,1\right] \) by \(B_{n}\left( f\circ \tau ^{-1}\right) \circ \tau \), \(B_{n}\) being the classical Bernstein operators and \(\tau \) being any function that is continuously differentiable \(\infty \) times on \(\left[ 0,1\right] ,\) such that \(\tau \left( 0\right) =0,\) \(\tau \left( 1\right) =1\) and \(\tau ^{\prime }\left( x\right) >0\) for \(x\in \left[ 0,1\right] \). They investigated the shape preserving and convergence properties, as well as the asymptotic behavior and saturation. A Durrmeyer type generalization of \(B_{n}\left( f\circ \tau ^{-1}\right) \circ \tau \) was also studied in [3]. The results of the aforementioned papers show that it is possible to obtain some improvements of the classical approximation by Bernstein and Bernstein-Durrmeyer operators in certain senses, simultaneously. Very recently Aral et al. [6] introduced similar modifications of the Szász–Mirakyan operators. Let us recall that construction.
Set \({{\mathbb {N}}}_{0}={{\mathbb {N}}}\cup \left\{ 0\right\} \) and let \({{\mathbb {R}}} ^{+}\) be the positive real semi-axis \(\left[ 0,\infty \right) .\) Assume that \(\rho \) is any function satisfying the conditions:
- \(\left( p_{1}\right) \) :
-
\(\rho \) is a continuously differentiable function on \({{\mathbb {R}}}^{+},\)
- \(\left( p_{2}\right) \) :
-
\(\rho \left( 0\right) =0,\) \(\inf _{x\in \left[ 0,\infty \right) }\rho ^{\prime }\left( x\right) \ge 1.\)
The generalized Szász–Mirakyan operators are defined by
where \({\mathcal {P}}_{n,\rho ,k}\left( x\right) :=\exp \left( -n\rho \left( x\right) \right) \left( n\rho \left( x\right) \right) ^{k}/k!\). \(S_{n}\) are the classical Szász–Mirakyan operators and can be obtained from \( S_{n}^{\rho }\) as a particular case \(\rho \left( x\right) =x.\) The weighted uniform convergence of \(S_{n}^{\rho }\) to f, the rate of convergence with the aid of weighted modulus of continuity and some shape preserving properties of \(S_{n}^{\rho }\) were studied.
The aim of this article is to introduce Durrmeyer type modifications of the operators (1.1). A Durrmeyer type generalization of Szász–Mirakyan operators was introduced in [18]. Later on, further Durrmeyer type generalizations of the Szász–Mirakyan operators have been studied intensively. Among others, we refer the reader to [5, 7, 15] and references therein. The general integral modification of ( 1.1) to approximate Lebesgue integrable functions on \({\mathbb {R}} ^{+}\) can be defined as
where \(n\in {{\mathbb {N}}}\), \(p_{n,k}\left( t\right) =\exp \left( -nx\right) \left( nx\right) ^{k}/k!\) and \(\rho \) is any function with the assumptions \( \left( \rho _{1}\right) \) and \(\left( \rho _{2}\right) .\) The operators \( D_{n}^{\rho }\) are linear and positive and in the case of \(\rho \left( x\right) =x\), the operators reduce to the classical ones. By considering the notion of \(\rho \)-convexity (a function \(f\in C^{k}\left( {\mathbb {R}} ^{+}\right) \) is said to be \(\rho \)-convex of order \(k\in {{\mathbb {N}}}\) whenever \(D_{\rho }^{k}f:=D^{k}\left( f\circ \rho ^{-1}\right) \circ \rho \ge 0,\) where D is the differential operator). Note that the operators \(D_{n}^{\rho }\) map \(\rho \)-convex functions of order k onto \( \rho \)-convex functions of order k, so they are said to be \(\rho \)-convex of order k, which means that \(D_{n}^{\rho }\) transform the so called \(\rho \)-polynomials into polynomials of the same degree, that is, if we consider the set \({\mathbb {P}}_{\rho ,k}:=\left\{ \rho ^{i}:i=0,1,...k,\text { }k\in {{\mathbb {N}}}\right\} ,\) then \(D_{n}^{\rho }\left( {\mathbb {P}}_{\rho ,k}\right) \subset {\mathbb {P}}_{\rho ,k}.\)
We shall first show that the operators (1.2) are an approximation process for functions belonging to a weighted space, we shall prove uniform convergence of the operators and determine the degree of this uniform convergence as well. In the next section, we obtain local approximation properties. The last section is devoted to a Voronovskaya type theorem in quantitative form. Quantitative Voronovskaya theorems have been studied intensively in the last decade. This kind of results are very useful to describe the rate of point-wise convergence and error of approximation simultaneously. In the paper [19], a quantitative Voronovskaya type theorem was presented for Bernstein operators in terms of usual modulus of continuity and in terms of the least concave majorant of usual modulus of continuity in [13, 14]. Since the modulus of continuity doesn’t work on unbounded intervals, we obtain the corresponding theorem with the weighted modulus of continuity. Some quantitative form of Voronvskaya’s theorem on bounded and unbounded intervals, we refer the readers to [1, 2, 4].
