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

$$\begin{aligned} S_{n}^{*}\left( g;x\right) =\frac{1}{e_{\nu }\left( nx\right) } \sum _{k=0}^{\infty }\frac{\left( nx\right) ^{k}}{\gamma _{\nu }\left( k\right) }g\left( \frac{k+2\nu \theta _{k}}{n}\right) \ ;\ n\in \mathbb {N} ,~\nu ,\ x\in [0,\infty ), \end{aligned}$$
(1.1)

for \(g\in C[0,\infty ),\) where Dunkl analogue of exponential function is defined by

$$\begin{aligned} e_{\nu }\left( x\right) =\sum _{k=0}^{\infty }\frac{x^{k}}{\gamma _{\nu }\left( k\right) } \end{aligned}$$
(1.2)

for \(k\in \mathbb {N} _{0}\) and \(\nu >-\frac{1}{2}\) and the coefficients \(\gamma _{\nu }\) are as follows

$$\begin{aligned} \gamma _{\nu }\left( 2k\right) =\frac{2^{2k}k!\Gamma \left( k+\nu +1/2\right) }{\Gamma \left( \nu +1/2\right) }\text { and }\gamma _{\nu }\left( 2k+1\right) =\frac{2^{2k+1}k!\Gamma \left( k+\nu +3/2\right) }{ \Gamma \left( \nu +1/2\right) } \nonumber \\ \end{aligned}$$
(1.3)

in [20]. Also, the coefficients \(\gamma _{\nu }\) verify the recursion relation

$$\begin{aligned} \frac{\gamma _{\nu }\left( k+1\right) }{\gamma _{\nu }\left( k\right) } =\left( 2\nu \theta _{k+1}+k+1\right) ,\text { }k\in \mathbb {N} _{0}, \end{aligned}$$
(1.4)

where

$$\begin{aligned} \theta _{k}=\left\{ \begin{array}{cc} 0, &{} if\text { }k=2p \\ 1, &{} if\text { }k=2p+1 \end{array} \right. \end{aligned}$$
(1.5)

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])

$$\begin{aligned} \sum \limits _{n=0}^{\infty }\frac{H_{n}(\xi ,\alpha )}{n!}t^{n}=e^{\xi t+\alpha t^{2}} \end{aligned}$$

from which, it follows

$$\begin{aligned} H_{n}(\xi ,\alpha )=n!\sum \limits _{k=0}^{\left[ \frac{n}{2}\right] }\frac{ \alpha ^{k}\xi ^{n-2k}}{k!(n-2k)!}. \end{aligned}$$

In a recent paper, Krech [13] has introduced the class of operators \( G_{n}^{\alpha }\) given by

$$\begin{aligned} G_{n}^{\alpha }\left( f;x\right)= & {} e^{-\left( nx+\alpha x^{2}\right) }\sum \limits _{k=0}^{\infty }\frac{x^{k}}{k!}H_{k}(n,\alpha )f\left( \frac{k}{ n}\right) ~~,~~x\in [0,\infty )~,\nonumber \\&~f\in C[0,\infty ),\ n\in \mathbb {N~} \text { , }\alpha \ge 0 \end{aligned}$$
(1.6)

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

$$\begin{aligned} \sum \limits _{n=0}^{\infty }\frac{H_{n}^{\mu }(\xi ,\alpha )}{n!} t^{n}=e^{\alpha t^{2}}e_{\mu }(\xi t) \end{aligned}$$
(2.1)

from which, we conclude

$$\begin{aligned} H_{n}^{\mu }(\xi ,\alpha )=n!\sum \limits _{k=0}^{\left[ \frac{n}{2}\right] } \frac{\alpha ^{k}\xi ^{n-2k}}{k!\gamma _{\mu }(n-2k)}, \end{aligned}$$

which gives the two-variable Hermite polynomials as \(\mu =0.\ \)For our purpose, we denote

$$\begin{aligned} h_{n}^{\mu }(\xi ,\alpha )=\dfrac{\gamma _{\mu }(n)H_{n}^{\mu }(\xi ,\alpha ) }{n!} \end{aligned}$$

and we can write that the polynomials \(h_{n}^{\mu }(\xi ,\alpha )\) are generated by

$$\begin{aligned} \sum \limits _{n=0}^{\infty }\frac{h_{n}^{\mu }(\xi ,\alpha )}{\gamma _{\mu }(n)}t^{n}=e^{\alpha t^{2}}e_{\mu }(\xi t), \end{aligned}$$
(2.2)

where

$$\begin{aligned} h_{n}^{\mu }(\xi ,\alpha )=\gamma _{\mu }(n)\sum \limits _{k=0}^{\left[ \frac{n }{2}\right] }\frac{\alpha ^{k}\xi ^{n-2k}}{k!\gamma _{\mu }(n-2k)}. \end{aligned}$$

