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Approximation by modified Kantorovich–Szász type operators involving Charlier polynomials
Advances in Difference Equations volume 2020, Article number: 192 (2020)
Abstract
In this paper, we give some direct approximation results by modified Kantorovich–Szász type operators involving Charlier polynomials. Further, approximation results are also developed in polynomial weighted spaces. Moreover, for the functions of bounded variation, approximation results are proved. Finally, some graphical examples are provided to show comparisons of convergence between old and modified operators towards a function under different parameters and conditions.
1 Introduction
In 1950, Szász [20] introduced positive linear operators in the sense of exponential growth on nonnegative semiaxes and exhaustively investigated them. These operators later became known as Szász operators. The Szász type operators involving Charlier polynomials were defined in [21] as
where \(b > 1\) and \(y \geq 0\), having the generating functions [5] of the form
where and \((k)_{0} = 1\), \((k)_{m} = k(k+1) \cdots (k+m-1)\), for \(m \geq 1\).
Motivated by the work done in [9], we define the Kantorovich generalization [10] of (1.2) as follows:
where \(\mu _{n}\) and \(\nu _{n}\) are sequences of positive numbers which are increasing and unbounded such that
If we take \(\mu _{n}=\nu _{n}=n\), we will have the operators defined in [9].
For some recent and interesting results on the various generalizations and corresponding approximation results, we refer to [1, 3, 6–8, 14–17, 22].
2 Auxiliary results
We first present some auxiliary results.
Lemma 2.1
Let\(\mathcal{Q}_{n,b}^{(\mu _{n},\nu _{n})}\)be defined by (1.3). Then, we have
- 1.
\(\mathcal{Q}_{n,b}^{(\mu _{n},\nu _{n})}(1;y)= 1\),
- 2.
\(\mathcal{Q}_{n,b}^{(\mu _{n},\nu _{n})}(t;y) = \frac{\mu _{n}}{\nu _{n}}y+\frac{3}{2\nu _{n}}\)
- 3.
\(\mathcal{Q}_{n,b}^{(\mu _{n},\nu _{n})}(t^{2};y) = \frac{\mu _{n}^{2}}{\nu _{n}^{2}}y^{2}+\frac{\mu _{n}}{\nu _{n}^{2}} (4+ \frac{1}{b-1} )y+\frac{10}{3\nu _{n}^{2}}\).
- 4.
\(\mathcal{Q}_{n,b}^{(\mu _{n},\nu _{n})}(t^{3};y) = \frac{\mu _{n}^{3}}{\nu _{n}^{3}}y^{3}+ \frac{\mu _{n}^{2}}{\nu _{n}^{3}} (\frac{15}{2}+\frac{3}{b-1} )y^{2}+\frac{\mu _{n}}{\nu _{n}^{3}} ( \frac{31}{2}+\frac{15}{2(b-1)} +\frac{2}{(b-1)^{2}} )y+\frac{37}{4\nu _{n}^{3}}\).
- 5.
\(\mathcal{Q}_{n,b}^{(\mu _{n},\nu _{n})}(t^{4};y) = \frac{\mu _{n}^{4}}{\nu _{n}^{4}}y^{4}+\frac{\mu _{n}^{3}}{\nu _{n}^{4}} (12+\frac{6}{b-1} )y^{3}+ \frac{\mu _{n}^{2}}{\nu _{n}^{4}} (46+ \frac{36}{b-1}+\frac{11}{(b-1)^{2}} )y^{2} + \frac{\mu _{n}}{\nu _{n}^{4}} (64+\frac{46}{b-1} + \frac{24}{(b-1)^{2}} +\frac{6}{(b-1)^{3}} )y+ \frac{151}{5\nu _{n}^{4}}\).
