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Approximation properties of generalized Baskakov–Schurer–Szasz–Stancu operators preserving \(e^{-2ax}, a>0\)
Journal of Inequalities and Applications volume 2019, Article number: 112 (2019)
Abstract
The current paper deals with a modified form of the Baskakov–Schurer–Szasz–Stancu operators which preserve \(e^{-2ax}\) for \(a>0\). The uniform convergence of the modified operators is shown. The rate of convergence is investigated by using the usual modulus of continuity and the exponential modulus of continuity. Then Voronovskaya-type theorem is given for quantitative asymptotic estimation.
1 Introduction
Use of linear positive operators has played a crucial role in approximation theory for the last seven decades. In 1950, Szász [22] defined
for \(x\in [0,\infty )\). In 1957, Baskakov [6] proposed
In 1962, Schurer [20] introduced
where \(x\in [0,1]\) and p is a non-negative integer. In 1983, Stancu [21] studied
satisfying the condition \(0\leq \alpha \leq \beta \). Various studies related to these operators, such as Baskakov–Szász type operators [12], Baskakov–Schurer–Szász operators [18], Baskakov–Szász–Stancu operators [17], and q-Baskakov–Schurer–Szász–Stancu operators [19], have been conducted.
In 2010, Aldaz and Render [4] introduced linear and positive operators preserving 1 and \(e^{x}\). In 2017, Acar et al. [2] examined a modified form of the Szász–Mirakyan operators which reproduces constant and \(e^{2ax}, a>0\). After that, in some studies Szász–Mirakyan operators [5], Baskakov–Szász–Mirakyan-type operators [10], Phillips operators [13], Szász–Mirakyan–Kantorovich operators [11], and Baskakov operators [24] preserving constant and exponential function were examined. In addition, Kajla [16] studied Srivastava–Gupta operators preserving linear functions. On the other hand, Gupta and Tachev [14] found the general estimation in terms of Pǎltǎneaś modulus of continuity.
In 2018, Bodur et al. [7] analyzed Baskakov–Szász–Stancu operators preserving exponential functions. Motivated by this paper, we construct a new generalization of the Baskakov–Schurer–Szász–Stancu operators
where s is a positive integer, p is a non-negative integer, and \(0\leq \alpha \leq \beta \). By taking \(\alpha =0\) and \(\beta =0\), we obtain Baskakov–Schurer–Szász operators [18]. In addition, by taking \(s+p=u\), we get Baskakov–Szász–Stancu operators [17]. Moreover, by taking \(s+p=u\), \(\alpha =0\) and \(\beta =0\), we have Baskakov–Szász operators [12]. We deal with the following modified form:
Assume that the operators (1) preserve \(e^{-2ax}, a>0\). In that case, we find the function \(v_{s,p}(x)\) satisfying \(M_{s}^{\alpha, \beta }(e^{-2at};x)=e^{-2ax}\) as follows:
By a simple computation, we have
2 Some auxiliary results
Here, we present some important equalities and auxiliary lemmas, necessary for the proof of the main theorems.
