Skip to main content

On modified Dunkl generalization of Szász operators via q-calculus

Abstract

The purpose of this paper is to introduce a modification of q-Dunkl generalization of exponential functions. These types of operators enable better error estimation on the interval \([\frac{1}{2},\infty)\) than the classical ones. We obtain some approximation results via a well-known Korovkin-type theorem and a weighted Korovkin-type theorem. Further, we obtain the rate of convergence of the operators for functions belonging to the Lipschitz class.

Introduction and preliminaries

In 1912, Bernstein [1] introduced the following sequence of operators \(B_{n}:C[0,1]\rightarrow C[0,1]\) defined by

$$ B_{n}(f;x)=\sum_{k=0}^{n} \binom{n}{k}x^{k}(1-x)^{n-k}f \biggl( \frac{k}{n} \biggr) , \quad x\in{}[0,1] $$
(1.1)

for \(n\in\mathbb{N}\) and \(f\in C[0,1]\).

In 1950, for \(x \geq0\), Szász [2] introduced the operators

$$ S_{n}(f;x)=e^{-nx}\sum _{k=0}^{\infty}\frac{(nx)^{k}}{k!} f \biggl( \frac{k}{n} \biggr),\quad f \in C[0,\infty). $$
(1.2)

In the field of approximation theory, the application of q-calculus emerged as a new area. The first q-analogue of well-known Bernstein polynomials was introduced by Lupaş by applying the idea of q-integers [3]. In 1997, Phillips [4] considered another q-analogue of the classical Bernstein polynomials. Later on, many authors introduced q-generalizations of various operators and investigated several approximation properties [514].

We now present some basic definitions and notations of the q-calculus which are used in this paper [15].

Definition 1.1

For \(|q|<1\), the q-number \([ \lambda ] _{q} \) is defined by

$$ [\lambda ]_{q}=\left \{ \textstyle\begin{array}{l@{\quad}l} \frac{1-q^{\lambda}}{1-q} & (\lambda\in\mathbb{C}), \\ \sum_{k=0}^{n-1}q^{k}=1+q+q^{2}+\cdots+q^{n-1} & (\lambda=n\in \mathbb{N}).\end{array}\displaystyle \right . $$
(1.3)

Definition 1.2

For \(|q|<1\), the q-factorial \([ n ] _{q}!\) is defined by

$$ [ n ] _{q}!=\left \{ \textstyle\begin{array}{l@{\quad}l} 1 & (n=0), \\ \prod_{k=1}^{n} [ k ] _{q} & (n\in\mathbb{N}).\end{array}\displaystyle \right . $$
(1.4)

Our investigation is to construct a linear positive operator generated by a generalization of the exponential function defined by (see [16])

$$ e_{\mu}(x)=\sum_{n=0}^{\infty} \frac{x^{n}}{\gamma_{\mu}(n)}, $$

where

$$ \gamma_{\mu}(2k)=\frac{2^{2k}k!\Gamma ( k+\mu+\frac{1}{2} ) }{\Gamma ( \mu+\frac{1}{2} ) }, $$

and

$$ \gamma_{\mu}(2k+1)=\frac{2^{2k+1}k!\Gamma ( k+\mu+\frac {3}{2} ) }{\Gamma ( \mu+\frac{1}{2} ) }. $$

The recursion formula for \(\gamma_{\mu}\) is given by

$$ \gamma_{\mu}(k+1)=(k+1+2\mu\theta_{k+1})\gamma_{\mu }(k), \quad k=0,1,2,\ldots, $$

where \(\mu>-\frac{1}{2}\) and

$$ \theta_{k}=\left \{ \textstyle\begin{array}{l@{\quad}l} 0 & \text{if }k\in2\mathbb{N}, \\ 1 & \text{if }k\in2\mathbb{N}+1.\end{array}\displaystyle \right . $$

Sucu [17] defined a Dunkl analogue of Szász operators via a generalization of the exponential function [16] as follows:

$$ S_{n}^{\ast}(f;x):=\frac{1}{e_{\mu}(nx)}\sum _{k=0}^{\infty}\frac {(nx)^{k}}{\gamma_{\mu}(k)}f \biggl( \frac{k+2\mu\theta_{k}}{n} \biggr) , $$
(1.5)

where \(x\geq0\), \(f\in C[0,\infty)\), \(\mu\geq0\), \(n\in\mathbb{N}\).

Cheikh et al. [18] stated the q-Dunkl classical q-Hermite-type polynomials and gave definitions of q-Dunkl analogues of exponential functions and recursion relations for \(\mu>-\frac{1}{2}\) and \(0< q<1\),

$$\begin{aligned}& e_{\mu,q}(x)=\sum_{n=0}^{\infty} \frac{x^{n}}{\gamma_{\mu,q}(n)}, \quad x\in {}[0,\infty), \end{aligned}$$
(1.6)
$$\begin{aligned}& E_{\mu,q}(x)=\sum_{n=0}^{\infty} \frac{q^{\frac {n(n-1)}{2}}x^{n}}{\gamma _{\mu,q}(n)}, \quad x\in{}[0,\infty), \end{aligned}$$
(1.7)

where

$$ \gamma_{\mu,q}(n)=\frac{(q^{2\mu+1},q^{2})_{[\frac{n+1}{2}]}(q^{2},q^{2})_{[\frac{n}{2}]}}{(1-q)^{n}},\quad n\in\mathbb{N}. $$
(1.8)

Some of the special cases of \(\gamma_{\mu,q}(n)\) are defined as follows:

$$\begin{aligned}& \gamma_{\mu,q}(0)=1,\qquad \gamma_{\mu,q}(1)=\frac{1-q^{2\mu+1}}{1-q}, \\& \gamma_{\mu,q}(2)= \biggl(\frac{1-q^{2\mu+1}}{1-q} \biggr) \biggl( \frac {1-q^{2}}{1-q} \biggr), \\& \gamma_{\mu,q}(3)= \biggl(\frac{1-q^{2\mu+1}}{1-q} \biggr) \biggl( \frac {1-q^{2}}{1-q} \biggr) \biggl(\frac{1-q^{2\mu+3}}{1-q} \biggr), \\& \gamma_{\mu,q}(4)= \biggl(\frac{1-q^{2\mu+1}}{1-q} \biggr) \biggl( \frac {1-q^{2}}{1-q} \biggr) \biggl(\frac{1-q^{2\mu+3}}{1-q} \biggr) \biggl( \frac {1-q^{4}}{1-q} \biggr). \end{aligned}$$

