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Certain approximation properties of Brenke polynomials using Jakimovski–Leviatan operators
Journal of Inequalities and Applications volume 2021, Article number: 104 (2021)
Abstract
In this article, we establish the approximation by Durrmeyer type Jakimovski–Leviatan operators involving the Brenke type polynomials. The positive linear operators are constructed for the Brenke polynomials, and thus approximation properties for these polynomials are obtained. The order of convergence and the weighted approximation are also considered. Finally, the Voronovskaya type theorem is demonstrated for some particular case of these polynomials.
1 Introduction and preliminaries
The Korovkin approximation process plays a crucial role in a wide variety of problems in measure theory, functional analysis, probability theory, and partial differential equations. Korovkin [11] established a well-known simple criterion to decide whether a given sequence \((K_{n})_{n\in \mathbb{N}}\) of positive linear operators on the space \(C[0,1]\) is an approximation process, i.e., \(K_{n}(f)\rightarrow f\) uniformly on \([0,1] \) for every \(f \in C[0,1]\). Taking into account this significant result, mathematicians all across the globe have extended this theorem named after Korovkin to other abstract spaces, such as Banach spaces, lattices, algebras, etc. The work of Korovkin laid a foundation and basis for a new theory, mainly referred to as Korovkin-type approximation theory.
Favard and Szasz [16] introduced the following example of positive linear operator:
where \(0\leqslant u\) and \(g\in \mathbb{C}[0,\infty )\). Numerous mathematicians dealt with the extension of these operators. For example, operators including generalized Appell polynomials [7], Stancu type Baskakov–Durrmeyer operators [10], Stancu type Dunkl generalization of Szász–Kantorovich operators [13].
Leviatan and Jakimovski [8] established the following generalization:
of expression (1.1) by considering the Appell polynomials \(p_{k}(x)\), \(k\geq 0\):
where an analytic function \(h(v)=\sum_{l=0}^{\infty }h_{l} v^{l}\); \(|v|< R\), \(R>1\), and \(0\neq g(1)\), and derived the approximation properties for these above operators.
Recently, Ali Karaisa [9] developed the Durrmeyer–Jakimovski–Leviatan operators of Appell polynomials \(p_{k}(y)\), \(k\geq 0\), ∀ h on \([0,\infty )\) as follows:
where beta function \(\mathbb{B}(k+1,m)\) is a given by
For related work on Jakimovski–Leviatan–beta type integral operators, we refer to [1]. The Szász operators involving Brenke type polynomials were studied in [17].
2 Construction of operators
Motivated by the work of Ali Karaisa [9], here we develop the positive linear operators containing the Brenke polynomials [2, 6], which possess generating relation of the form
where analytic functions g and B are given by
and possess explicit expansions:
Restraining ourselves to \(p_{k}(y)\), i.e., the Brenke polynomials satisfying:
Further, the positive linear operators involving \(p_{k}(y)\) polynomials are introduced while keeping in consideration the above restrictions by the following manner:
where \(y\geq 0\) and \(n\in \mathbb{N}\).
Remark 2.1
For \(B(t)=e^{t}\), expressions (2.6) and (2.1) reduce to the expressions represented by (1.4) and (1.2).
3 Approximation properties of \(\mathbb{T}_{n}\) operators
Korovkin [11, 12] derived the results concerning the convergence of sequences \((K_{m}(g,y))_{m=1}^{\infty }\), where \(K_{m}(g,y) \) are positive linear operators. For instance, if \(K_{m}(g,y)\) uniformly converges to g for some particular cases \(1,t, t^{2} \equiv g(t)\), likewise it performs such activity for each g, being continuous and real. Shisha and Mond in [14, 15] described the rate of convergence for \(K_{m}(g, y)\) in terms of the moduli of continuity of g.
Our aim is to derive the convergence theorem and the order of convergence of operators \(\mathbb{T}_{n}(f;y)\) given by expression (2.6).