Negative binomial series is given as follows:
Lemma 1
Let \(v_{s,p}(x)\) be given by (2), then we have
Proof
Take \(f(t)=e^{-At}\), then by using (3) and (4) we obtain
□
Lemma 2
Let \(e_{k}(t)=t^{k}, k=0,1,2,3,4\). Then we get the following equalities:
Proof
Take \(f(t)=e_{1}\), then by using (3) and (4) we have
By the same manner, other results can be obtained. □
Lemma 3
Let us briefly denote \(\phi _{x}^{k}(t)=(t-x)^{k}\) for \(k=0,1,2,4\). Then we obtain the following equalities for the central moments:
Proof
We use the linearity of the \(M_{s}^{\alpha,\beta }\) operators and Lemma 2 \(M_{s}^{\alpha,\beta }(\phi _{x}^{0};x)=M_{s}^{\alpha,\beta }(e_{0};x)\), \(M_{s}^{\alpha,\beta }(\phi _{x}^{1};x)= M_{s}^{\alpha,\beta }(e_{1};x)-x M_{s}^{\alpha,\beta }(e_{0};x) \), \(M_{s}^{\alpha,\beta }(\phi _{x}^{2};x)= M_{s}^{\alpha,\beta }(e_{2};x)-2x M_{s}^{\alpha,\beta }(e_{1};x)+ x ^{2}M_{s}^{\alpha,\beta }(e_{0};x) \), \(M_{s}^{\alpha,\beta }(\phi _{x}^{4};x)= M_{s}^{\alpha,\beta }(e_{4};x)-4x M_{s}^{\alpha,\beta }(e_{3};x)+6x ^{2}M_{s}^{\alpha,\beta }(e_{2};x)-4x^{3}M_{s}^{\alpha,\beta }(e _{1};x)+x^{4}M_{s}^{\alpha,\beta }(e_{0};x) \). □
Remark 1
Considering the definition of \(v_{s,p}(x)\), we obtain the following limits for every \(x\in [0,\infty )\) and \(0\leq \alpha \leq \beta \):
and
3 Main results
Let \({C^{*}[0,\infty )}\) denote the subspace of all real-valued continuous functions on \({[0,\infty )}\) with the condition that \(\lim_{m\rightarrow \infty }f(x)\) exists and is finite, equipped with the uniform norm. The uniform convergence of a sequence of linear positive operators is demonstrated by Boyanov and Veselinov [8]. We present the following theorem according to [8] for the newly constructed operators (1).
Theorem 1
If the linear positive operators (1) satisfy
uniformly in \([0,\infty )\), then for each \(f\in {C^{*}[0,\infty )}\)
uniformly in \([0,\infty )\).
Proof
We have already known that \(\lim_{s\rightarrow \infty } M _{s}^{\alpha,\beta } (1;x)=1\). Considering equality (5) with \(v_{s,p}(x)\) given in (2), we have
and
Hence, we prove that
uniformly in \([0,\infty )\). This means that, for any \(f\in {C^{*}[0, \infty )}\), \(\lim_{s\rightarrow \infty } M_{s}^{\alpha, \beta } (f;x)=f(x) \) uniformly in \([0,\infty )\). □
After about four decades later than Boyanov and Veselinov [8], Holhoş [15] studied the uniform convergence of a sequence of linear positive operators. He obtained the following theorem for an effective estimation of the linear positive operators.
Theorem 2
([15])
For a sequence of linear positive operators \(A_{s}: C ^{*}[0,\infty )\longrightarrow C^{*}[0,\infty )\), we have
for every function \(f\in C^{*}[0,\infty )\), where
and \(\omega ^{*}(f,\eta )=\sup_{|e^{-x}-e^{-t}| \leq \eta, x,t>0} |f(t)-f(x)|\) denotes the modulus of continuity. Here, \(\delta _{s}, \sigma _{s}\), and \(\rho _{s}\) tend to zero as \(s\rightarrow \infty \).
In the same manner with the above theorem, we present a quantitative estimation of the Baskakov–Schurer–Szasz–Stancu operators which preserve \(e^{-2ax}, a>0\), as follows.
Theorem 3
For \(f\in C^{*}[0,\infty )\), we have the following inequality:
where
Here, \(\sigma _{s,p}\) and \(\rho _{s,p}\) tend to zero as \(s\rightarrow \infty \). So, \(M_{s}^{\alpha,\beta }f\) converges to f uniformly.
Proof
The Baskakov–Schurer–Szasz–Stancu operators \(M_{s}^{\alpha,\beta }\) preserve constant. Thus, \(\delta _{s,p}= \Vert M_{s}^{\alpha,\beta }(e_{0})-1 \Vert _{[0,\infty )}=0\). In order to calculate \(\sigma _{s,p}\), we take into consideration equality (10). So, we obtain
Since
we achieve
In the same way, with the help of equality (11), we have
By using
we get
Consequently, \(\sigma _{s,p}\) and \(\rho _{s,p}\) tend to zero as \(s\rightarrow \infty \). □
In Sect. 4, we investigate the rate of convergence by using the usual modulus of continuity.
