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Approximation by Jakimovski–Leviatan-beta operators in weighted space
Advances in Difference Equations volume 2020, Article number: 393 (2020)
Abstract
The main purpose of this article is to introduce a more generalized version of Jakimovski–Leviatan-beta operators through the Appell polynomials. We present some uniform convergence results of these operators via Korovkin’s theorem and obtain the rate of convergence by using the modulus of continuity and Lipschitz class. Moreover, we obtain the approximation with the help of Peetre’s K-functional and give some direct theorems.
1 Introduction
Approximation theory basically deals with problems to find approximation of functions by simpler functions like polynomial. Bernstein [8] was first to construct a sequence of positive linear operators to provide a constructive proof of the well-known Weierstrass approximation theorem. Since then several operators have been defined to study approximation properties in different spaces.
In [13], Jakimovski and Leviatan introduced the following operators and obtained some of their approximation properties:
For all \(h\in E[0,\infty)\), the set of functions of exponential type on \([0,\infty)\) with \(|{h}(x)|\leq\beta e^{\alpha x}\), \(\alpha,\beta>0\), where \(Q_{k}(y)=\sum_{j=0}^{k}b_{{j}}\frac{y^{k-j}}{(k-j)!}\) (\(k\in\mathbb{N}\)), are Appell polynomials [5] defined by the identity
such that \(P(w)=\sum_{m=0}^{\infty}b_{m}w^{m}\), \(P(1)\neq0\) is an analytic function in the disk \(|w|< r\) (\(r>1\)). Note that, for \(P(1)=1\), \(Q_{{m}}(y)=\frac{y}{{m!}}\) and operators (1.1) are reduced to Favard–Szász operators:
Recently, Büyükyazıcı et al. [9] studied the following operators:
In this paper, we generalize the above operators and study their several approximation properties. We investigate a Korovkin-type theorem and obtain the order of convergence by using the modulus of continuity. Furthermore, we obtain the approximation with the help of Lipschitz continuous functions and give some direct theorems.
For more details on the related work, we refer to [1–4, 6, 7, 11, 14, 15, 17–19, 21, 23–26].
We define an integral type modification of Jakimovski–Leviatan operators by introducing the sequences of unbounded and increasing functions \(\{u_{m}\}\), \(\{v_{m}\}\) such that
Let \(m\in\mathbb{N}\), \(\phi>m\) and
For \(Q_{r}(y)\geq0\) and \(P(1)\neq0\), we define
2 Moments
Lemma 2.1
Suppose\(e_{i}=t^{i-1}\) (\(i=1,2,3\)). Then the following hold true for operators (1.7):
-
(1)
\(L_{m}^{u_{m},v_{m} }(e_{1};y) =1\);
-
(2)
\(L_{m}^{u_{m},v_{m} }(e_{2};y)= \frac{u_{m}}{v_{m}}\frac{m}{(m-1)} y+\frac{1}{v_{m}} \frac{m}{(m-1)} \bigl(\frac{P^{\prime}(1)}{P(1)} +1\bigr)\);
-
(3)
\(L_{m}^{u_{m},v_{m} }(e_{3};y)= (\frac{u_{m}}{v_{m}} )^{2}\frac{m^{2}}{(m-2)(m-1)} y^{2}+2\frac{u_{m}}{v_{m}^{2}}\frac{m^{2}}{(m-2)(m-1)} \bigl( \frac {P^{\prime }(1)}{P(1)}+2 \bigr)y+ \frac{1}{v_{m}^{2}}\frac{m^{2}}{(m-2)(m-1)} \bigl( \frac{P^{\prime \prime }(1)}{P(1)}+4\frac{P^{\prime}(1)}{P(1)}+2 \bigr)\).