In [19], Içöz and Çekim gave the Dunkl generalization of Szász operators via q-calculus as follows:

$$ D_{n,q}(f;x)=\frac{1}{e_{\mu,q}([n]_{q}x)}\sum_{k=0}^{\infty} \frac{([n]_{q}x)^{k}}{\gamma_{\mu,q}(k)}f \biggl( \frac{1-q^{2\mu\theta _{k}+k}}{1-q^{n}} \biggr) $$
(1.9)

for \(\mu>\frac{1}{2}\), \(x\geq0\), \(0< q<1\) and \(f\in C[0,\infty)\).

Previous studies demonstrate that providing a better error estimation for positive linear operators plays an important role in approximation theory, which allows us to approximate much faster to the function being approximated.

Motivated essentially by Içöz and Çekim’s [19] recent investigation of Dunkl generalization of Szász-Mirakjan operators via q-calculus, we show that our modified operators have better error estimation than those in [19]. We also prove several approximation results and successfully extend the results of [19]. Several other related results are also discussed.

Construction of operators and moments estimation

Let \(\{r_{[n]_{q}}\}\) be a sequence of real-valued continuous functions defined on \([0,\infty)\) with \(0\leq r_{[n]_{q}}(x)<\infty\) such that

$$ r_{[n]_{q}}(x)=x-\frac{1}{2[n]_{q}}, \quad \mbox{where } \frac{1}{2n} \leq x< \frac{1}{1-q^{n}} \mbox{ and } n\in\mathbb{N}. $$
(2.1)

Then, for any \(\frac{1}{2n} \leq x < \frac{1}{1-q^{n}}\), \(0< q<1\), \(\mu>\frac{ 1}{2n}\) and \(n \in\mathbb{N}\), we define

$$ D_{n,q}^{\ast}(f;x)=\frac{1}{e_{\mu,q}([n]_{q}r_{[n]_{q}}(x))}\sum _{k=0}^{\infty}\frac{([n]_{q}r_{[n]_{q}}(x))^{k}}{\gamma_{\mu ,q}(k)}f \biggl( \frac{1-q^{2\mu\theta_{k}+k}}{1-q^{n}} \biggr) , $$
(2.2)

where \(e_{\mu,q}(x)\), \(\gamma_{\mu,q}\) are defined in (1.6), (1.8) by [17] and \(f\in C_{\zeta}[0,\infty)\) with \(\zeta \geq0\) and

$$ C_{\zeta}[0,\infty)=\bigl\{ f\in C[0,\infty):\bigl\vert f(t)\bigr\vert \leq M(1+t)^{\zeta } \mbox{ for some } M>0, \zeta>0\bigr\} . $$
(2.3)

Lemma 2.1

Let \(D_{n,q}^{\ast}(\cdot ; \cdot)\) be the operators given by (2.2). Then, for each \(\frac{1}{2n}\leq x<\frac{1}{1-q^{n}}\), \(n\in \mathbb{N}\), we have the following identities/estimates:

  1. (1)

    \(D_{n,q}^{\ast}(1;x)=1\),

  2. (2)

    \(D_{n,q}^{\ast}(t;x)=x-\frac{1}{2[n]_{q}}\),

  3. (3)

    \(x^{2}+ ( q^{2 \mu}[1-2 \mu]_{q} \frac{e_{\mu,q}(q [n]_{q} r_{[n]_{q}}(x))}{e_{\mu,q}([n]_{q} r_{[n]_{q}} (x))}-1 )\frac {x}{[n]_{q}}-\frac{1}{4[n]_{q}^{2}} (2q^{2 \mu}[1-2 \mu]_{q} \frac{e_{\mu,q}(q [n]_{q} r_{[n]_{q}}(x))}{e_{\mu,q}([n]_{q} r_{[n]_{q}} (x))}-1 )\leq D_{n,q}^{\ast}(t^{2};x) \leq x^{2}+ ( [1+2 \mu]_{q} -1 )\frac {x}{[n]_{q}}-\frac{1}{4[n]_{q}^{2}} (2[1+2 \mu]_{q} -1 ) \).

Proof

As \(D_{n,q}^{\ast}(1;x)=\frac{1}{e_{\mu,q}([n]_{q}r_{[n]_{q}}(x))}\sum_{k=0}^{\infty}\frac{([n]_{q}r_{[n]_{q}}(x))^{k}}{\gamma_{\mu }(k)}=1\), and

$$\begin{aligned} D_{n,q}^{\ast}(t;x) =&\frac{1}{e_{\mu,q}([n]_{q}r_{[n]_{q}}(x))}\sum _{k=0}^{\infty}\frac{([n]_{q}r_{[n]_{q}}(x))^{k}}{\gamma_{\mu }(k)} \biggl( \frac{1-q^{2\mu\theta_{k}+k}}{1-q^{n}} \biggr) \\ =&\frac{1}{[n]_{q}e_{\mu,q}([n]_{q}r_{[n]_{q}}(x))}\sum_{k=1}^{\infty }\frac{([n]_{q}r_{[n]_{q}}(x))^{k}}{\gamma_{\mu}(k-1)} \\ =&x-\frac{1}{2[n]_{q}}, \end{aligned}$$