4 The usual modulus of continuity
The class of all bounded and uniform continuous functions f on \([0,\infty )\) is denoted by \(C_{B}[0,\infty )\) endowed with the norm \(\Vert f \Vert _{C_{B}}=\sup_{x\geq 0} |f(x)|\). For \(f\in C_{B}[0,\infty )\), the modulus of continuity is given by
The second order modulus of continuity of the function \(f\in C_{B}[0, \infty )\) is defined by
where \(\delta >0\). Peetre’s K-functionals are described as
Here, \(C_{B}^{2}[0,\infty )\) denotes the space of the functions f, for which \(f'\) and \(f''\) belong to \(C_{B}[0,\infty )\). The relation between the second order modulus of continuity and Peetre’s K-functional is given by [9]
where \(M>0\).
Lemma 4
For \(f\in C_{B}[0,\infty )\), we have \(|M_{s}^{\alpha,\beta } (f;x)| \leq \Vert f \Vert \).
Theorem 4
Let \(f\in C_{B}[0,\infty ) \). Then, for all \(x \in {[0,\infty )}\), there exists a positive constant M such that
where
Here, \(v_{s,p}(x)\) is the same as in (2).
Proof
We define the auxiliary operators \(\tilde{M}_{s}^{\alpha,\beta }:C _{B}[0,\infty )\rightarrow C_{B}[0,\infty )\)
where \(v_{s,p}(x)\) is as given by (2). Note that the operators (15) are positive and linear. By using the Taylor expansion for \(g \in C_{B}^{2}[0,\infty )\), we have
Applying \(\tilde{M}_{s}^{\alpha,\beta }\) operators to the both sides of equation (16) and using Lemma 3, we obtain
Further,
and
Rewrite (18) and (19) in (17), then we have
where
By using the auxiliary operators (15) and Lemma 4, we get
From (15), (20), and (22), for every \(g\in {C_{B} ^{2}[0,\infty )}\), we obtain
□
Remark 2
We see that \(\mu _{s,p}=\frac{x(2+x)}{s+p}+O((s+p)^{-2})\rightarrow 0\), when \(s\rightarrow \infty \). This result guarantees the convergence of Theorem 4.
In Sect. 5, we obtain the rate of convergence by using the exponential modulus of continuity.
5 The exponential modulus of continuity
For \(f\in C[0,\infty )\), the exponential growth of order \(B>0\) is given by
The first order modulus of continuity of functions with exponential growth is defined as
Let \(f\in \operatorname{Lip}(c,B)\) for some \(0< c\leq 1\). Then, for each \(\delta <1\),
Let K be a subspace of \(C[0,\infty )\) which contains functions f with exponential growth, \(\Vert f \Vert _{B}<\infty \).
Theorem 5
Let \({M}_{s}^{\alpha,\beta }:K\rightarrow C[0,\infty )\) be the sequence of linear positive operators preserving \(e^{-2ax}\), \(a>0\). We assume that \({M}_{s}^{\alpha,\beta }\) satisfy
for fixed \(x\in {[0,\infty )}\) and for \(B>0\). Additionally, if \(f\in C^{2}[0,\infty )\cap K\), \(0< c\leq 1\), and \(f''\in \operatorname{Lip}(c,B)\), then for fixed \(x\in {[0,\infty )}\), we have
Proof
We begin with the Taylor expansion of the function \(f\in C^{2}[0,\infty )\) at \(x\in {[0,\infty )}\).