Proof
With the help of the Charlier polynomials’ generating function given by (1.2), after some simple calculations, we obtain
From the above equalities, the claims of the lemma can be obtained. □
Lemma 2.2
For the operator\(\mathcal{Q}_{n,b}^{(\mu _{n},\nu _{n})}\)given by (1.3), we have the following equalities:
3 Local approximation results
In what follows, let \(\mathcal{Q}_{n,b}^{(\mu _{n},\nu _{n})}(t - y; y)=\chi_{\mu _{n},\nu _{n}}(y)\) and \(\mathcal{Q}_{n,b}^{(\mu _{n},\nu _{n})}((t - y)^{2}; y)=\xi_{\mu _{n},\nu _{n}}(y)\). We will now give two theorems on the uniform convergence and the order of approximation.
Theorem 3.1
Let\(f\in C[0,\infty )\cap G\). Then\(\lim_{n\to \infty }\mathcal{Q}_{n,b}^{(\mu _{n},\nu _{n})}(f;y)=f(y)\), the sequence of operators Eq. (1.3) converges uniformly in each compact subset of\([0,\infty )\), where
Proof
From Lemma 2.1(1)–(3), we get
The proof of the theorem is established by taking advantage of the above uniform convergence in each compact subset of \([0,\infty )\) and the famous Korovkin’s theorem. □
Suppose \(f\in \tilde{C}[0,\infty )\), i.e., f belongs to the space of uniformly continuous functions on \([0,\infty )\). If \(\delta >0\), then the modulus of continuity \(\omega (f,\delta )\) is defined by
Theorem 3.2
Let\(f\in \tilde{C}[0,\infty )\cap E\). For the operators\(\mathcal{Q}_{n,b}^{(\mu _{n},\nu _{n})}(f;y)\)given by (1.3) the following estimate holds:
Proof
From (1.3) and the property of modulus of continuity, the left-hand side of (3.1) leads to
Using Cauchy–Schwarz inequality for the integral, we get
In the above sum, we apply Cauchy–Schwarz inequality, and then in view of Lemma 2.1, (3.2) becomes
where, taking \(\delta =\frac{1}{\nu _{n}}\), we get (3.1). □
Let \(a_{1},a_{2} > 0\) be fixed. We now consider the following space of Lipschitz type (see [18]):
where M is a positive constant and \(0< r\leq 1\).
Theorem 3.3
Let\(f\in \mathrm{Lip}_{M}^{(a_{1},a_{2})}(r)\)and\(r\in (0,1]\), then\(\forall y>0\), we have
Proof
Since
one has
Applying Hölder’s inequality with \(p=\frac{2}{r}\) and \(\frac{2}{2-r}\), we find that
Since \(f\in \mathrm{Lip}_{M}^{(a_{1},a_{2})}(r)\) and \(\frac{1}{t+a_{1}y^{2}+a_{2}y}<\frac{1}{a_{1}y^{2}+a_{2}y}\), \(\forall y\in (0,\infty )\), we have
Our proof is now completed. □
We denote the space of all functions h on \([0,1)\) which are real-valued, uniformly continuous, as well as bounded by \(\tilde{C}_{B}[0,\infty )\) and endow it with the norm \(\Vert h \Vert _{\infty }=\sup_{y\in [0,1)}|h(y)|\). Further, we obtain a local direct estimate for the operators (1.3), using the Lipschitz maximal function of order r introduced by Lenze [13] as:
where \(y \in [0,1)\) and \(r\in (0,1]\).