Lemma 3.1
From the generating function of the Brenke type polynomials given by (2.1), we obtain
Lemma 3.2
For all \(y\in [0,\infty )\), we have
where
and
Proof
Putting \(f(t)=1\) in operator equation (2.6), we have
Now, putting \(f(t)=t\) in operator equation (2.6), we have
Now, putting \(f(t)=t^{2}\) in operator equation (2.6), we have
Similarly, putting \(f(t)=t^{3}\) in operator equation (2.6), we have
Again, putting \(f(t)=t^{4}\) in operator equation (2.6), we have
□
Lemma 3.3
For \(\mathbb{T}_{n}(f;y)\) operators and for \(y\in [0,\infty )\), the following identities are satisfied:
where \(A_{0}\), \(B_{0}\), \(C_{0}\), \(A_{1}\), \(A_{2}\), \(A_{3}\), \(B_{1}\), \(B_{2}\), \(B_{3}\), \(B_{4}\), and \({\digamma _{1}^{-}}\) are given by equations (3.6)–(3.10) respectively.
Proof
In view of the linearity property of \(\mathbb{T}_{n}\), it follows that
which, on applying Lemma 3.2, yields assertions (3.17), (3.18), and (3.19), respectively. □
Theorem 3.1
Let
If \(f\in C[0,\infty )\cap E\), then \(\lim_{n\rightarrow \infty } \mathbb{T}_{n}(f;y)=f(y)\) and the operators \(\mathbb{T}_{n}\) in each compact subset of \([0,\infty )\) converge uniformly, where \(E:=\{f: {\textit{ for all }} y\in [0,\infty ), |f(y)|\leqslant ce^{Ay}, A \in \mathbb{R} {\textit{ and }} c\in \mathbb{R^{+}}\}\).
Proof
By Lemma 3.2, we find
The above convergence is verified uniformly on each compact subset of \([0,\infty )\). Applying Korovkin’s theorem, the desired result is achieved. □
4 Order of convergence
Now we recall the following definitions.
Definition 4.1
The II modulus of continuity of the function \(h \in \mathbb{C}_{\mathbb{B}}[0,\infty )\) is defined by
where \(\mathbb{C}_{\mathbb{B}}[0,\infty )\) is the class of real-valued functions defined on \([0,\infty )\), which are bounded and uniformly continuous with the norm
Definition 4.2
Peetre’s \(\mathbb{K}\)-functional of the function \(g\in \mathbb{C}_{\mathbb{B}}[0,\infty )\) is defined by
where
and the norm
It is clear that the following inequality
holds for all \(\delta >0\). The constant M is independent of g and δ.
Lemma 4.1
(Gavrea and Raşa [5])
Let \(h\in \mathbb{C}^{2}[0,a]\) and \((\mathbb{K}_{m})_{m\geq 0}\) be a sequence of positive linear operators with the property \(\mathbb{K}_{m}(1;y)=1\). Then
Lemma 4.2
(Zhuk [18])
Let \(g\in \mathbb{C}[a,b]\) and \(h \in (0,\frac{a-b}{2})\). Let \(g_{h}\) be the second-order Steklov function attached to the function g. Then the following inequalities are satisfied:
Now, we compute the rates of convergence of the operators \(\mathbb{T}_{n}(f;y)\) to f by means of a classical approach, the modulus of continuity, and Peetre’s \(\mathbb{K}\)-functional. The following result gives the rates of convergence of the sequence \(\mathbb{T}_{n}(f;y)\) to f by means of modulus of continuity.
Theorem 4.1
For \(f\in C[0,a]\), the following inequality is satisfied:
where
and the second order modulus of continuity is given by \(w_{2}(f;\delta )\) with the norm \(\|f\|=\max_{{y\in [a,b]}}|f(y)|\)
Proof
Let \(f_{h}\) be the second-order Steklov function attached to the function f. In view of identity (3.1), we have
which on using inequality (4.7) becomes
Taking into account that \(f_{h} \in C^{2}[0,a]\), from Lemma 4.1, it follows that
which in view of inequality (4.8) becomes
Further, the Landau inequality
combined with inequality (4.8) gives
Using inequality (4.14) in inequality (4.13) and taking \(h=\sqrt[4]{{\mathbb{T}_{n}((s-y)^{2});y}}\), we find
Making use of inequality (4.15) in inequality (4.11) can lead to assertion (4.9). □
Theorem 4.2
Let \(f \in C^{2}_{B}\), \([0,\infty )\), then
where
Proof
Using Taylor’s expansion of f, the linearity property of the operator \(\mathbb{T}_{n}\), and (3.1), it follows that
From Lemma (3.2), it is evident that
for \(s\geq y\), thus by considering Lemma (3.2) and (3.3) in (4.17), we can write
which completes the proof. □
Theorem 4.3
If \(f\in C_{B}[0,\infty )\), then one has
where
and \(M\geq 0\) is a constant, which is independent of the functions f and δ. Also, \({\xi }_{n}(y)\) is the same as in Theorem 4.2.