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

$$\begin{aligned} \mathbb {D}_{\mu }(\varphi (x))=\varphi ^{^{\prime }}(x)+\frac{\mu }{x} (\varphi (x)-\varphi (-x)),\ x\in \mathbb {C}. \end{aligned}$$
(2.3)

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

$$\begin{aligned} \begin{array}{cl} (i) &{} \mathbb {D}_{\mu }^{j}:x^{n}\rightarrow \frac{\gamma _{\mu }(n)}{\gamma _{\mu }(n-j)}x^{n-j},j=0,1,2,...,n\ (n\in \mathbb {N)};~\mathbb {D}_{\mu }^{j}:1\rightarrow 0, \\ &{} \\ (ii) &{} \mathbb {D}_{\mu }(\varphi \psi )=\mathbb {D}_{\mu }(\varphi )\psi +\varphi \mathbb {D}_{\mu }(\psi ),\ \text {where }\varphi \text { is an even function,} \\ &{} \\ (iii) &{} \mathbb {D}_{\mu }:e_{\mu }(\lambda x)\rightarrow \lambda e_{\mu }(\lambda x). \end{array} \end{aligned}$$

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

$$\begin{aligned} \begin{array}{cl} (i) &{} \sum \limits _{n=0}^{\infty }\frac{h_{n+1}^{\mu }(\xi ,\alpha )}{\gamma _{\mu }(n)}t^{n}=(\xi +2\alpha t)e^{\alpha t^{2}}e_{\mu }(\xi t), \\ (ii) &{} \sum \limits _{n=0}^{\infty }\frac{h_{n+2}^{\mu }(\xi ,\alpha )}{\gamma _{\mu }(n)}t^{n}=(\xi ^{2}+4\xi \alpha t+4\alpha ^{2}t^{2}+2\alpha )e^{\alpha t^{2}}e_{\mu }(\xi t)+4\alpha \mu e^{\alpha t^{2}}e_{\mu }(-\xi t). \end{array} \end{aligned}$$

Proof

Applying the linear operator \(\mathbb {D}_{\mu }\) in view of Lemma 1 ,  we have

$$\begin{aligned} \begin{array}{l} \mathbb {D}_{\mu }(te_{\mu }(\xi t))=(t\xi +1)e_{\mu }(\xi t)+2\mu e_{\mu }(-\xi t), \\ \mathbb {D}_{\mu }(e^{\alpha t^{2}})=2\alpha te^{\alpha t^{2}}. \end{array} \end{aligned}$$
(2.4)

Also applying the linear operator \(\mathbb {D}_{\mu }\) to both side of generating function (2.2), we have

$$\begin{aligned} \sum \limits _{n=0}^{\infty }\frac{h_{n}^{\mu }(\xi ,\alpha )}{\gamma _{\mu }(n)}\mathbb {D}_{\mu }(t^{n})=\mathbb {D}_{\mu }(e^{\alpha t^{2}}e_{\mu }(\xi t)). \end{aligned}$$

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

$$\begin{aligned} \sum \limits _{n=0}^{\infty }\frac{h_{n+1}^{\mu }(\xi ,\alpha )}{\gamma _{\mu }(n)}\mathbb {D}_{\mu }(t^{n})=\mathbb {D}_{\mu }\left[ (\xi +2\alpha t)e^{\alpha t^{2}}e_{\mu }(\xi t)\right] . \end{aligned}$$