Theorem 3.4
Let\(f\in \tilde{C}_{B}[0,\infty )\)and\(0< r\le 1\), then\(\forall y\in [0,\infty )\)
Proof
By equation (3.4),
Applying \(\mathcal{Q}_{n,b}^{(\mu _{n},\nu _{n})}\) on both sides of the above inequality, then using Lemma 2.1, as well as Hölder’s inequality with \(p=2/r\), \(q=2/(2-r)\), we obtain
Thus, we have our desired result. □
The Peetre’s K-functional is given by
where \(\tilde{C}_{B}^{2}[0,\infty )= \{ h\in \tilde{C}_{B}[0,\infty ):h^{ \prime },h^{\prime \prime }\in \tilde{C}_{B}[0,\infty ) \} \) with the norm \(\Vert h\Vert _{\tilde{C}_{B}^{2}}=\Vert h\Vert _{\infty }+\Vert h^{ \prime }\Vert _{\infty }+\Vert h^{\prime \prime }\Vert _{\infty }\). Also, the inequality
holds for all \(\delta >0\), where \(\omega _{2}\) is the second-order modulus of smoothness of \(g\in \tilde{C}_{B}[0,\infty )\), which is defined by
Theorem 3.5
If\(f\in \tilde{C}_{B}[0,\infty )\), then
where\(\zeta_{\mu _{n},\nu _{n}}(y)=(\xi_{\mu _{n},\nu _{n}}(y)+\chi^{2}_{\mu _{n},\nu _{n}}(y))/4\). Furthermore,
Proof
For \(f\in \tilde{C}_{B}[0,\infty )\), we define the auxiliary operator as follows:
After taking the absolute value of both sides,
By Lemma 2.1, we have \(\tilde{\mathcal{Q}}_{n,b}^{(\mu _{n},\nu _{n})}(t;y)=y\), and therefore \(\tilde{\mathcal{Q}}_{n,b}^{(\mu _{n},\nu _{n})}(t-y;y)=0\).
Let \(g\in \tilde{C}_{B}^{2}[0,\infty )\), using Taylor’s theorem, we can write
Applying operator \(\tilde{\mathcal{Q}}_{n,b}^{(\mu _{n},\nu _{n})}\) to the above equation, we get
Now taking the absolute value of both sides, we obtain
Therefore, by using the norm on g, we have
Now, using the definition of auxiliary operators (3.5), we get
Combining (3.6) and (3.7) with the above equation, we get
and after taking the infimum on the right-hand side over all \(g\in {\tilde{C}_{B}^{2}}\), we have
This completes the proof of the theorem. □
Theorem 3.6
Let\(f\in \tilde{C}^{1}_{B}[0,\infty )\), then\(\forall y\geq 0\)and\(\delta >0\),
Proof
Since \(f\in \tilde{C}^{1}_{B}[0,\infty )\), we can write
Now, using the well-known property of the modulus of continuity for \(\delta >0\) and \(f\in \tilde{C}^{1}_{B}[0,\infty )\),
hence
Therefore, from (3.8) and the above equation, we have
After applying the Cauchy–Schwarz inequality, we get
Choosing \(\delta =\delta _{n}(y)\), we get our desired result. □
For \(f\in \tilde{C}_{B}[0,\infty )\), the Ditzian–Totik modulus of smoothness [4] of the first order is given by
and an appropriate Peetre’s K-functional is defined by
where \(W_{\varphi }[0,\infty ):=\{g: g\in \mathrm{AC}_{\mathrm{loc}}[0,\infty ), \Vert \varphi g^{\prime } \Vert _{\infty }<\infty \}\) where \(g\in \mathrm{AC}_{\mathrm{loc}}[0,\infty )\) means g is absolutely continuous on every compact subset \([a,b]\) of \([0,\infty )\). It is known from [4] that there exists a constant M such that
Now, we find the order of approximation of the sequence of operators (1.3) by means of Ditzian–Totik modulus of smoothness.
Theorem 3.7
For any\(f\in \tilde{C}_{B}[0,\infty )\)and\(y\in [0,\infty )\),
Proof
Let \(\varphi (y)=\sqrt{y}\), then by Taylor’s theorem, for any \(g\in W_{\varphi }[0,\infty )\), we get
therefore,
which gives
Using Lemma 2.1 and the above equation, for any \(g\in W_{\varphi }[0,\infty )\), we get
Applying the Cauchy–Schwarz inequality yields
Taking infimum on the right-hand side over all \(g\in W_{\varphi }[0,\infty )\), we get
which leads to the required result with the help of the relation between Peetre’s K-functional and Ditzian–Totik modulus of smoothness as given by the relation (3.9). □
4 Approximation results in weighted spaces
Let \(\nu >0\). We denote \(C_{\nu }[0,\infty ):=\{f\in C[0,\infty ):|f(t)|\leq M_{f}(1+t^{\nu }), \forall t\geq 0\}\) equipped with the norm
Further, let \(C_{2}^{*}[0,\infty )\) be the subspace of \(C_{2}[0,\infty )\) consisting of functions f such that \(\lim_{t\to \infty }\frac{f(t)}{1+t^{2}}\) exists.