2 Preliminary results
In what follows, we give the moments and recurrence relation for the central moments of the operators without proofs since they are similar to the corresponding results for the Szász-Durrmeyer operators. Also they can be verified just by taking \(\rho \left( x\right) =x.\) We also recall the weighted modulus of continuity and its properties.
Lemma 2.1
We have
Lemma 2.2
If we define the central moment of degree m,
then we have
Also, using the above recurrence relation we get
Throughout the paper we shall consider the following class of functions. \( C_{B}\left( {\mathbb {R}}^{+}\right) \) is the space of all real valued continuous and bounded functions f on \({\mathbb {R}}^{+}.\) Let \(\varphi \left( x\right) =1+\rho ^{2}\left( x\right) .\)
where \(M_{f}\) is a constant depending only on f. \(C_{B}\left( {\mathbb {R}} ^{+}\right) \) is the linear normed space with the norm \(\left\| f\right\| =\sup _{x\in {\mathbb {R}}^{+}}\left| f\left( x\right) \right| \) and the other spaces are normed linear spaces with the norm \( \left\| f\right\| _{\varphi }=\sup _{x\in {\mathbb {R}}^{+}}\left| f\left( x\right) \right| /\varphi \left( x\right) .\)
The weighted modulus of continuity defined in [16] is given by
for each \(f\in C_{\varphi }\left( {\mathbb {R}}^{+}\right) \) and for every \( \delta >0.\) We observe that \(\omega _{\rho }\left( f;0\right) =0\) for every \( f\in C_{\varphi }\left( {\mathbb {R}}^{+}\right) \) and the function \(\omega _{\rho }\left( f;\delta \right) \) is nonnegative and nondecreasing with respect to \(\delta \) for \(f\in C_{\varphi }\left( {\mathbb {R}}^{+}\right) \) and also \(\lim _{\delta \rightarrow 0}\omega _{\rho }\left( f;\delta \right) =0\) for every \(f\in U_{\varphi }\left( {\mathbb {R}}^{+}\right) \)(For more details see [16]).
Let \(\delta >0\) and \({\mathcal {W}}_{\infty }^{2}=\left\{ g\in C_{B}\left[ 0,\infty \right) ;g^{\prime },g^{\prime \prime }\in C_{B}\left[ 0,\infty \right) \right\} .\) The Peetre’s K-functional is defined by
where
It was shown in [9] that there exists an absolute constant \(C>0\) such that
where the second order modulus of continuity is defined by
The usual modulus of continuity for \(f\in C_{B}\left( {\mathbb {R}}^{+}\right) \) is given by
3 Uniform convergence of \({\mathcal {D}}_{n}^{\rho }\)
In this section, we obtain the uniform convergence of the operators \( {\mathcal {D}}_{n}^{\rho }\) in terms of the weighted Korovkin theorem [11, 12] and we describe the rate of the corresponding uniform convergence. Foremost, we recall the weighted form of the Korovkin theorem.
Lemma 3.1
([11]) The positive linear operators \(L_{n}\), \(n\ge 1\), act from \( C_{\varphi }\left( {\mathbb {R}}^{+}\right) \) to \(B_{\varphi }\left( {\mathbb {R}} ^{+}\right) \) if and only if the inequality
holds, where \(K_{n}\) is a positive constant depending on n.