From (2.4) and Lemma 1, it follows

$$\begin{aligned} \sum \limits _{n=0}^{\infty }\frac{h_{n+2}^{\mu }(\xi ,\alpha )}{\gamma _{\mu }(n)}t^{n}=(\xi ^{2}+4\xi \alpha t+4\alpha ^{2}t^{2}+2\alpha )e^{\alpha t^{2}}e_{\mu }(\xi t)+4\alpha \mu e^{\alpha t^{2}}e_{\mu }(-\xi t). \end{aligned}$$

\(\square \)

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

$$\begin{aligned} T_{n}(f;x):=\frac{1}{e^{\alpha x^{2}}e_{\mu }(nx)}\sum \limits _{k=0}^{\infty } \frac{h_{k}^{\mu }(n,\alpha )}{\gamma _{\mu }(k)}x^{k}f\left( \frac{k+2\mu \theta _{k}}{n}\right) , \end{aligned}$$
(2.5)

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:

$$\begin{aligned} \begin{array}{cl} (i) &{} T_{n}(1;x)=1, \\ (ii) &{} T_{n}(t;x)=x+\frac{2\alpha x^{2}}{n}, \\ (iii) &{} T_{n}(t^{2};x)=x^{2}+\frac{4\alpha }{n^{2}}x^{2}+\frac{4\alpha }{n} x^{3}+\frac{4\alpha ^{2}}{n^{2}}x^{4}+\frac{x}{n}+\frac{2\mu x}{n}\frac{ e_{\mu }(-nx)}{e_{\mu }(nx)}. \end{array} \end{aligned}$$

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

$$\begin{aligned} T_{n}(t;x)=\frac{1}{ne^{\alpha x^{2}}e_{\mu }(nx)}\sum \limits _{k=1}^{\infty } \frac{h_{k}^{\mu }(n,\alpha )}{\gamma _{\mu }(k-1)}x^{k}. \end{aligned}$$

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

$$\begin{aligned} T_{n}(t^{2};x)=\frac{x}{n^{2}e^{\alpha x^{2}}e_{\mu }(nx)} \sum \limits _{k=0}^{\infty }(k+1+2\mu \theta _{k+1})\frac{h_{k+1}^{\mu }(n,\alpha )}{\gamma _{\mu }(k)}x^{k}. \end{aligned}$$

From the equation

$$\begin{aligned} \theta _{k+1}=\theta _{k}+(-1)^{k}, \end{aligned}$$
(2.6)

it yields

$$\begin{aligned} T_{n}(t^{2};x)= & {} \frac{x}{n^{2}e^{\alpha x^{2}}e_{\mu }(nx)} \sum \limits _{k=0}^{\infty }(k+2\mu \theta _{k})\frac{h_{k+1}^{\mu }(n,\alpha )}{\gamma _{\mu }(k)}x^{k} \\&+\frac{x}{n^{2}e^{\alpha x^{2}}e_{\mu }(nx)}\sum \limits _{k=0}^{\infty }(1+2\mu (-1)^{k})\frac{h_{k+1}^{\mu }(n,\alpha )}{\gamma _{\mu }(k)}x^{k}. \end{aligned}$$

Using the recursion relation in (1.4) in the first series, it follows

$$\begin{aligned} T_{n}(t^{2};x)= & {} \frac{x^{2}}{n^{2}e^{\alpha x^{2}}e_{\mu }(nx)} \sum \limits _{k=0}^{\infty }\frac{h_{k+2}^{\mu }(n,\alpha )}{\gamma _{\mu }(k) }x^{k}+\frac{x}{n^{2}e^{\alpha x^{2}}e_{\mu }(nx)}\sum \limits _{k=0}^{\infty } \frac{h_{k+1}^{\mu }(n,\alpha )}{\gamma _{\mu }(k)}x^{k} \\&\\&+\frac{2\mu x}{n^{2}e^{\alpha x^{2}}e_{\mu }(nx)}\sum \limits _{k=0}^{\infty }(-x)^{k}\frac{h_{k+1}^{\mu }(n,\alpha )}{\gamma _{\mu }(k)}. \end{aligned}$$

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

$$\begin{aligned} \Delta _{1}= & {} T_{n}(t-x;x)=\frac{2\alpha x^{2}}{n}, \nonumber \\ \Delta _{2}= & {} T_{n}(\left( t-x\right) ^{2};x)=\frac{1}{n^{2}}x\left( 4x^{3}\alpha ^{2}+4\alpha x+n\right) +\frac{2\mu x}{n}\frac{e_{\mu }(-nx)}{ e_{\mu }(nx)}. \end{aligned}$$
(2.7)