Theorem 4.1
For each\(f\in C_{2}^{*}[0,\infty )\)and\(r>0\), the following relation holds:
Proof
Let \(y_{0}>0\) be arbitrary but fixed, then by (4.1), we can write
Since \(|f(t)|\leq \Vert f \Vert _{2}(1+y^{2})\), we get
By Korovkin’s theorem, we can see that the sequence of operators \(\{\mathcal{Q}_{n,b}^{(\mu _{n},\nu _{n})}(f;y)\}\) converges uniformly to the function f on every closed interval \([0,a]\) as \(n\to \infty \), (cf. [12, p. 149]). Therefore, for a given \(\epsilon >0\), \(\exists n_{1}\in \mathbb{N}\) such that
By using Lemma 2.1, we can find \(n_{2}\in \mathbb{N}\) such that
Hence
Now, using (4.1),
Let us denote \(n_{0}=\max \{n_{1},n_{2}\}\), then by (4.4), (4.5), and (4.6), we get
Choose \(y_{0}\) so large that
Then, combining (4.3), (4.7), and (4.8), we obtain
Hence, the proof is completed. □
Now, we will obtain the rate of convergence of the operators \(\mathcal{Q}_{n,b}^{(\mu _{n},\nu _{n})}(f;y)\) defined by (1.3) for the functions having derivatives of bounded variation. Let \(\operatorname{DBV}[0,\infty )\) be the space of functions in \(C_{2}[0,\infty )\), which have the derivative of bounded variation on every finite subinterval of \([0,\infty )\). Here, we show at the point y, where \(f^{\prime }(y+)\) and \(f^{\prime }(y-)\) exist, the operators \(\mathcal{Q}_{n,b}^{(\mu _{n},\nu _{n})}(f;y)\) converge to the function \(f(y)\). A function \(f\in \operatorname{DBV}[0,\infty )\) can be represented as
where g denotes a function of bounded variation on every finite subinterval \([0,\infty )\). Many researchers studied in this direction and their work pertaining to this area is described in the papers [2, 11, 19], etc.
In order to study the order of convergence of the operators \(\mathcal{Q}_{n,b}^{(\mu _{n},\nu _{n})}(f;y)\) for the functions having a derivative of bounded variation, we rewrite the operator (1.3) as follows:
where \(W(t,y)\) is a kernel given by
\(\chi _{I}(t)\) being the characteristic function of \(I= [\frac{l}{\nu _{n}},\frac{l+1}{\nu _{n}} ]\).
Lemma 4.2
Let for all\(x>0\)and sufficiently largen,
- (1)
\(\lambda _{\mu _{n},\nu _{n}}(t,y)=\int _{0}^{t}W(u,y)\,du\leq \frac{\xi _{\mu _{n},\nu _{n}}^{2}(x)}{(y-t)^{2}}\), \(0\leq t< y\),
- (2)
\(1-\lambda _{\mu _{n},\nu _{n}}(t,y)=\int _{t}^{\infty }W(u,y)\,du\leq \frac{\xi _{\mu _{n},\nu _{n}}^{2}(y)}{(t-y)^{2}}\), \(y\leq t< \infty \).
