Abstract
The aim of this paper is to introduce a Dunkl generalization of the operators including two-variable Hermite polynomials which are defined by Krech and to investigate approximating properties for these operators by means of the classical modulus of continuity, second modulus of continuity and Peetre’s K-functional.
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1 Introduction
Up to now, linear positive operators and their approximation properties have been studied by many research workers, see for example [3,4,5,6, 8, 9, 14, 15, 22, 25, 27] and references therein. Also, linear positive operators defined via generating functions and their further extensions are intensively studied by a large number of authors. For various extensions and further properties, we refer for example Altin et al. [1], Dogru et al. [7], Olgun et al. [18], Sucu et al. [24], Tasdelen et al. [26], Varma et al. [28, 29].
Recently, linear positive operators generated by a Dunkl generalization of the exponential function have been stated by many authors. In [23], Dunkl analogue of Szász operators by using Dunkl analogue of exponential function was given as follows
for \(g\in C[0,\infty ),\) where Dunkl analogue of exponential function is defined by
for \(k\in \mathbb {N} _{0}\) and \(\nu >-\frac{1}{2}\) and the coefficients \(\gamma _{\nu }\) are as follows
in [20]. Also, the coefficients \(\gamma _{\nu }\) verify the recursion relation
where
for \(p\in \mathbb {N} _{0}.\) Similarly, Stancu-type generalization of Dunkl analogue of Szá sz-Kantorovich operators and Dunkl generalization of Szász operators via q-calculus have been defined in [10, 11] and for other research see [16, 17].
The two-variable Hermite Kampe de Feriet polynomials \(H_{n}(\xi ,\alpha )\) are defined by (see [2])
from which, it follows
In a recent paper, Krech [13] has introduced the class of operators \( G_{n}^{\alpha }\) given by
in terms of two-variable Hermite polynomials and investigated approximation properties of \(G_{n}^{\alpha }\) .
In the present paper, we first give the Dunkl generalization of two-variable Hermite polynomials and then we define a class of operators by using the Dunkl generalization of two-variable Hermite polynomials. We give the rates of convergence of the operators \(T_{n}\) to f by means of the classical modulus of continuity, second modulus of continuity and Peetre’s K -functional and in terms of the elements of the Lipschitz class \( Lip_{M}\left( \alpha \right) .\)
2 The Dunkl Generalization of Two-Variable Hermite Polynomials
The Dunkl generalization of two-variable Hermite polynomials is defined by
from which, we conclude
which gives the two-variable Hermite polynomials as \(\mu =0.\ \)For our purpose, we denote
and we can write that the polynomials \(h_{n}^{\mu }(\xi ,\alpha )\) are generated by
where
In order to obtain some properties of \(h_{n}^{\mu }(\xi ,\alpha ),\) we remind the following definition and lemma given in [20].
Definition 1
[20] Let \( {\mu } \in \mathbb {C}_{0}~(\mathbb {C}_{0}:=\mathbb {C\setminus }\left\{ -\frac{1}{2} ,-\frac{3}{2},...\right\} ,~x\in \mathbb {C}\) and let \(\varphi \) be entire function. The linear operator \(\mathbb {D}_{\mu }\) is defined on all entire functions \(\varphi \) on \(\mathbb {C}\) by
We use the notation \(\mathbb {D}_{\mu ,x}\) since \(\mathbb {D}_{\mu }\) is acting on functions of the variable x. Thus, \(\mathbb {D}_{\mu ,x}(\varphi (x))=\left( \mathbb {D}_{\mu }\varphi \right) (x).\)
Lemma 1
[20] Let \(\varphi ,\psi \) be entire functions. For the linear operator \(\mathbb {D}_{\mu }\), the following statements hold
By using these definition and lemma, we can state the next result.
Lemma 2
For the Dunkl generalization of two-variable Hermite polynomials \(h_{n}^{\mu }(\xi ,\alpha )\), the following results hold true
Proof
Applying the linear operator \(\mathbb {D}_{\mu }\) in view of Lemma 1 , we have
Also applying the linear operator \(\mathbb {D}_{\mu }\) to both side of generating function (2.2), we have
By using (2.4) and Lemma 1 (i), we get the first relation. Similarly, if we apply the linear operator \(\mathbb {D}_{\mu }\) to the relation in (i), we get
From (2.4) and Lemma 1, it follows
\(\square \)
Definition 2
With the help of the Dunkl generalization of two-variable Hermite polynomials given in (2.2), we introduce the operators \(T_{n}(f;x),\)\( n\in \mathbb {N} \) given by
where \(\alpha \ge 0,\mu \ge 0,\ f\in C[0,\infty )\) and \(x\in \left[ 0,\infty \right) .\) Operators (2.5) are linear and positive. In the case of \(\mu =0,\) it gives \(G_{n}^{\alpha }\) given by (1.6).