then (1) and (2) hold. Similarly,

$$\begin{aligned} D_{n,q}^{\ast}\bigl(t^{2};x\bigr) =& \frac{1}{e_{\mu,q}([n]_{q}r_{[n]_{q}}(x))}\sum_{k=0}^{\infty} \frac{([n]_{q}r_{[n]_{q}}(x))^{k}}{\gamma_{\mu }(k)} \biggl( \frac{1-q^{2\mu\theta_{k}+k}}{1-q^{n}} \biggr) ^{2} \\ =&\frac{1}{[n]_{q}^{2}e_{\mu,q}([n]_{q}r_{[n]_{q}}(x))}\sum_{k=0}^{\infty}\frac{([n]_{q}r_{[n]_{q}}(x))^{k}}{\gamma_{\mu}(k-1)} \biggl( \frac{1-q^{2\mu\theta_{k}+k}}{1-q} \biggr) \\ =&\frac{1}{[n]_{q}^{2}e_{\mu,q}([n]_{q}r_{[n]_{q}}(x))}\sum_{k=0}^{\infty}\frac{([n]_{q}r_{[n]_{q}}(x))^{k+1}}{\gamma_{\mu}(k)} \biggl( \frac{1-q^{2\mu\theta_{k+1}+k+1}}{1-q} \biggr) . \end{aligned}$$

From [19] we know that

$$ [2\mu\theta_{k+1}+k+1]_{q}=[2\mu\theta_{k}+k]_{q}+q^{2\mu \theta _{k}+k} \bigl[2\mu(-1)^{k}+1\bigr]_{q}. $$
(2.4)

Now, by separating to the even and odd terms and using (2.4), we get

$$\begin{aligned} D_{n,q}^{\ast}\bigl(t^{2};x\bigr) =& \frac{1}{[n]_{q}^{2}e_{\mu ,q}([n]_{q}r_{[n]_{q}}(x))}\sum_{k=0}^{\infty} \frac{([n]_{q}r_{[n]_{q}}(x))^{k+1}}{\gamma_{\mu}(k)} \biggl( \frac{1-q^{2\mu \theta_{k+1}+k+1}}{1-q} \biggr) \\ &{}+\frac{[1+2\mu]_{q}}{[n]_{q}^{2}e_{\mu,q}([n]_{q}r_{[n]_{q}}(x))}\sum_{k=0}^{\infty} \frac{([n]_{q}r_{[n]_{q}}(x))^{2k+1}}{\gamma_{\mu }(2k)}q^{2\mu\theta_{2k}+2k} \\ &{}+\frac{[1-2\mu]_{q}}{[n]_{q}^{2}e_{\mu,q}([n]_{q}r_{[n]_{q}}(x))}\sum_{k=0}^{\infty} \frac{([n]_{q}r_{[n]_{q}}(x))^{2k+2}}{\gamma_{\mu }(2k)}q^{2\mu\theta_{2k+1}+2k+1}. \end{aligned}$$

Since

$$ {}[1-2\mu]_{q}\leq{}[1+2\mu]_{q}, $$
(2.5)

we have

$$\begin{aligned} D_{n,q}^{\ast}\bigl(t^{2};x\bigr) \geq& \bigl(r_{[n]_{q}}(x)\bigr)^{2}+\frac{r_{[n]_{q}}(x)[1-2\mu]_{q}}{[n]_{q}e_{\mu,q}([n]_{q}r_{[n]_{q}}(x))}\sum _{k=0}^{\infty}\frac{(q[n]_{q}r_{[n]_{q}}(x))^{2k}}{\gamma_{\mu}(2k)} \\ &{}+\frac{q^{2\mu}r_{[n]_{q}}(x)[1-2\mu]_{q}}{[n]_{q}e_{\mu ,q}([n]_{q}r_{[n]_{q}}(x))}\sum_{k=0}^{\infty} \frac{(q[n]_{q}r_{[n]_{q}}(x))^{2k+1}}{\gamma_{\mu}(2k+1)} \\ \geq&\bigl(r_{[n]_{q}}(x)\bigr)^{2}+q^{2\mu}[1-2 \mu]_{q}\frac{e_{\mu ,q}(q[n]_{q}r_{[n]_{q}}(x))}{e_{\mu,q}([n]_{q}r_{[n]_{q}}(x))}\frac{r_{[n]_{q}}(x)}{[n]_{q}}. \end{aligned}$$

On the other hand, we have

$$ D_{n,q}^{\ast}\bigl(t^{2};x\bigr)\leq \bigl(r_{[n]_{q}}(x)\bigr)^{2}+[1+2\mu]_{q} \frac{r_{[n]_{q}}(x)}{[n]_{q}}. $$

This completes the proof. □

Lemma 2.2

Let the operators \(D_{n,q}^{\ast}(\cdot ; \cdot)\) be given by (2.2). Then, for each \(x\geq\frac{1}{2n}\), \(n \in\mathbb{N} \), we have

  1. (1)

    \(D_{n,q}^{\ast}(t-x;x)=-\frac{1}{2 [n]_{q}}\),

  2. (2)

    \(D_{n,q}^{\ast}((t-x)^{2};x)\leq[1+2\mu ]_{q}\frac{x}{[n]_{q}}-\frac{1}{4[n]_{q}^{2}} ( 2[1+2\mu]_{q}-1 ) \).

Proof

For the proof of this lemma, we use Lemma 2.1. In view of

$$ D_{n}^{\ast}(t-x;x)=D_{n}^{\ast}(t;x)-D_{n}^{\ast}(1;x), $$

(1) follows immediately.

Also

$$\begin{aligned} D_{n}^{\ast}\bigl((t-x)^{2};x\bigr) =&D_{n}^{\ast}\bigl(t^{2};x\bigr)-2xD_{n}^{\ast }(t;x)+x^{2}D_{n}^{\ast}(1;x) \\ \leq&x^{2}+ \bigl( [1+2\mu]_{q}-1 \bigr) \frac{x}{[n]_{q}}-\frac{1}{4[n]_{q}^{2}} \bigl( 2[1+2\mu]_{q}-1 \bigr) \\ &{}-2x \biggl( x-\frac {1}{2[n]_{q}} \biggr) +x^{2} \\ \leq&[1+2\mu]_{q}\frac{x}{[n]_{q}}-\frac{1}{4[n]_{q}^{2}} \bigl( 2[1+2 \mu ]_{q}-1 \bigr) . \end{aligned}$$

This proves (2). □

Main results

We obtain the Korovkin-type approximation properties for our operators (see [2022]).

