Abstract
The main purpose of this paper is to introduce a generalized class of Dunkl type Szász operators via post quantum calculus on the interval \([ \frac{1}{2},\infty )\). This type of modification allows a better estimation of the error on \([ \frac{1}{2},\infty ) \) rather than \([ 0,\infty )\). We establish Korovkin type result in weighted spaces and also study approximation properties with the help of modulus of continuity of order one, Lipschitz type maximal functions, and Peetre’s K-functional. Furthermore, we estimate the degrees of approximations of the operators by modulus of continuity of order two.
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1 Introduction and preliminaries
The first most elegant and easiest proof of Weierstrass approximation theorem was given by S.N Bernstein by introducing positive linear operators [8] known as Bernstein operators. The q-analogue of the Bernstein operators was studied by Lupaş [17] and Phillips [33].
For all \(g\in C[0,\infty ), x\geq 0\), and \(n\in \mathbb{N}\), Szász introduced positive linear operators called Szász operators [38] which are defined by
Recently Szász operators have been studied via Dunkl modification such as the classical Dunkl Szász operators [37], q-Dunkl–Szász operators [14], and \((p,q)\)-Dunkl–Szász operators [7] (see also [1, 6, 32] and [9, 15, 19, 25, 27, 31, 34, 35]). The \((p,q)\)-analogue of Bernstein operators was given in [24] and the Dunkl type modification was studied in [7] (see also [2,3,4,5, 18, 20, 21, 26, 28,29,30, 36, 39]). For some recent work on statistical approxiation of positive linear operators, we refer to [12, 22, 23].
The \((p,q)\)-integer \([n]_{p,q}\) is given by \([n]_{p,q}=\frac{p^{n}-q ^{n}}{p-q}\) for \(n=0,1,2,\ldots \) ; for more details on \([n]_{p,q}\)-integers, see [16]. For the exponential function on \((p,q)\)-analogues one has \(e_{p,q}(x)=\sum_{n=0}^{\infty }p^{ \frac{n(n-1)}{2}}\frac{x^{n}}{[n]_{p,q}!}\) and \(E_{p,q}(x)=\sum_{n=0} ^{\infty }q^{\frac{n(n-1)}{2}}\frac{x^{n}}{[n]_{p,q}!}\).
The q-Hermite type polynomials on q-Dunkl were given in [10] and a recursion formula was obtained by applying a relation. For \(\vartheta >\frac{1}{2}, x\geq 0, 0< q<1\), and \(g\in C[0,\infty )\), Içöz gave a Dunkl generalization of Szász operators via q-calculus [14] as follows:
Recently, the \((p,q)\)-approximation of Szász operators on Dunkl analogue has been studied in [7] by using the following exponential function:
for \(\vartheta >\frac{1}{2}\), \(0< q< p\leq 1\), \(x\in {}[ 0,\infty )\), and \(u\in \mathbb{N}\). The explicit formula for \(\varTheta _{\vartheta,p,q}(u)\) is given by
where \([\frac{u}{2}]\) denotes the greatest integer functions for \(u\in \mathbb{N}\cup \{0\}\). Also
and
2 Auxiliary results
Let \(\{\zeta _{n}(x)\}_{n\geq 1}\) be a sequence of nonnegative continuous functions on \([0,\infty )\) such that
where
Moreover, suppose
Let \(x\in {}[ 0,\infty ), g\in C[0,\infty ), n\in \mathbb{N}, 0< q< p \leq 1\), and \(\vartheta >\frac{1}{2}\). We define the new operators by
If we put \(\zeta _{n}(x)=x\), then these operators are reduced to the operators studied in [7] and, in addition, if \(p=1\), then we get the operators studied in [14].
Lemma 2.1
Suppose that the operators \(A_{n,p,q}^{\ast }(\cdot;\cdot)\) are given by (2.4). Then, for all \(x\geq \frac{1}{2[n]_{p,q}}\) and \(n\in \mathbb{N}\), one obtains
-
(1)
\(A_{n,p,q}^{\ast }(1;x)=1\);
-
(2)
\(A_{n,p,q}^{\ast }(t;x)=x-\frac{1}{2[n]_{p,q}}\);
-
(3)
\(x^{2}+\frac{1}{[n]_{p,q}} (q^{2\vartheta }[1-2 \vartheta ]_{p,q}\mathcal{H}_{n,\vartheta }(x) -1 )x+ \frac{1}{4[n]_{p,q} ^{2}} (1-2q^{2\vartheta }[1-2 \vartheta ]_{p,q}\mathcal{H}_{n, \vartheta }(x) ) \leq A_{n,p,q}^{\ast }(t^{2};x)\leq x^{2}+ \frac{1}{[n]_{p,q}} ( {}[ 1+2\vartheta ]_{p,q}-1 )x+\frac{1}{4[n]_{p,q} ^{2}} (1-2 [1+2\vartheta ]_{p,q} )\).