Lemma 3
For the operators \(T_{n}(f;x),\) we can obtain the following equations:
Proof
By using the generating function in (2.2), the relation (i) holds. For the proof of (ii), in view of the recursion relation in (1.4), we get
When we replace k by \(k+1\), we obtain (ii) by use of Lemma 2 (i). For the proof of (iii), by using (1.4), we have
From the equation
it yields
Using the recursion relation in (1.4) in the first series, it follows
From Lemma 2 (i) and (ii), we complete the proof of (iii). \(\square \)
Lemma 4
As a consequence of Lemma 3, we can give the next results for \(T_{n}\) operators
Theorem 1
For \(T_{n}\) operators and any uniformly continuous bounded function g on the interval \([0,\infty )\), we can give
on each compact set \(A\subset \)\([0,\infty )\) when \(n\rightarrow \infty \).
Proof
From Korovkin Theorem in [12], when \(n\rightarrow \infty ,\ \)we have \(T_{n}\left( g;x\right) \overset{\text {uniformly}}{\rightrightarrows } g\left( x\right) \) on \(A\subset [0,\infty )\) which is each compact set because \(\lim _{n\rightarrow \infty }T_{n}(e_{i};x)=x^{i},\) for\(\ i=0,1,2, \) which is uniformly on \(A\subset [0,\infty )\) with the help of using Lemma 4. \(\square \)
Theorem 2
The operator \(T_{n}\) maps \(C_{B}[0,\infty )\) into \(C_{B}[0,\infty )\) and \( \left\| T_{n}\left( f\right) \right\| \le \left\| f\right\| \) for each \(f\in C_{B}[0,\infty )\) where \(C_{B}\) is the space of uniformly continuous and bounded functions on \([0,\infty )\).
3 Convergence of Operators in (2.5)
In what follows, we give some rates of convergence of the operators \(T_{n}\). Firstly, we recall some definitions as follows. Let \(Lip_{M}\left( \alpha \right) \) Lipschitz class of order \(\alpha .\) If \(g\in Lip_{M}\left( \alpha \right) \), the inequality
holds where \(s,t\in [0,\infty ),\ 0<\alpha \le 1\) and \(M>0.\)\( \widetilde{C}[0,\infty )\) is the space of uniformly continuous on \([0,\infty ).\) The modulus of continuity \(g\in \widetilde{C}[0,\infty )\) is denoted by
We first estimate the rates of convergence of the operators \(T_{n}\) by using modulus of continuity and in terms of the elements of the Lipschitz class \( Lip_{M}\left( \alpha \right) .\)
Theorem 3
If \(h\in Lip_{M}\left( \alpha \right) \), we have
where \(\Delta _{2}\) is given in Lemma 4.
Proof
Since \(h\in Lip_{M}\left( \alpha \right) \), it follows from linearity
From Lemma 4 and Hölder’s famous inequality, we can write
Thus, we find the required inequality. \(\square \)
Theorem 4
The operators in (2.5) verify the inequality
where \(g\in \widetilde{C}[0,\infty ).\)
Proof
The proof is clear from the result of Shisha and Mond in [21]. \(\square \)
Let \(C_{B}[0,\infty )\) denote the space of uniformly continuous and bounded functions on \([0,\infty )\). Also
with the norm
for all g in \(C_{B}^{2}[0,\infty ).\)
Lemma 5
For \(h\in C_{B}^{2}[0,\infty )\), the following inequality holds true
where \(\Delta _{1}\) and \(\Delta _{2}\) are given by in Lemma 4.
Proof
From the Taylor’s series of the function h,
Applying the operator \(T_{n}\) to both sides of this equality and then using the linearity of the operator, we have
From Lemma 4, it yields
which finishes the proof. \(\square \)
Now we recall that the second order of modulus continuity of f on \( C_{B}[0,\infty )\) is given as
Peetre’s K-functional of the function \(f\in C_{B}\left[ 0,\infty \right) \) is as follows
The relation between K and \(\omega _{2}\) is as
for all \(\delta >0.\) Here M is a positive constant. Now, we can give the important theorem.
Theorem 5
For the operators defined by (2.5), the following inequality holds
where for all g in \(C_{B}[0,\infty ),\ x\in [0,\infty )\), M is a positive constant which is independent of n and \(\chi _{n}\left( x\right) =\Delta _{1}+\Delta _{2}\).
Proof
For any \(f\in C_{B}^{2}[0,\infty )\), from the triangle inequality, we can write
from Lemma 5, which follows
From (3.4), we have
which holds
from (3.5). \(\square \)
Similar to the proof of above theorem, simple computations give the next theorem.
Theorem 6
If \(g\in C_{B}[0,\infty )\) and \(x\in [0,\infty )\), we get
where M is a positive constant.
Remark 1
Similar results to Theorem 6 can be obtained using the theorem by Paltanea (see [19]).
Remark 2
The case of \(\mu =0\) in Theorem 6 gives the result given in [13].
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Communicated by Rosihan M. Ali.
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Aktaş, R., Çekim, B. & Taşdelen, F. A Dunkl Analogue of Operators Including Two-Variable Hermite polynomials. Bull. Malays. Math. Sci. Soc. 42, 2795–2805 (2019). https://doi.org/10.1007/s40840-018-0631-z
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DOI: https://doi.org/10.1007/s40840-018-0631-z