Proof
Using Lemma 2.1 and the definition of the kernel, we get
Hence, we have
In the same fashion, we can prove the other inequality, therefore, we omit the details. □
Let \(\bigvee_{a}^{b}f\) be the total variation of f on \([a,b]\), i.e.,
where \(\mathbb{P}\) is the set of all partitions \(P=\{a=y_{0},y_{1},\dots ,y_{n}=b\} \) of \([a,b]\), whick also has the property
Let
Theorem 4.3
Let\(f\in \operatorname{DBV}[0,\infty )\), \(y>0\), andnbe sufficiently large, then we get
Proof
By (4.11), we obtain
where
Now using Lemma 2.1, equations (4.9) and (4.12), we get
Since \(\int _{y}^{t}\delta _{y}(u)\,du=0\), we have
Now, we break the second term on the right-hand side of the above equation as follows:
where
Taking the absolute value on both sides of (4.13), we have
After applying the Cauchy–Schwarz inequality, we obtain
Now applying Lemma 4.2 and integration by parts, \(I_{1}\) can be written as
On taking the absolute value of \(I_{1}\), we have
Since \(f_{y}^{\prime }(y)=0\), by (4.11), we have
Now, using Lemma 4.2,
By the definition of total variation (4.10) and taking \(t=y-y/u\), we obtain
Now, after breaking the integral into a sum, we have
Since by Lemma 4.2, \(\lambda _{\mu _{n},\nu _{n}}(t,y)\leq 1\) and using (4.11), we get
Thus, we get
Using Lemma 4.2, we can write
Now, applying integration by parts and (4.11), we get
Now, by (4.11), we get
and
Using Lemma 4.2, \(1-\lambda _{\mu _{n},\nu _{n}}(t,y)\leq 1\) and (4.11), we get
Now, again with the help of Lemma 4.2 and (4.11), we obtain
By using (4.10) and \(t=y+y/u\), we get
Hence, we derive
Now, we estimate \(P_{3}\). As \(t\geq 2y\), then using \(2(t-y)\geq t\) and \(t-y\geq y\), we get
We can compute \(P_{4}\) as follows:
Hence, we get
Now, from (4.14)–(4.16), we obtain
which gives the desired result. □
5 Graphical examples
Example 5.1
Let us take \(f(x)=5x^{4}-11x^{3}+2x^{2}\). The convergence of the sequence of operators defined by Eq. (1.3) when \(\mu _{n}=\nu _{n}=n\) towards the function \(f(x)\) (cyan) is shown for \(n=10,50,100\), respectively, in Figs. 1–3 taking \(b=2\) (blue), \(b=6\) (black), and \(b=15\) (red). Figures 4–6 illustrate the convergence of the sequence of operators defined by Eq. (1.3) taking \(\mu _{n}=n+\sqrt{n+1}\), \(\nu _{n}=n+12 \) towards the function \(f(x)\) (cyan) for \(n=10,50,100\), keeping the value of b the same.
Also, a direct comparison between the convergence of the old operator applied to f (when \(\mu _{n}=\nu _{n}=n\) discussed in [9]) (blue) and the new operator (red) defined in Eq. (1.3) towards \(f(x)\) (cyan) is shown in Figs. 7–9, respectively, for \(n=10,50,100\), and \(b=10\). It is clear that the new operator exhibits faster convergence towards the limit than the old operator. Also, the new operator is giving flexibility in choosing parameters in the form of the sequences \(\mu _{n}\) and \(\nu _{n}\).
6 Conclusions
We have modified the sequence of operators discussed in [9] and developed many approximation properties such as direct theorems, rate of convergence in weighted spaces, and approximation for functions of bounded variation. Moreover, we have also shown the convergence of old and modified new operators graphically.
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The author (K.J. Ansari) extends his appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through research groups program under grant number G.R.P-93-41.
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Ansari, K.J., Mursaleen, M., Shareef KP, M. et al. Approximation by modified Kantorovich–Szász type operators involving Charlier polynomials. Adv Differ Equ 2020, 192 (2020). https://doi.org/10.1186/s13662-020-02645-6
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DOI: https://doi.org/10.1186/s13662-020-02645-6