Let \(C_{B}(\mathbb{R}^{+})\) be the set of all bounded and continuous functions on \(\mathbb{R}^{+}=[0,\infty)\), which is a linear normed space with

$$ \| f\|_{C_{B}}=\sup_{x\geq0}\bigl\vert f(x)\bigr\vert . $$

Let

$$ H:=\biggl\{ f:x\in{}[0,\infty),\frac{f(x)}{1+x^{2}} \mbox{ is convergent as } x \rightarrow\infty\biggr\} . $$

Theorem 3.1

Let \(D_{n,q}^{\ast}(\cdot ; \cdot)\) be the operators defined by (2.2). Then, for any function \(f\in C_{\zeta}[0,\infty)\cap H\), \(\zeta\geq2\),

$$ \lim_{n\rightarrow\infty}D_{n,q}^{\ast}(f;x)=f(x) $$

is uniform on each compact subset of \([0,\infty)\), where \(x \in[ \frac{1}{2},b)\), \(b>\frac{1}{2}\).

Proof

The proof is based on Lemma 2.1 and the well-known Korovkin theorem regarding the convergence of a sequence of linear positive operators, so it is enough to prove the conditions

$$ \lim_{n\rightarrow\infty}D_{n,q}^{\ast}\bigl(t^{j};x \bigr)=x^{j},\quad j=0,1,2\ (\mbox{as } n\rightarrow\infty) $$

uniformly on \([0,1]\).

Clearly, \(\frac{1}{[n]_{q}}\rightarrow0\) (\(n\rightarrow\infty\)) we have

$$ \lim_{n\rightarrow\infty}D_{n,q}^{\ast}(t;x)=x, \qquad \lim _{n\rightarrow \infty}D_{n,q}^{\ast}\bigl(t^{2};x \bigr)=x^{2}. $$

This completes the proof. □

We recall the weighted spaces of the functions on \(\mathbb{R}^{+}\), which are defined as follows:

$$\begin{aligned}& P_{\rho}\bigl(\mathbb{R}^{+}\bigr) = \bigl\{ f:\bigl\vert f(x)\bigr\vert \leq M_{f}\rho (x) \bigr\} , \\& Q_{\rho}\bigl(\mathbb{R}^{+}\bigr) = \bigl\{ f:f\in P_{\rho}\bigl(\mathbb {R}^{+}\bigr)\cap C[0,\infty) \bigr\} , \\& Q_{\rho}^{k}\bigl(\mathbb{R}^{+}\bigr) = \biggl\{ f:f\in Q_{\rho}\bigl(\mathbb {R}^{+}\bigr) \mbox{ and } \lim_{x\rightarrow\infty}\frac{f(x)}{\rho(x)}=k\ (k \mbox{ is a constant}) \biggr\} , \end{aligned}$$

where \(\rho(x)=1+x^{2}\) is a weight function and \(M_{f}\) is a constant depending only on f. Note that \(Q_{\rho}(\mathbb{R}^{+})\) is a normed space with the norm \(\| f\|_{\rho}=\sup_{x\geq0}\frac {|f(x)|}{\rho(x)}\).

Lemma 3.2

[23]

The linear positive operators \(L_{n}\), \(n\geq1\) act from \(Q_{\rho}(\mathbb{R}^{+})\to P_{\rho}(\mathbb{R}^{+}) \) if and only if

$$ \bigl\Vert L_{n}(\varphi;x) \bigr\Vert \leq K\varphi(x), $$

where \(\varphi(x)=1+x^{2}\), \(x \in\mathbb{R}^{+}\) and K is a positive constant.

Theorem 3.3

[23]

Let \(\{L_{n}\}_{n \geq1}\) be a sequence of positive linear operators acting from \(Q_{\rho}(\mathbb{R}^{+})\to P_{\rho}(\mathbb{R}^{+}) \) and satisfying the condition

$$ \lim_{n\to\infty} \bigl\Vert L_{n}\bigl( \rho^{\tau}\bigr)-\rho^{\tau}\bigr\Vert _{\varphi}=0, \quad \tau=0,1,2. $$

Then, for any function \(f\in Q_{\rho}^{k}(\mathbb{R}^{+})\), we have

$$ \lim_{n\to\infty} \bigl\Vert L_{n}(f;x)-f \bigr\Vert _{\varphi}=0. $$

Theorem 3.4

Let \(D_{n,q}^{\ast}(\cdot ; \cdot)\) be the operators defined by (2.2). Then, for each function \(f \in Q^{k}_{\rho}(\mathbb{R}^{+})\), we have

$$ \lim_{n\to\infty} \bigl\Vert D_{n,q}^{\ast}(f;x)-f \bigr\Vert _{\rho}=0. $$

Proof

From Lemma 2.1 and Theorem 3.3, for \(\tau=0\), the first condition is fulfilled. Therefore,

$$ \lim_{n\rightarrow\infty}\bigl\Vert D_{n,q}^{\ast}(1;x)-1 \bigr\Vert _{\rho }=0. $$

Similarly, from Lemma 2.1 and Theorem 3.3, for \(\tau =1,2\) we have that

$$\begin{aligned} \sup_{x\in{}[0,\infty)}\frac{| D_{n,q}^{\ast}(t;x)-x| }{1+x^{2}} \leq&\frac{1}{2[n]_{q}}\sup _{x\in{}[0,\infty)}\frac {1}{1+x^{2}} \\ =&\frac{1}{2[n]_{q}}, \end{aligned}$$

which implies that

$$\begin{aligned}& \lim_{n\rightarrow\infty}\bigl\Vert D_{n,q}^{\ast}(t;x)-x \bigr\Vert _{\rho }=0, \\& \sup_{x\in{}[0,\infty)}\frac{| D_{n,q}^{\ast }(t^{2};x)-x^{2}| }{1+x^{2}} \leq\frac{|{}[1+2\mu]_{q}-1|}{[n]_{q}}\sup _{x\in {}[0,\infty)}\frac{x}{1+x^{2}} \\& \hphantom{\sup_{x\in{}[0,\infty)}\frac{| D_{n,q}^{\ast }(t^{2};x)-x^{2}| }{1+x^{2}} \leq{}}{}+\frac{1}{4[n]_{q}^{2}}\bigl\vert [1+2\mu]_{q}-1\bigr\vert \sup_{x\in {}[ 0,\infty)}\frac{1}{1+x^{2}}. \end{aligned}$$