Theorem 3.2
([11]) Let the sequence of linear positive operators \(\left( L_{n}\right) \), \(n\ge 1,\) acting from \(C_{\varphi }\left( {\mathbb {R}} ^{+}\right) \) to \(B_{\varphi }\left( {\mathbb {R}}^{+}\right) \) satisfy the following three conditions
Then for any function \(f\in C_{\varphi }^{*}\left( {\mathbb {R}}^{+}\right) \),
Therefore, we have the following result.
Theorem 3.3
For each function \(f\in C_{\varphi }^{*}\left( {\mathbb {R}}^{+}\right) \)
Proof
We first have to show that \({\mathcal {D}}_{n}^{\rho }:C_{\varphi }\left( {\mathbb {R}}^{+}\right) \rightarrow B_{\varphi }\left( {\mathbb {R}}^{+}\right) .\) In fact, using (2.1)–(2.2) we have
Since
and
we get
which verifies our assertion by Lemma 3.1. On the other hand, since
we deduce
by Theorem 3.2. \(\square \)
Let us describe the rate of the above convergence. To do this, we consider the following theorem proved in [16].
Theorem 3.4
([16]) Let \(L_{n}\): \(C_{\varphi }\left( {\mathbb {R}} ^{+}\right) \rightarrow B_{\varphi }\left( {\mathbb {R}} ^{+}\right) \) be a sequence of positive linear operators with
where \(a_{n},\) \(b_{n},\) \(c_{n}\) and \(d_{n}\) tend to zero as \(n\rightarrow \infty .\) Then
for all \(f\in C_{\varphi }\left( {\mathbb {R}} ^{+}\right) \), where
Theorem 3.5
For all \(f\in C_{\varphi }\left( {\mathbb {R}} ^{+}\right) \) we have
Proof
In order to apply Theorem 3.4, we should calculate the sequences \(a_{n},\) \(b_{n},\) \(c_{n}\) and \(d_{n}.\) In light of (2.1) and (2.2) we obtain
and
Also by (3.1) we have
Finally using (2.3), we get
Since all conditions of Theorem 3.4 are satisfied, the desired result follows. \(\square \)
4 Local approximation
Theorem 4.1
Let \(\rho \) be a function satisfying the conditions (\(p_{1}\)), (\(p_{2}\)) and \(\left\| \rho ^{\prime \prime }\right\| \) be finite. If \(f\in C_{B}\left( {\mathbb {R}} ^{+}\right) ,\) then we have
where C is a constant independent of n.
Proof
Let us consider the auxiliary operator
It is clear by Lemma 2.1 that
and
The classical Taylor’s expansion of \(g\in {\mathcal {W}}_{\infty }^{2}\) yields for \(t\in {\mathbb {R}} ^{+}\) that
On the other hand, with the change of variable \(u=\rho \left( y\right) \) we get
Using the equality
one can write
So (4.3) can be written as
Since \(\rho \) is strictly increasing on \(\left[ 0,\infty \right) \) and with the condition (\(p_{2}\)), we get
Also, it is clear that
Hence we have
and choosing \(C:=\max \left\{ 1, \Vert \rho ^{^{\prime \prime }} \Vert \right\} \) we have
Taking the infimum on the right hand side over all \(g\in {\mathcal {W}}_{\infty }^{2}\) we obtain
Furthermore, for \(x\in {\mathbb {R}} ^{+}\) we have
Since \(0\le \rho \left( \bar{y}\right) -\rho \left( \bar{x}\right) \le t,\) then \(0\le \left( \bar{y}-\bar{x}\right) \rho ^{\prime }\left( u\right) \le t\) for some \(u\in \left( \bar{x},\bar{y}\right) ,\) i.e., \(0\le \bar{y}- \bar{x}\le t/\rho ^{\prime }\left( u\right) \le t.\) Thus we have
Hence we have
\(\square \)
5 Pointwise convergence of \({\mathcal {D}}_{n}^{\rho }\)
In this section, we shall focus on pointwise convergence of \({\mathcal {D}} _{n}^{\rho }\) by obtaining the Voronovskaya theorem in quantitative form. We need the following lemma.