Proof
We can easily see that
(1) Take \(h=e_{1}\)
(2) Take \(h=e_{2}\)
(3) Take \(h=e_{3}\)
□
Lemma 2.2
If\(\varUpsilon_{j}=(e_{2}-y)^{j}\)for\(j=1,2\), then we have
- 1∘:
-
\(L_{m}^{u_{m},v_{m} }(\varUpsilon_{1};y)= \bigl( \frac{u_{m}}{v_{m}}\frac {m}{(m-1)}-1 \bigr) y+\frac{1}{v_{m}}\frac{m}{(m-1)} \bigl(\frac{P^{\prime}(1)}{P(1)}+ 1\bigr) \);
- 2∘:
-
\(L_{m}^{u_{m},v_{m} }(\varUpsilon_{2};y) = \bigl[ (\frac {u_{m}}{v_{m}} )^{2}\frac{m^{2}}{(m-2)(m-1)}-2\frac{u_{m}}{v_{m}}\frac{m}{(m-1)}+1 \bigr] y^{2} + \bigl[ 2\frac{u_{m}}{v_{m}^{2}}\frac{m^{2}}{(m-2)(m-1)} \bigl( \frac{P^{\prime }(1)}{P(1)}+2 \bigr)-2\frac{u_{m}}{v_{m}}\frac{m}{(m-1)} \bigl( \frac{P^{\prime}(1)}{P(1)}+1 \bigr) \bigr]y+ \frac{1}{v_{m}^{2}}\frac{m^{2}}{(m-2)(m-1)} \bigl( \frac{P^{\prime\prime }(1)}{P(1)}+\frac{P^{\prime}(1)}{P(1)}+2\bigr )\).
3 Approximation in a weighted space
In this section we present approximation results in the space \(C_{B}[0,\infty)\) of all bounded and continuous functions on \([0,\infty)\), which is a linear normed space with the norm
For our convenience, we rewrite
Let
Theorem 3.1
Let\(h\in C_{{\gamma} }[0,\infty)\cap E\)and\({\gamma} \geq2\). Then
Proof
In the lighting of Korovkin’s theorem and Lemma 2.1, it is obvious that
□
Following Gadžiev [12], we recall the weighted spaces for which the analogues of Korovkin’s theorem hold [20]. Let ϕ be a continuous and strictly increasing function and \(\varrho(y)=1+\phi ^{2}(y)\), \(\lim_{y\rightarrow\infty}\varrho(y)=\infty\). Moreover,
with \(\Vert h\Vert_{\varrho}=\sup_{y\geq0}\frac{|h(y)|}{\varrho(y)}\), where the constant \(K_{h}\) depends only on h. The sequence of positive linear operators \(\{L_{m}^{u_{m},v_{m}}\}_{m\geq1}\) maps \(C_{\varrho }[0,\infty)\) into \(B_{\varrho}[0,\infty)\) if and only if
where \(y\in{}[0,\infty)\) and \(K>0\) is a constant. Finally, let \(C_{\varrho}^{0}[0,\infty)\subset C_{\varrho}[0,\infty)\) satisfying
Theorem 3.2
(cf. [16])
Let the sequence of positive linear operators\(K_{m}\), acting from\(C_{\varrho}[0,\infty) \)to\(B_{\varrho}[0,\infty)\), satisfy the conditions
Then, for each\(h\in C_{\varrho}^{0}[0,\infty)\),
Theorem 3.3
For every\(h\in C_{\varrho}^{0}[0,\infty)\), operators\(L_{m}^{u_{m},v_{m}}\)satisfy
Proof
In view of Theorem 3.2, suppose \({\phi(y)}=y\), then \(\varrho(y)=1+y^{2}\). Since
For i = 1, from Lemma 2.1 we get
For \(i=2\),
Therefore
For \(i=3\),
Therefore,
Lemma 2.1 implies that
□
4 Rate of convergence
We write \({C^{*}_{B}}[0,\infty)\) for the set of all uniformly continuous and bounded functions on \([0,\infty)\) with \(\Vert f\Vert _{C_{B}[0,\infty)}=\sup_{y\in{}[0,\infty)}| f(y)|\). For
we have
and \(\lim_{\delta^{\circ}\rightarrow0+}\omega^{\circ}(h,\delta ^{\circ })=0\).
For \(C>0\) and \(0<\zeta\leq1\), the Lipschitz class is defined by
Theorem 4.1
Let\(m\in\mathbb{N}\)and\(m>2\), then for all\(h\in{C^{*}_{B}}[0,\infty)\)
where\((\varPsi_{m})^{2}= \bigl{[} ( \frac{u_{m}}{v_{m}} ) ^{2}\frac{m^{2}}{(m-2)(m-1)}-2\frac{u_{m}}{v_{m}}\frac{m}{(m-1)}+1 \bigr{]} y^{2} + \bigl{[} \frac{u_{m}}{v_{m}^{2}}\frac{m^{2}}{(m-2)(m-1)} \bigl( 2\frac {P^{\prime }(1)}{P(1)}+4 \bigr) -\frac{1}{v_{m}}\frac{m}{(m-1)}\bigl( \frac{2P^{\prime }(1)}{P(1)}+2 \bigr) \bigr{]} y+\frac{1}{v_{m}^{2}}\frac{m^{2}}{(m-2)(m-1)} \bigl( \frac{P^{\prime \prime}(1)}{P(1)}+\frac{P^{\prime}(1)}{P(1)}+2 \bigr) \).