Hence

$$ \lim_{n\rightarrow\infty}\bigl\Vert D_{n,q}^{\ast } \bigl(t^{2};x\bigr)-x^{2}\bigr\Vert _{\rho}=0. $$

This completes the proof. □

Rate of convergence

Let \(f\in C_{B}[0,\infty]\), the space of all bounded and continuous functions on \([0,\infty)\) and \(x\geq\frac{1}{2n}\), \(n\in\mathbb{N}\). Then, for \(\delta>0\), the modulus of continuity of f denoted by \(\omega (f,\delta)\) gives the maximum oscillation of f in any interval of length not exceeding \(\delta>0\), and it is given by

$$ \omega(f,\delta)=\sup_{| t-x|\leq\delta}\bigl\vert f(t)-f(x)\bigr\vert , \quad t\in{}[0,\infty). $$
(4.1)

It is known that \(\lim_{\delta\rightarrow0+}\omega(f,\delta)=0\) for \(f\in C_{B}[0,\infty)\), and for any \(\delta>0\) we have

$$ \bigl\vert f(t)-f(x)\bigr\vert \leq \biggl( \frac{| t-x|}{\delta}+1 \biggr) \omega (f,\delta). $$
(4.2)

Now we calculate the rate of convergence of operators (2.2) by means of modulus of continuity and Lipschitz-type maximal functions.

Theorem 4.1

Let \(D_{n,q}^{\ast}(\cdot ; \cdot)\) be the operators defined by (2.2). Then, for \(f\in C_{B}[0,\infty)\), \(x\geq\frac{1}{2n}\) and \(n\in\mathbb{N}\), we have

$$ \bigl\vert D_{n,q}^{\ast}(f;x)-f(x)\bigr\vert \leq2\omega ( f;\delta _{n,x} ) , $$

where

$$ \delta_{n,x}=\sqrt{[1+2\mu]_{q}\frac{x}{[n]_{q}}- \frac {1}{4[n]_{q}^{2}} \bigl( 2[1+2\mu]_{q}-1 \bigr) }. $$
(4.3)

Proof

We prove it by using (4.1), (4.2) and the Cauchy-Schwarz inequality. We can easily get

$$ \bigl\vert D_{n,q}^{\ast}(f;x)-f(x)\bigr\vert \leq \biggl\{ 1+\frac{1}{\delta} \bigl( D_{n,q}^{\ast}(t-x)^{2};x \bigr) ^{\frac{1}{2}} \biggr\} \omega (f;\delta) $$

if we choose \(\delta=\delta_{n,x}\), and by applying the result (2) of Lemma 2.2, we get the result. □

Remark 4.2

For the operators \(D_{n,q}(\cdot ; \cdot)\) defined by (1.9) we may write that, for every \(f\in C_{B}[0,\infty)\), \(x\geq0\) and \(n \in\mathbb{N}\),

$$ \bigl\vert D_{n,q}(f;x)-f(x)\bigr\vert \leq2\omega (f;\lambda_{n,x} ), $$
(4.4)

where by [19] we have

$$ \lambda_{n,x}=\sqrt{D_{n,q} \bigl((t-x)^{2};x\bigr)}\leq\sqrt{[1+2\mu]_{q} \frac{x}{[n]_{q}}}. $$
(4.5)

Now we claim that the error estimation in Theorem 4.1 is better than that of (4.4) provided \(f\in C_{B}[0,\infty)\) and \(x\geq\frac {1}{2n}\), \(n\in\mathbb{N}\). Indeed, for \(x\geq\frac{1}{2n}\), \(\mu\geq\frac {1}{2n}\) and \(n\in\mathbb{N}\), it is guaranteed that

$$\begin{aligned}& D_{n,q}^{\ast}\bigl((t-x)^{2};x\bigr)\leq D_{n,q}\bigl((t-x)^{2};x\bigr), \end{aligned}$$
(4.6)
$$\begin{aligned}& {}[1+2\mu]_{q}\frac{x}{[n]_{q}}-\frac{1}{4[n]_{q}^{2}} \bigl( 2[1+2\mu ]_{q}-1 \bigr) \leq{}[1+2\mu]_{q}\frac{x}{[n]_{q}}, \end{aligned}$$
(4.7)

which implies that

$$ \sqrt{[1+2\mu]_{q}\frac{x}{[n]_{q}}-\frac{1}{4[n]_{q}^{2}} \bigl( 2[1+2\mu]_{q}-1 \bigr) }\leq\sqrt{[1+2\mu]_{q} \frac{x}{[n]_{q}}}. $$
(4.8)

Now we give the rate of convergence of the operators \({D}_{n,q}^{\ast }(f;x) \) defined in (2.2) in terms of the elements of the usual Lipschitz class \(\operatorname{Lip}_{M}(\nu)\).

Let \(f\in C_{B}[0,\infty)\), \(M>0\) and \(0<\nu\leq1\). The class \(\operatorname{Lip}_{M}(\nu) \) is defined as

$$ \operatorname{Lip}_{M}(\nu)= \bigl\{ f:\bigl\vert f( \zeta_{1})-f(\zeta_{2})\bigr\vert \leq M| \zeta_{1}-\zeta_{2}|^{\nu}\ \bigl( \zeta_{1},\zeta_{2}\in{}[ 0,\infty)\bigr) \bigr\} . $$
(4.9)

Theorem 4.3

Let \(D_{n,q}^{\ast}(\cdot ; \cdot)\) be the operators defined in (2.2).Then, for each \(f\in \operatorname{Lip}_{M}(\nu)\) (\(M>0\), \(0<\nu\leq1\)) satisfying (4.9), we have

$$ \bigl\vert D_{n,q}^{\ast}(f;x)-f(x)\bigr\vert \leq M ( \delta_{n,x} ) ^{\frac{\nu}{2}}, $$

where \(\delta_{n,x}\) is given in Theorem  4.1.