Lemma 5.1
([16]) For every \(f\in C_{\varphi }\left( {\mathbb {R}} ^{+}\right) ,\) for \(\delta >0\) and for all \(x,y\ge 0,\)
holds.
Theorem 5.2
If the function \(\rho \) satisfies the conditions (\(p_{1}\)), (\(p_{2}\)) and \( f^{\prime \prime }/\left( \rho ^{^{\prime }}\right) ^{2},f^{\prime }.\rho ^{^{\prime \prime }}/\left( \rho ^{^{\prime }}\right) ^{3}\in C_{\varphi }\left( {\mathbb {R}} ^{+}\right) \), then we have for any \(x\in \left[ 0,\infty \right) \) that
where \(\delta _{n}^{\rho }(x)=\left( 144\frac{\left( 1+n\rho \left( x\right) \right) ^{2}}{n^{4}}\right) ^{\frac{1}{3}}.\)
Proof
By the Taylor expansion of \(f\circ \rho ^{-1}\) we can write
where
and \(\xi \) is a number between \(\rho \left( x\right) \) and \(\rho \left( t\right) .\) Applying the operator \({\mathcal {D}}_{n}^{\rho }\) to both sides of equality (5.2), we immediately have
Let us estimate \(\left| h\left( t,x\right) \right| .\) Using (4.4) and (5.1), respectively, we have
On the other hand, since \(\varphi \left( t\right) \,+\,\varphi \left( x\right) \le \delta ^{2}+2\rho ^{2}\left( x\right) +2\rho \left( x\right) \delta +2\) whenever \(\left| \rho \left( t\right) -\rho \left( x\right) \right| \le \delta \), we have
and since \(\varphi \left( t\right) +\varphi \left( x\right) \le \left( \frac{\rho \left( t\right) -\rho \left( x\right) }{\delta }\right) ^{2}\left( \delta ^{2}+2\rho ^{2}\left( x\right) +2\rho \left( x\right) \delta +2\right) \) whenever \(\left| \rho \left( t\right) \right. \left. -\rho \left( x\right) \right| >\delta \), we have
Choosing \(\delta <1\) we deduce
So using Lemma 2.2 and the Cauchy–Schwarz inequality we get
and if we choose \(\delta =\left( \sqrt{\frac{\mu _{n,4}\left( x\right) \mu _{n,6}\left( x\right) }{\mu _{n,2}\left( x\right) }}\right) ^{\frac{1}{3}}\) we get
On the other hand, straightforward calculations give
Hence we have
If the above estimates are substituted in (5.3), we get the desired result. \(\square \)
Corollary 5.3
We have the following particular cases:
\(\mathrm{\left( i\right) } \) Suppose that \(\rho \left( x\right) =x.\) If \(f^{\prime \prime }\in C_{x^{2}}\left( {\mathbb {R}} ^{+}\right) \) (where \(C_{x^{2}}\left( {\mathbb {R}} ^{+}\right) \) is the analogues one of \(C_{\varphi }\left( {\mathbb {R}} ^{+}\right) \)), then we have for any \(x\in \left[ 0,\infty \right) \) that
where \(\Omega \left( f;\delta \right) \) is another weighted modulus of continuity defined in [17] and \(\delta _{n}(x)=\left( 144\frac{ \left( 1+nx\right) ^{2}}{n^{4}}\right) ^{\frac{1}{3}}\).
\(\mathrm{\left( ii\right) }\) If \(f^{\prime \prime }/\left( \rho ^{^{\prime }}\right) ^{2},f^{\prime }.\rho ^{^{\prime \prime }}/\left( \rho ^{^{\prime }}\right) ^{3}\in U_{\varphi }\left( {\mathbb {R}} ^{+}\right) \), then we have for any \(x\in \left[ 0,\infty \right) \) that
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The authors are thankful to the referee(s) for making valuable suggestions leading to a better presentation of the paper. Thanks are also due to Prof. András Kroó for sending the reports timely.
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Acar, T., Ulusoy, G. Approximation by modified Szász-Durrmeyer operators. Period Math Hung 72, 64–75 (2016). https://doi.org/10.1007/s10998-015-0091-2
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DOI: https://doi.org/10.1007/s10998-015-0091-2