Lemma 2.2
For all \(x\geq \frac{1}{2[n]_{p,q}}\) and \(n\in \mathbb{N}\), the operators \(A_{n,p,q}^{\ast }(\cdot;\cdot)\) satisfy
-
(1)
\(A_{n,p,q}^{\ast }(t-x;x)=-\frac{1}{2[n]_{p,q}} \);
-
(2)
\(A_{n,p,q}^{\ast }((t-x)^{2};x)\leq \frac{1}{[n]_{p,q}}[1+2 \vartheta ]_{p,q}x+\frac{1}{4[n]_{p,q}^{2}} (1-[1+2\vartheta ]_{p,q} )\).
3 Approximation in weighted spaces
This section deals with the approximation properties of the operators \(A_{n,p,q}^{\ast }\) in weighted spaces. We evaluate the order of approximation by using the modulus of continuity and Lipschitz class and study some direct theorems. We also obtain the approximation results by modulus of continuity of order two. We denote \(C_{B}(\mathbb{R^{+}})\) for the set of all bounded and continuous functions on \(\mathbb{R^{+}} \) equipped with the norm
where \(\mathbb{R^{+}}=[0,\infty )\). We suppose \(F:=\{g:x\in {}[ 0, \infty )\}\) such that \(\frac{g(x)}{1+x^{2}}\) is convergent when \(x\rightarrow \infty \). Let \(B_{\varsigma }(\mathbb{R}^{+})\) be the set of all functions satisfying \(g(x)\leq u_{g}\varsigma (x)\) with \(\varsigma (x)=1+\xi ^{2}(x)\) and \(\xi (x)\rightarrow x\) in which \(u_{g}\) is a constant depending on g (see Gadžiev [13]). Moreover, take \(C_{\varsigma }(\mathbb{R}^{+})=B_{\varsigma }( \mathbb{R}^{+})\cap C(\mathbb{R}^{+})\). Note that \(B_{\varsigma }( \mathbb{R}^{+})\) is a normed space with the norm given by
Let \(C_{\varsigma }^{0}(\mathbb{R}^{+})\) be a subset of \(C_{\varsigma }(\mathbb{R}^{+})\) such that
We consider the positive sequences \(q=q_{n}\) and \(p=p_{n}\) with \(0< q_{n}<1\) and \(q_{n}< p_{n}\leq 1\) such that
where \(0< c,d\leq 1\).
Theorem 3.1
Let the sequences of positive numbers \(p_{n}\) and \(q_{n}\) be such that \(0< q_{n}< p_{n}\leq 1\). Then, for all \(f\in C[0,\infty )\cap F\), the operators \(A_{n,p_{n},q_{n}}^{\ast }(\cdot;\cdot)\) are uniformly convergent on each compact subset of \([0,\infty )\).