Proof

We prove it by using (4.9) and Hölder’s inequality. We have

$$\begin{aligned} \bigl\vert D_{n,q}^{\ast}(f;x)-f(x)\bigr\vert \leq&\bigl\vert D_{n,q}^{\ast }\bigl(f(t)-f(x);x\bigr)\bigr\vert \\ \leq&D_{n,q}^{\ast} \bigl( \bigl\vert f(t)-f(x)\bigr\vert ;x \bigr) \\ \leq&MD_{n,q}^{\ast} \bigl( \vert t-x\vert ^{\nu};x \bigr) . \end{aligned}$$

Therefore,

$$\begin{aligned}& \bigl\vert D_{n,q}^{\ast}(f;x)-f(x)\bigr\vert \\& \quad \leq M\frac{[n]_{q}}{e_{\mu,q}([n]_{q}r_{[n]_{q}}(x))}\sum_{k=0}^{\infty}\frac{([n]_{q}r_{[n]_{q}}(x))^{k}}{\gamma_{\mu,q}(k)}\biggl\vert \frac{1-q^{2\mu\theta_{k}+k}}{1-q^{n}}-x\biggr\vert ^{\nu} \\& \quad \leq M\frac{[n]_{q}}{e_{\mu,q}([n]_{q}r_{[n]_{q}}(x))}\sum_{k=0}^{\infty } \biggl( \frac{([n]_{q}r_{[n]_{q}}(x))^{k}}{\gamma_{\mu,q}(k)} \biggr) ^{\frac{2-\nu}{2}} \\& \qquad {} \times \biggl( \frac{([n]_{q}r_{[n]_{q}}(x))^{k}}{\gamma_{\mu,q}(k)} \biggr) ^{\frac{\nu}{2}} \biggl\vert \frac{1-q^{2\mu\theta _{k}+k}}{1-q^{n}}-x\biggr\vert ^{\nu} \\& \quad \leq M \Biggl( \frac{n}{e_{\mu,q}([n]_{q}r_{[n]_{q}}(x))}\sum_{k=0}^{\infty } \frac{([n]_{q}r_{[n]_{q}}(x))^{k}}{\gamma_{\mu,q}(k)} \Biggr) ^{\frac{ 2-\nu}{2}} \\& \qquad {} \times \Biggl( \frac{[n]_{q}}{e_{\mu,q}([n]_{q}r_{[n]_{q}}(x))}\sum _{k=0}^{\infty}\frac{([n]_{q}r_{[n]_{q}}(x))^{k}}{\gamma_{\mu ,q}(k)}\biggl\vert \frac{1-q^{2\mu\theta_{k}+k}}{1-q^{n}}-x\biggr\vert ^{2} \Biggr) ^{ \frac{\nu}{2}} \\& \quad = M \bigl( D_{n,q}^{\ast}(t-x)^{2};x \bigr) ^{\frac{\nu}{2}}. \end{aligned}$$

This completes the proof. □

Let

$$ C_{B}^{2}\bigl(\mathbb{R}^{+}\bigr)=\bigl\{ g\in C_{B}\bigl(\mathbb{R}^{+}\bigr):g^{\prime },g^{\prime \prime} \in C_{B}\bigl(\mathbb{R}^{+}\bigr)\bigr\} , $$
(4.10)

with the norm

$$ \Vert g\Vert _{C_{B}^{2}(\mathbb{R}^{+})}=\Vert g\Vert _{C_{B}(\mathbb{R}^{+})}+\bigl\Vert g^{\prime}\bigr\Vert _{C_{B}(\mathbb {R}^{+})}+\bigl\Vert g^{\prime\prime}\bigr\Vert _{C_{B}(\mathbb{R}^{+})}, $$
(4.11)

also

$$ \| g\|_{C_{B}(\mathbb{R}^{+})}=\sup_{x\in\mathbb {R}^{+}}\bigl\vert g(x)\bigr\vert . $$
(4.12)

Theorem 4.4

Let \(D_{n,q}^{\ast}(\cdot ; \cdot)\) be the operators defined in (2.2). Then for any \(g\in C_{B}^{2}(\mathbb{R}^{+})\) we have

$$ \bigl\vert D_{n,q}^{\ast}(f;x)-f(x)\bigr\vert \leq \biggl\{ \biggl( -\frac {1}{2[n]_{q}} \biggr) +\frac{\delta_{n,x}}{2} \biggr\} \| g \| _{C_{B}^{2}(\mathbb{R}^{+})}, $$

where \(\delta_{n,x}\) is given in Theorem  4.1.

Proof

Let \(g\in C_{B}^{2}(\mathbb{R}^{+})\). Then, by using the generalized mean value theorem in the Taylor series expansion, we have

$$ g(t)=g(x)+g^{\prime}(x) (t-x)+g^{\prime\prime}(\psi)\frac {(t-x)^{2}}{2}, \quad \psi\in(x,t). $$

By applying the linearity property on \(D_{n,q}^{\ast}\), we have

$$ D_{n,q}^{\ast}(g,x)-g(x)=g^{\prime}(x)D_{n,q}^{\ast} \bigl( (t-x);x \bigr) +\frac{g^{\prime\prime}(\psi)}{2}D_{n,q}^{\ast} \bigl( (t-x)^{2};x \bigr) , $$

which implies that

$$\begin{aligned}& \bigl\vert D_{n,q}^{\ast}(g;x)-g(x)\bigr\vert \\& \quad \leq \biggl( -\frac {1}{2[n]_{q}} \biggr) \bigl\Vert g^{\prime}\bigr\Vert _{C_{B}(\mathbb{R}^{+})}+ \biggl( [1+2\mu ]_{q}\frac{x}{[n]_{q}}- \frac{1}{4[n]_{q}^{2}} \bigl( 2[1+2\mu]_{q}-1 \bigr) \biggr) \frac{\Vert g^{\prime\prime} \Vert _{C_{B}(\mathbb {R}^{+})}}{2}. \end{aligned}$$