Proof
We have
In the light of (4.1) and (4.2), we get
Apply the Cauchy–Schwarz inequality
so that
Choosing \(\delta^{\circ}=\varPsi_{m}=\sqrt{L_{m}^{u_{m},v_{m}} ( \varUpsilon_{2};y ) }\) yields the result. □
Remark 4.2
For \(u_{m}=v_{m}=1\), the above estimate is reduced to [22], i.e.,
where \((\varPhi_{m})^{2}= \bigl( \frac{m^{2}}{(m-2)(m-1)}- \frac{2m}{(m-1)}+1\bigr) y^{2} +\bigl[\frac{2m}{(m-2)(m-1)}\bigl( \frac{P^{\prime}(1)}{P(1)} ) +2 ) - \frac{2}{(m-1)} \bigl( \frac{P^{\prime}(1)}{P(1)} +1\bigr)\bigr]y + \frac{1}{(m-2)(m-1)} \bigl( \frac{P^{\prime\prime}(1)}{P(1)}+\frac{P^{\prime}(1)}{P(1)}+2 \bigr) \).
Theorem 4.3
For every\(h\in \operatorname{Lip}_{C}(\zeta)\), we have
where\(m>2\), \(m \in\mathbb{N} \), and\(\varPsi_{m}=\sqrt{L_{m}^{u_{m},v_{m}} (\varUpsilon_{2};y )}\)by Theorem4.1.
Proof
We use (4.3) and Hölder’s inequality to get
This completes the proof. □
5 Direct theorems
Let
with the norm
and let \(\varOmega^{\circ}= \{ f\in C_{B}[0,\infty ):f^{\prime},f^{\prime\prime}\in C_{B}[0,\infty) \}\). For \(h\in C_{B}[0,\infty)\), Peetre’s K-functional is defined by
For a positive constant M, one has \(K_{2}^{\circ}(h,\delta^{\circ })\leq M\omega_{2}^{\circ}(h,\sqrt{\delta^{\circ}})\), where \({\delta^{\circ}}>0\) and the second order modulus of continuity \(\omega _{2}^{\circ}\) is defined by
Theorem 5.1
Let\(m>2\), \(m\in\mathbb{N}\). Then for all\(f\in C_{B}^{\kappa}[0,\infty )\)we have
where\(\varPsi_{m}\)is defined by Theorem4.1.
Proof
By Taylor’s formula, one has
where
Therefore,
and
Thus, we get
where
Hence the result. □
Theorem 5.2
For every\(h\in C_{B}[0,\infty)\)and\(m>2\), \(m\in\mathbb{N}\), we have
Proof
As previously, we easily conclude that
By taking infimum and using (5.3), we get
Now, for an absolute constant \(M>0\) [10], we use the relation
This completes the proof. □
6 Conclusion
We have constructed an integral type modification of Jakimovski–Leviatan operators by using beta function and two sequences of unbounded and increasing functions \(\{u_{m}\}\), \(\{v_{m}\}\) such that \(\lim_{m\rightarrow\infty}\frac{u_{m}}{v_{m}}=1+O ( \frac{1}{v_{m}} ) \) and \(\lim_{m\rightarrow\infty}\frac{1}{v_{m}}{=0}\). We derived some uniform convergence results of these operators via Korovkin’s theorem and obtained the rate of convergence by using the modulus of continuity and Lipschitz class. Furthermore, we obtained some direct theorems with the help of Peetre’s K-functional.
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Nasiruzzaman, M., Mursaleen, M. Approximation by Jakimovski–Leviatan-beta operators in weighted space. Adv Differ Equ 2020, 393 (2020). https://doi.org/10.1186/s13662-020-02848-x
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DOI: https://doi.org/10.1186/s13662-020-02848-x