Proof
In the light of Korovkin’s theorem, we prove the uniform convergence of a sequence of \(A_{n,p_{n},q_{n}}^{\ast }\) on \([0,1]\) as \(n\rightarrow \infty \) by
Clearly, from (3.1) and \(\frac{1}{[n]_{p_{n},q_{n}}}\rightarrow 0\) \((n\rightarrow \infty )\), we have
□
Theorem 3.2
Let \(A_{n,p_{n},q_{n}}^{\ast }:C_{\varsigma }(\mathbb{R}^{+})\rightarrow B_{\varsigma }(\mathbb{R}^{+})\). Then, for all \(g\in C_{\varsigma } ^{0}(\mathbb{R}^{+})\),
if and only if
Proof
Consider \(\xi (x)=x\), \(\varsigma =1+\xi ^{2}(x)\) and
From Korovkin’s theorem, easily we obtain \(\lim_{n\rightarrow \infty } \vert \vert A_{n,p_{n},q_{n}}^{\ast } ( t^{\ell };x ) -x^{\ell } \vert \vert _{\varsigma }=0\) for \(\ell =0,1,2\). Hence, for any \(g\in C_{\varsigma }^{0}(\mathbb{R} ^{+})\), we get
□
Theorem 3.3
For every \(g \in C_{\varsigma }^{0}(\mathbb{R}^{+})\), we have
Proof
We prove this theorem in the light of Theorem 3.2. Take \(f(t)=t ^{\ell } \) for \(\ell =0,1,2\) in Lemma 2.1. Then Korovkin’s theorem allows for every \(g(t)\in C_{\varsigma }^{0}(\mathbb{R}^{+})\) if it satisfies \(A_{n,p_{n},q_{n}}^{\ast }(t^{\ell };x)\rightarrow x^{ \ell }\) uniformly. Then, for \(\ell =0\), Lemma 2.1 gives \(A_{n,p_{n},q_{n}}^{\ast }(1;x)=1\), which implies that
If \(\ell =1\)
then
Similarly, for \(\ell =2\), we have
Hence,
This completes the proof. □
4 Rate of convergence
Here, we compute the rate of convergence of our new operators (2.4) with the help of modulus of continuity and Lipschitz type maximal functions.
Let \(g\in C[0,\infty ]\). The modulus of continuity of g is given by
for any \(\delta >0\). It is known that \(\lim_{\delta \rightarrow 0+} \omega _{\varrho }(g;\delta )=0\), and one has
Theorem 4.1
Let \(\omega _{\varrho }(g;\delta )\) be defined on the interval \([0,\varrho +1]\subset {}[ 0,\infty )\) with \(\varrho >0\). Then, for every \(g\in C_{\varsigma }^{u}\) on \([0,\infty )\), we have
Proof
To prove this theorem, we use the Cauchy–Schwarz inequality and apply (4.1) and (4.2). Thus, we have
if we choose \(\delta =\sqrt{\frac{1}{[n]_{p,q}}}\), then we get our result. □
We now give the rate of convergence of \(A_{n,p,q}^{\ast }\) in terms of the elements of the usual Lipschitz class \(\mathrm{Lip}_{K}(\mu )\).
Let \(g\in C[0,\infty )\), \(K>0\), and \(0<\mu \leq 1\). The Lipschitz class \(\mathrm{Lip}_{K}(\mu )\) is given by
Theorem 4.2
Let \(A_{n,p,q}^{\ast }(\cdot;\cdot)\) be the operator defined in (2.4). Then, for each \(g\in \mathrm{Lip}_{K}(\mu )\) with \(K>0\), \(0<\mu \leq 1\) and satisfying (4.3), we have
Proof
We apply Hölder’s inequality.
Therefore
which proves the theorem. □
We consider the following space:
which is equipped with the norm
also
Theorem 4.3
Let us consider the operators \(A_{n,p,q}^{\ast }(\cdot;\cdot)\) given in (2.4). Then, for any \(g \in C_{B}^{2}(\mathbb{R}^{+})\), we have
Proof
Suppose that \(g\in C_{B}^{2}(\mathbb{R}^{+})\). It follows from Taylor series expansion that
Since the operator \(A_{n,p,q}^{\ast }\) is linear, by operating \(A_{n,p,q}^{\ast }\) on both sides of the last equality, we have
which yields
From (4.5), we have
Consequently,
which completes the proof. □
Peetre’s K-functional is defined by
where
Then there exists a constant \(M>0\) such that
where \(\omega _{2}(g;\delta ^{\frac{1}{2}})\) (second order modulus of continuity) is given by
Theorem 4.4
For every \(g\in C_{B}(\mathbb{R}^{+})\), there exists a positive constant M such that
Proof
We prove this by using Theorem (4.3)
Considering the infimum over all \(f\in C_{B}^{2}(\mathbb{R}^{+})\) and using (4.7), we obtain
where
Now, for an absolute constant \(M>0\) in [11], we use the relation
which proves our theorem. □
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Alotaibi, A. Approximation by a generalized class of Dunkl type Szász operators based on post quantum calculus. J Inequal Appl 2019, 241 (2019). https://doi.org/10.1186/s13660-019-2182-8
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DOI: https://doi.org/10.1186/s13660-019-2182-8