Theorem 1

For \(T_{n}\) operators and any uniformly continuous bounded function g on the interval \([0,\infty )\), we can give

$$\begin{aligned} T_{n}\left( g;x\right) \overset{\text {uniformly}}{\rightrightarrows }g\left( x\right) \end{aligned}$$

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

$$\begin{aligned} \left| g\left( s\right) -g\left( t\right) \right| \le M\left| s-t\right| ^{\alpha } \end{aligned}$$

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

$$\begin{aligned} \omega \left( g;\delta \right) :=\sup \limits _{\begin{array}{c} s,t\in [0,\infty ) \\ \left| s-t\right| \le \delta \end{array}}\left| g\left( s\right) -g\left( t\right) \right| . \end{aligned}$$
(3.1)

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

$$\begin{aligned} \left| T_{n}\left( h;x\right) -h\left( x\right) \right| \le M\left( \Delta _{2}\right) ^{\alpha /2}, \end{aligned}$$

where \(\Delta _{2}\) is given in Lemma 4.

Proof

Since \(h\in Lip_{M}\left( \alpha \right) \), it follows from linearity

$$\begin{aligned} \left| T_{n}\left( h;x\right) -h\left( x\right) \right| \le T_{n}\left( \left| h\left( t\right) -h\left( x\right) \right| ;x\right) \le MT_{n}\left( \left| t-x\right| ^{\alpha };x\right) . \end{aligned}$$

From Lemma 4 and Hölder’s famous inequality, we can write

$$\begin{aligned} \left| T_{n}\left( h;x\right) -h\left( x\right) \right| \le M\left[ \Delta _{2}\right] ^{\frac{\alpha }{2}}. \end{aligned}$$

Thus, we find the required inequality. \(\square \)

Theorem 4

The operators in (2.5) verify the inequality

$$\begin{aligned} \left| T_{n}\left( g;x\right) -g\left( x\right) \right| {\le } \left( 1+\sqrt{\frac{1}{n}x\left( 4x^{3}\alpha ^{2}+4x\alpha +n\right) +2\mu x\frac{ e_{\mu }(-nx)}{e_{\mu }(nx)}}\right) \omega \left( g;\frac{1}{\sqrt{n}} \right) , \end{aligned}$$

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

$$\begin{aligned} C_{B}^{2}[0,\infty )=\{g\in C_{B}[0,\infty ):g^{\prime },g^{\prime \prime }\in C_{B}[0,\infty )\} \end{aligned}$$
(3.2)

with the norm

$$\begin{aligned} \left\| g\right\| _{C_{B}^{2}[0,\infty )}=\left\| g\right\| _{C_{B}[0,\infty )}+\left\| g^{\prime }\right\| _{C_{B}[0,\infty )}+\left\| g^{\prime \prime }\right\| _{C_{B}[0,\infty )} \end{aligned}$$

for all g in \(C_{B}^{2}[0,\infty ).\)

Lemma 5

For \(h\in C_{B}^{2}[0,\infty )\), the following inequality holds true

$$\begin{aligned} \left| T_{n}\left( h;x\right) -h\left( x\right) \right| \le \left[ \Delta _{1}+\Delta _{2}\right] \left\| h\right\| _{C_{B}^{2}[0,\infty )}, \end{aligned}$$
(3.3)

where \(\Delta _{1}\) and \(\Delta _{2}\) are given by in Lemma 4.

Proof

From the Taylor’s series of the function h,

$$\begin{aligned} h\left( s\right) =h\left( x\right) +\left( s-x\right) h^{\prime }\left( x\right) +\frac{\left( s-x\right) ^{2}}{2!}h^{\prime \prime }\left( \varrho \right) ,\text { }\varrho \in \left( x,s\right) . \end{aligned}$$

Applying the operator \(T_{n}\) to both sides of this equality and then using the linearity of the operator, we have

$$\begin{aligned} T_{n}\left( h;x\right) -h\left( x\right) =h^{\prime }\left( x\right) \Delta _{1}+\frac{h^{\prime \prime }\left( \varrho \right) }{2}\Delta _{2}. \end{aligned}$$