Proof
Suppose that \(g\in C^{2}_{B}[0,\infty )\), from previous Theorem 4.2, we have
Since the l.h.s of the above inequality does not depend on the function \(g\in C^{2}_{B}[0,\infty )\),
where K is Peetre’s functional defined by (4.3). By using relation (4.5) in (4.22), the inequality
holds. □
5 Weighted approximation
Here, some properties of approximation for the operator \(\mathbb{T}_{n}\) of a space of weighted continuous functions are given, for which the succeeding class of functions is defined on \([0,\infty )\).
Consider \(B_{y^{2}}[0,\infty )\) defined on \([0,\infty )\) as the set of all functions h which satisfies \(|h(y)|\leq M_{h}(1+y^{2})\), where \(M_{h}\) depends on h is a constant. Also, consider \(C_{y^{2}}[0,\infty )\) as the subspace of \(B_{y}^{2}[0,\infty )\) of all continuous functions. Further, \(C^{*}_{y^{2}}[0,\infty ) \) as the subspace of \(h\in C_{y^{2}}[0,\infty )\) for which \(\lim_{|y|\rightarrow \infty }\frac{h(y)}{1+y^{2}}\) is finite. It is evident that \(C^{*}_{y^{2}}[0,\infty ) \subset C_{y^{2}}[0,\infty ) \subset B_{y}^{2}[0, \infty ) \). The norm on \(C^{*}_{y^{2}}[0,\infty )\) is given as follows:
Lemma 5.1
Let the weight function \(\rho (y)=1+y^{2}\). If \(h\in C_{y^{2}}[0,\infty )\), then
Proof
By expressions (3.1) and (3.3) of Lemma 3.2, for \(n > 1\), we get
Then we deduce
As we know already,
also, we know that
thus in view of these assumptions there exists a positive constant M such that
which concludes the proof. □
By using Lemma 5.1, one can see that the operator \(\mathbb{T}_{n}\) defined by (2.6) acts from \(C_{y^{2}}[0,\infty )\) to \(B_{y^{2}}[0,\infty )\).
Theorem 5.1
Lthe5.1 Let \(\mathbb{T}_{n}\) be the sequence of positive linear operators defined by (2.6) and \(\rho (y)=1+y^{2}\) be a weight function, then for each \(f\in C^{*}_{y^{2}}[0,\infty )\),
Proof
By using the weighted Korovkin theorem presented by Gadzhiev [4], it is enough to verify the following conditions:
By equation (3.2), we have
therefore, keeping in view the fact \(\lim_{n\rightarrow \infty }\frac{B'(ny)}{B(ny)}=1\), we get
By equation (3.3), we have
therefore, keeping in view the fact \(\lim_{n\rightarrow \infty }\frac{B''(ny)}{B(ny)}=1\) and \(\lim_{n\rightarrow \infty }\frac{n}{n-1}=1\), we get
Hence the proof is completed. □
6 Special cases of the operators \(\mathbb{T}_{n}\) and further properties
The Gould–Hopper polynomials \(p_{k}^{d+1}(y;h)\) given by the identity
possess the generating expression
These \(p_{k}^{d+1}(y;h)\) polynomials are \(p_{k}(y)\) Brenke polynomials for \(g(t)=e^{ht^{d+1}}\) and \(B(t)=e^{t}\) in expression (2.1). Thus, for \(g(t)=e^{ht^{d+1}}\) and \(B(t)=e^{t}\) in equation (2.6) gives the following Durrmeyer type Jakimovski–Leviatan operators \(\mathbb{T}_{n}^{*}(f;y)\) involving the \(p_{k}^{d+1}(y;h)\) polynomials:
beneath the presumption \(h \geq 0\).
Now, to prove the Voronovskaya theorem for operators (6.3), first we prove the succeeding results.