where \(H_{2}(f;t,x)= \frac{ ( f''(\eta )-f''(x) ) (t-x)^{2}}{2}\) is the remainder term. Here, η is between t and x. Applying the operators \({M}_{s}^{\alpha,\beta }\) to equality (28), we obtain
Here,
Tachev et al. [23] proved that, for each \(h>0\) and \(k\in \mathbb{N}\),
By using inequality (30), we get
Therefore,
Applying the operators \({M}_{s}^{\alpha,\beta }\) to inequality (31), we write
By some computations we obtain
Since \(s+p\geq 1\),
We have the following inequalities with the help of Cauchy–Schwarz inequality:
Thus, by using inequalities (32), (33), and (34) in (29), we write
Finally, when we choose \(h=\sqrt{\frac{{M}_{s}^{\alpha,\beta } ( \phi _{x}^{4};x )}{ {M}_{s}^{\alpha,\beta } ( \phi _{x}^{2};x )}}\) and substitute it in (35), we obtain
Note that, for fixed \(x\in {[0,\infty )}\), \(\frac{{M}_{s}^{\alpha, \beta } ( \phi _{x}^{4};x )}{ {M}_{s}^{\alpha,\beta } ( \phi _{x}^{2};x )}=\frac{5x(2+x)}{s+p}+O((s+p)^{-2})\rightarrow 0\) as \(s\rightarrow \infty \). This result guarantees the convergence of Theorem 5. □
In Sect. 6, we give the Voronovskaya-type theorem to examine the asymptotic behavior of the constructed operators (1). For the quantitative Voronovskaya-type theorems, we refer to the pioneering works [1] and [3].
6 Voronovskaya-type theorem
Theorem 6
For \(f,f''\in C^{*}[0,\infty )\) and \(x\in {[0,\infty )}\), we have the inequality
where
Proof
By the Taylor expansion for a function f, we write
where
Here, \(k(t,x)\) is the remainder term and ξ is a number between x and t. Applying the \({M}_{s}^{\alpha,\beta }\) operators to (36), we obtain
Then
We briefly denote that \(r_{s,p}(x):=s{M}_{s}^{\alpha,\beta }(\phi _{x}^{1};x)-(2ax+ax^{2})\) and \(t_{s,p}(x):=\frac{s}{2} {M}_{s}^{\alpha,\beta }(\phi _{x}^{2};x)- (x+\frac{x^{2}}{2} )\). Thus,
Note that by using equalities (6) and (7), \(r_{s,p}(x)\) and \(t_{s,p}(x)\) go to zero as \(s\rightarrow \infty \). Now, we deal with the term \(\vert s{M}_{s}^{\alpha,\beta }(k(t,x)\phi _{x}^{2};x) \vert \).
Using this inequality, we have
For \(\eta >0\), if \(|e^{-x}-e^{-t}|>\eta \), then \(\vert k(t,x) \vert \leq \frac{2(e^{-x}-e^{-t})^{2}}{\eta ^{2}} \omega ^{*}(f'',\eta )\) and if \(|e^{-x}-e^{-t}|\leq \eta \), then \(\vert k(t,x) \vert \leq 2 \omega ^{*}(f'',\eta )\). Thus, we write \(\vert k(t,x) \vert \leq 2 (1+\frac{(e^{-x}-e^{-t})^{2}}{\eta ^{2}} ) \omega ^{*}(f'',\eta )\). Therefore,
If we choose \(\eta =1/\sqrt{s}\) and \(z_{s,p}:=\sqrt{s^{2}{M}_{s} ^{\alpha,\beta } ( (e^{-x}-e^{-t})^{4};x ) }\sqrt{s^{2} {M}_{s}^{\alpha,\beta } ( \phi _{x}^{4};x ) }\), we get
□
Remark 3
We obtain the following result by some calculations:
Additionally, we get the following result:
We give the following corollary as a result of Theorem 6 and Remark 3.
Corollary 1
Suppose that \(f,f''\in C^{*}[0,\infty )\) and \(x\in {[0,\infty )}\). Then the equality
holds.
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Sofyalıoğlu, M., Kanat, K. Approximation properties of generalized Baskakov–Schurer–Szasz–Stancu operators preserving \(e^{-2ax}, a>0\). J Inequal Appl 2019, 112 (2019). https://doi.org/10.1186/s13660-019-2062-2
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DOI: https://doi.org/10.1186/s13660-019-2062-2
MSC
- 41A25
- 41A36
- 47A58
Keywords
- Exponential functions
- Modulus of continuity
- Voronovskaya-type theorem