From (4.11) we have \(\| g^{\prime}\| _{C_{B}[0,\infty)}\leq\| g\|_{C_{B}^{2}[0,\infty)}\),

$$\begin{aligned}& \bigl\vert D_{n,q}^{\ast}(g;x)-g(x)\bigr\vert \\& \quad \leq \biggl( -\frac {1}{2[n]_{q}} \biggr) \| g\|_{C_{B}^{2}(\mathbb{R}^{+})}+ \biggl( [1+2\mu ]_{q}\frac{x}{[n]_{q}}-\frac{1}{4[n]_{q}^{2}} \bigl( 2[1+2 \mu]_{q}-1 \bigr) \biggr) \frac{\| g\|_{C_{B}^{2}(\mathbb{R}^{+})}}{2}. \end{aligned}$$

The proof follows from (2) of Lemma 2.2. □

The Peetre’s K-functional is defined by

$$ K_{2}(f,\delta)=\inf_{C_{B}^{2}(\mathbb{R}^{+})} \bigl\{ \bigl( \Vert f-g\Vert _{C_{B}(\mathbb{R}^{+})}+\delta\bigl\Vert g^{\prime\prime }\bigr\Vert _{C_{B}^{2}(\mathbb{R}^{+})} \bigr) :g\in\mathcal {W}^{2} \bigr\} , $$
(4.13)

where

$$ \mathcal{W}^{2}= \bigl\{ g\in C_{B}\bigl( \mathbb{R}^{+}\bigr):g^{\prime},g^{\prime \prime}\in C_{B}\bigl(\mathbb{R}^{+}\bigr) \bigr\} . $$
(4.14)

There exists a positive constant \(C>0\) such that \(K_{2}(f,\delta)\leq C\omega_{2}(f,\delta^{\frac{1}{2}})\), \(\delta>0\), where the second-order modulus of continuity is given by

$$ \omega_{2}\bigl(f,\delta^{\frac{1}{2}}\bigr)=\sup _{0< h< \delta^{\frac{1}{2}}}\sup_{x\in\mathbb{R}^{+}}\bigl\vert f(x+2h)-2f(x+h)+f(x)\bigr\vert . $$
(4.15)

Theorem 4.5

For \(x\geq\frac{1}{2n}\), \(n\in\mathbb{N}\) and \(f\in C_{B}( \mathbb{R}^{+})\), we have

$$\begin{aligned}& \bigl\vert D_{n,q}^{\ast}(f;x)-f(x)\bigr\vert \\& \quad \leq2M \biggl\{ \omega_{2} \biggl( f;\sqrt{\frac{ ( -\frac{1}{ [n]_{q}} )+ \delta_{n,x}}{4}} \biggr) +\min \biggl( 1,\frac{ ( -\frac{1}{ [n]_{q}} )+ \delta_{n,x}}{4} \biggr) \| f\| _{C_{B}(\mathbb{R}^{+})} \biggr\} , \end{aligned}$$

where M is a positive constant, \(\delta_{n,x}\) is given in Theorem  4.3 and \(\omega_{2}(f;\delta)\) is the second-order modulus of continuity of the function f defined in (4.15).

Proof

We prove this by using Theorem 4.4

$$\begin{aligned} \bigl\vert D_{n,q}^{\ast}(f;x)-f(x)\bigr\vert \leq& \bigl\vert D_{n,q}^{\ast}(f-g;x)\bigr\vert +\bigl\vert D_{n,q}^{\ast}(g;x)-g(x)\bigr\vert +\bigl\vert f(x)-g(x) \bigr\vert \\ \leq& 2 \| f-g \|_{C_{B}(\mathbb{R}^{+})}+ \frac{\delta _{n,x}}{2}\| g \|_{C_{B}^{2}(\mathbb{R}^{+})} + \biggl( -\frac{1}{2 [n]_{q}} \biggr)\| g \|_{C_{B}(\mathbb{R}^{+})}. \end{aligned}$$

From (4.11), clearly, we have \(\| g \|_{C_{B}[0,\infty)}\leq\| g \|_{C_{B}^{2}[0,\infty)}\).

Therefore,

$$ \bigl\vert D_{n,q}^{\ast}(f;x)-f(x)\bigr\vert \leq2 \biggl( \| f-g \| _{C_{B}(\mathbb{R}^{+})}+\frac{ ( -\frac{1}{ [n]_{q}} )+ \delta_{n,x}}{4}\| g \|_{C_{B}^{2}(\mathbb{R}^{+})} \biggr), $$

where \(\delta_{n,x}\) is given in Theorem 4.1.

By taking infimum over all \(g\in C_{B}^{2}(\mathbb{R}^{+})\) and by using (4.13), we get

$$ \bigl\vert D_{n,q}^{\ast}(f;x)-f(x)\bigr\vert \leq2K_{2} \biggl( f;\frac{ ( -\frac{1}{ [n]_{q}} )+ \delta_{n,x}}{4} \biggr). $$

Now, for an absolute constant \(Q>0\) in [24], we use the relation

$$ K_{2}(f;\delta)\leq Q\bigl\{ \omega_{2}(f;\sqrt{\delta})+ \min(1,\delta )\| f\|\bigr\} . $$

This completes the proof. □

Conclusion

The purpose of this paper is to provide a better error estimation of convergence by modification of the q-Dunkl analogue of Szász operators. Here we have defined a Dunkl generalization of these modified operators. This type of modification enables better error estimation on the interval \([1/2,\infty)\) if compared to the classical Dunkl-Szász operators [19]. We obtained some approximation results via the well-known Korovkin-type theorem. We have also calculated the rate of convergence of operators by means of modulus of continuity and Lipschitz-type maximal functions.