Lemma 6.1
\(\forall y\in [0,\infty )\), it follows that
Lemma 6.2
and
Lemma 6.3
Here, we possess the limits:
and
Proof
In view of equation (6.9), we have
where \(\lim_{n\rightarrow \infty }\frac{n}{n-1}=1\) and \(\lim_{n\rightarrow \infty }\frac{1}{n-1}=0\), and in view of equation (6.10), we have
and
□
Theorem 6.1
For any \(f\in C^{*}_{y^{2}}[0,\infty )\) such that \(f', f'' \in C^{*}_{y^{2}}[0,\infty )\), it follows that
Proof
Using Taylor’s expansion of f, we obtain
where
By the linearity of the operator \(\mathbb{T}_{n}^{*}(f;y)\), we get
From Lemma 6.2, we have
For the last term of equation (6.14) or (6.15), using the Cauchy–Schwarz inequality, we get
Because of \(\lim_{n\rightarrow \infty }\mathbb{T}_{n}^{*} (\eta ^{2} ((s,y);y) )=0\) and by Lemma 6.3(ii), \(\lim_{n\rightarrow \infty }n^{2}\mathbb{T}_{n}^{*} (((s-y)^{4};y) )\) is finite, we have \(\lim_{n\rightarrow \infty }n\mathbb{T}_{n}^{*} (\eta (s,y)((s-y)^{2};y) )=0\). Therefore we obtain
□
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References
Alotaibi, A., Mursaleen, M.: Approximation of Jakimovski–Leviatan–Beta type integral operators via q-calculus. AIMS Math. 5(4), 3019–3034 (2020)
Chihara, T.S.: An Introduction to Orthogonal Polynomials. Gordon & Breach, New York (1978)
Ditzian, Z., Totik, V.: Moduli of Smoothness. Springer Series in Computational Mathematics, vol. 9. Springer, New York (1987)
Gadzhiev, A.D.: The convergence problem for a sequence of positive linear operators on unbounded sets and theorems analogous to that of P. P. Krovokin. Sov. Math. Dokl. 15(5), 1433–1436 (1974)
Gavrea, I., Raşa, I.: Remarks on some quantitative Korovkin-type results. Rev. Anal. Numér. Théor. Approx. 22(2), 173–176 (1993)
İçöz, G., Tasdelen Yesildal, F., Dog̈ru, O.: Kantrovich process of linear positive operators via biorthogonal polynomials. J. Inequal. Spec. Funct. 3, 77–84 (2012)
İçöz, G., Varma, S., Sucu, S.: Approximation by operators including generalized Appell polynomials. Filomat 30(2), 429–440 (2016)
Jakimovski, A., Leviatan, D.: Generalized Szász operators for the approximation in the infinite interval. Mathematica 11, 97–103 (1969)
Karaisa, A.: Approximation by Durrmeyer type Jakimovski–Leviatan operators. Math. Methods Appl. Sci. 39(9), 2401–2410 (2015). https://doi.org/10.1002/mma.3650
Kilicman, A., Ayman Mursaleen, M., Al-Abied, A.A.H.: Stancu type Baskakov–Durrmeyer operators and approximation properties. Mathematics 8, Article ID 1164 (2020). https://doi.org/10.3390/math8071164
Korovkin, P.P.: On convergence of linear positive operators in the space of continuous functions. Dokl. Akad. Nauk SSSR 90, 961–964 (1953)
Korovkin, P.P.: Linear Operators and Approximation Theory, Chapters I, II. Hindustan Publ., Delhi (1960) (Russian, 1959) English translation
Milovanovic, G.V., Mursaleen, M., Nasiruzzaman, M.: Modified Stancu type Dunkl generalization of Szász–Kantorovich operators. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 112(1), 135–151 (2018)
Shisha, O., Mond, B.: The degree of convergence of linear positive operators. Proc. Natl. Acad. Sci. USA 60, 1196–1200 (1968)
Shisha, O., Mond, B.: The degree of approximation to periodic functions by linear positive operators. J. Approx. Theory 1, 335–339 (1968)
Szasz, O.: Generalization of S. Bernstein’s polynomials to the infinite interval. J. Res. Natl. Bur. Stand. 45, 239–245 (1950)
Varma, S., Sucu, S., İçöz, G.: Generalization of Szász operators involving Brenke type polynomials. Comput. Math. Appl. 64, 121–127 (2012)
Zhuk, V.V.: Functions of the Lip 1 class and S. N. Bernstein’s polynomials. Vestn. Leningr. Univ., Mat. Meh. Astron. 1, 25–30 (1989)
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Wani, S.A., Mursaleen, M. & Nisar, K.S. Certain approximation properties of Brenke polynomials using Jakimovski–Leviatan operators. J Inequal Appl 2021, 104 (2021). https://doi.org/10.1186/s13660-021-02639-2
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DOI: https://doi.org/10.1186/s13660-021-02639-2