From Lemma 4, it yields

$$\begin{aligned} \left| T_{n}\left( h;x\right) -h\left( x\right) \right|\le & {} \frac{ 2\alpha x^{2}}{n}\left\| h^{\prime }\right\| _{C_{B}[0,\infty )} \\&+\left[ \frac{1}{n^{2}}x\left( 4x^{3}\alpha ^{2}+4\alpha x+n\right) +\frac{ 2\mu x}{n}\frac{e_{\mu }(-nx)}{e_{\mu }(nx)}\right] \left\| h^{\prime \prime }\right\| _{C_{B}[0,\infty )} \\\le & {} [\Delta _{1}+\Delta _{2}]\left\| h\right\| _{C_{B}^{2}[0,\infty )}, \end{aligned}$$

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

$$\begin{aligned} \omega _{2}\left( f;\delta \right) :=\sup _{0<s\le \delta }\left\| f\left( .+2s\right) -2f\left( .+s\right) +f\left( .\right) \right\| _{C_{B}[0,\infty )}. \end{aligned}$$

Peetre’s K-functional of the function \(f\in C_{B}\left[ 0,\infty \right) \) is as follows

$$\begin{aligned} K\left( f;\delta \right) :=\inf _{g\in C_{B}^{2}\left[ 0,\infty \right) }\left\{ \left\| f-g\right\| _{C_{B}}+\delta \left\| g\right\| _{C_{B}^{2}}\right\} . \end{aligned}$$
(3.4)

The relation between K and \(\omega _{2}\) is as

$$\begin{aligned} K\left( f;\delta \right) \le M\left\{ w_{2}\left( f;\sqrt{\delta }\right) +\min \left( 1,\delta \right) \left\| f\right\| _{C_{B}}\right\} \end{aligned}$$
(3.5)

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

$$\begin{aligned} \left| T_{n}\left( g;x\right) -g\left( x\right) \right| \le 2M\left\{ \min \left( 1,\frac{\chi _{n}\left( x\right) }{2}\right) \left\| g\right\| _{C_{B}[0,\infty )}+\omega _{2}\left( g;\sqrt{\frac{ \chi _{n}\left( x\right) }{2}}\right) \right\} \nonumber \\ \end{aligned}$$
(3.6)

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

$$\begin{aligned} \Theta =\left| T_{n}\left( g;x\right) -g\left( x\right) \right| \le \left| T_{n}\left( g-f;x\right) \right| +\left| T_{n}\left( f;x\right) -f\left( x\right) \right| +\left| g\left( x\right) -f\left( x\right) \right| \end{aligned}$$

from Lemma 5, which follows

$$\begin{aligned} \Theta\le & {} 2\left\| g-f\right\| _{C_{B}[0,\infty )}+\chi _{n}\left( x\right) \left\| f\right\| _{C_{B}^{2}[0,\infty )} \\= & {} 2\left\{ \left\| g-f\right\| _{C_{B}[0,\infty )}+\frac{\chi _{n}}{2} \left( x\right) \left\| f\right\| _{C_{B}^{2}[0,\infty )}\right\} . \end{aligned}$$

From (3.4), we have

$$\begin{aligned} \Theta \le 2K\left( g;\frac{\chi _{n}\left( x\right) }{2}\right) , \end{aligned}$$

which holds

$$\begin{aligned} \Theta \le 2M\left\{ \min \left( 1,\frac{\chi _{n}\left( x\right) }{2} \right) \left\| g\right\| _{C_{B}[0,\infty )}+\omega _{2}\left( g; \sqrt{\frac{\chi _{n}\left( x\right) }{2}}\right) \right\} \end{aligned}$$

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

$$\begin{aligned} \left| T_{n}\left( g;x\right) -g\left( x\right) \right|\le & {} M\omega _{2}\left( g;\frac{1}{2}\sqrt{\frac{1}{n^{2}}x\left( 8x^{3}\alpha ^{2}+4x\alpha +n\right) +\frac{2\mu x}{n}\frac{e_{\mu }(-nx)}{e_{\mu }(nx)}} \right) \\&+\omega \left( g;\frac{2\alpha x^{2}}{n}\right) \end{aligned}$$

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].