References

  1. Bernstein, SN: Démonstration du théoréme de Weierstrass fondée sur le calcul des probabilités. Commun. Soc. Math. Kharkow 2(13), 1-2 (2012)

    Google Scholar 

  2. Szász, O: Generalization of S. Bernstein’s polynomials to the infinite interval. J. Res. Natl. Bur. Stand. 45, 239-245 (1950)

    MathSciNet  Article  Google Scholar 

  3. Lupaş, A: A q-analogue of the Bernstein operator. In: Seminar on Numerical and Statistical Calculus, vol. 9, pp. 85-92. University of Cluj-Napoca, Cluj-Napoca (1987)

    Google Scholar 

  4. Phillips, GM: Bernstein polynomials based on the q-integers. Ann. Numer. Math. 4, 511-518 (1997)

    MathSciNet  MATH  Google Scholar 

  5. Mursaleen, M, Ansari, KJ: Approximation of q-Stancu-beta operators which preserve \(x^{2}\). Bull. Malays. Math. Sci. Soc. (2015). doi:10.1007/s40840-015-0146-9

    Google Scholar 

  6. Mursaleen, M, Khan, A: Statistical approximation properties of modified q-Stancu-beta operators. Bull. Malays. Math. Sci. Soc. (2) 36(3), 683-690 (2013)

    MathSciNet  MATH  Google Scholar 

  7. Mursaleen, M, Khan, A: Generalized q-Bernstein-Schurer operators and some approximation theorems. J. Funct. Spaces Appl. 2013, Article ID 719834 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  8. Mursaleen, M, Khan, F, Khan, A: Approximation properties for modified q-Bernstein-Kantorovich operators. Numer. Funct. Anal. Optim. 36(9), 1178-1197 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  9. Mursaleen, M, Khan, F, Khan, A: Approximation properties for King’s type modified q-Bernstein-Kantorovich operators. Math. Methods Appl. Sci. 38, 5242-5252 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  10. Mursaleen, M, Khan, T, Nasiruzzaman, M: Approximating properties of generalized Dunkl analogue of Szász operators. Appl. Math. Inf. Sci. 10(6), 1-8 (2016)

    MathSciNet  Article  Google Scholar 

  11. Örkcü, M, Doğru, O: Weighted statistical approximation by Kantorovich type q-Szász Mirakjan operators. Appl. Math. Comput. 217, 7913-7919 (2011)

    MathSciNet  MATH  Google Scholar 

  12. Örkcü, M, Doğru, O: q-Szász-Mirakyan-Kantorovich type operators preserving some test functions. Appl. Math. Lett. 24, 1588-1593 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  13. Wafi, A, Rao, N, Rai, D: Approximation properties by generalized-Baskakov-Kantorovich-Stancu type operators. Appl. Math. Inf. Sci. Lett. 4(3), 111-118 (2016)

    Article  Google Scholar 

  14. Wafi, A, Rao, N: A generalization of Szász-type operators which preserves constant and quadratic test functions. Cogent Math. 3, 1227023 (2016)

    Article  Google Scholar 

  15. Aral, A, Gupta, V, Agarwal, RP: Applications of q-Calculus in Operator Theory. Springer, New York (2013)

    Book  MATH  Google Scholar 

  16. Rosenblum, M: Generalized Hermite polynomials and the Bose-like oscillator calculus. Oper. Theory, Adv. Appl. 73, 369-396 (1994)

    MathSciNet  MATH  Google Scholar 

  17. Sucu, S: Dunkl analogue of Szász operators. Appl. Math. Comput. 244, 42-48 (2014)

    MathSciNet  MATH  Google Scholar 

  18. Cheikh, B, Gaied, Y, Zaghouani, M: A q-Dunkl-classical q-Hermite type polynomials. Georgian Math. J. 21(2), 125-137 (2014)

    MathSciNet  MATH  Google Scholar 

  19. İçōz, G, Çekim, B: Dunkl generalization of Szász operators via q-calculus. J. Inequal. Appl. 2015, 284 (2015)

    Article  MATH  Google Scholar 

  20. Braha, NL, Srivastava, HM, Mohiuddine, SA: A Korovkin’s type approximation theorem for periodic functions via the statistical summability of the generalized de la Vallée Poussin mean. Appl. Math. Comput. 228, 162-169 (2014)

    MathSciNet  Google Scholar 

  21. Edely, OHH, Mohiuddine, SA, Noman, AK: Korovkin type approximation theorems obtained through generalized statistical convergence. Appl. Math. Lett. 23(11), 1382-1387 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  22. Mohiuddine, SA: An application of almost convergence in approximation theorems. Appl. Math. Lett. 24(11), 1856-1860 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  23. Gadzhiev, AD: The convergence problem for a sequence of positive linear operators on unbounded sets and theorems analogues to that of PP Korovkin. Sov. Math. Dokl. 15(5), 1433-1436 (1974)

    Google Scholar 

  24. Ciupa, A: A class of integral Favard-Szász type operators. Stud. Univ. Babeş-Bolyai, Math. 40(1), 39-47 (1995)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors gratefully acknowledge the financial support from King Abdulaziz University, Jeddah, Saudi Arabia.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to M Mursaleen.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors of the manuscript have read and agreed to its content and are accountable for all aspects of the accuracy and integrity of the manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Mursaleen, M., Nasiruzzaman, M. & Alotaibi, A. On modified Dunkl generalization of Szász operators via q-calculus. J Inequal Appl 2017, 38 (2017). https://doi.org/10.1186/s13660-017-1311-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/s13660-017-1311-5

MSC

  • 41A25
  • 41A36
  • 33C45

Keywords

  • q-integers
  • Dunkl analogue
  • Szász operator
  • q-Szász-Mirakjan-Kantorovich
  • modulus of continuity
  • Peetre’s K-functional