Gamma Generalization Operators Involving Analytic Functions

Abstract

In the present paper, we give an operator with the help of a generalization of Boas–Buck type polynomials by means of Gamma function. We have approximation properties and moments. The rate of convergence is given by the Ditzian–Totik first order modulus of smoothness and the K-functional. Furthermore, we obtain the point-wise estimations for this operator.

1. Introduction

The positive approximation processes introduced by Korovkin to the literature are one of the cornerstones of the approximation theory. This process comes as a problem in many fields of mathematics, such as measure theory, function analysis, harmonic analysis, partial differential equations and probability theory. Szász operators are one of the most common operators in such processes.

Szász operator [1] is a well known operator in the approximation theory on $[0,\infty )$ as follows:

$$S_n(f;x)=e^{-nx}\sum _{j=0}^{\infty }\frac{{\left(nx\right)}^j}{j!}f\left(\frac{j}{n}\right)$$

where $f\in C[0,\infty )$ and $x\ge 0.$ Many authors have investigated and studied the operator and its generalization. Let us talk briefly about these generalizations. In 1974, Ismail [2] considered a new generalization of this operator, and Jakimovski and Leviatan [3] introduced the one of the best known generalizations in the literature with the help of the formulae of the generating function of Sheffer polynomials. With the same method, Varma, Sucu and İçöz [4] investigated the generalization of Szász operator obtained by the generating function Brenke-type polynomials. The details of these polynomials can be found in [5]. These authors gave the generalizations of Szász operator including Boas–Buck-type polynomials in [6] and generalized Appell polynomials in [7], respectively. Furthermore, the details of these polynomials can be found in [8,9]. In [10], Sucu defined Dunkl analogue of Szász operator via the generalization of exponential function. In 2015, the authors [11] gave a new type generalization of Jakimovski–Leviatan-type Szász operators. Krech obtained the generalization of Szász operators with the help of two variable Hermite polynomials in [12], and then the authors gave a Kantorovich type generalization of this operator and investigated the approximation properties of this operator in [13]. Lastly, Sucu and Varma defined the sequence of operators involving analytic functions as follows:

$$H_n\left(f;x\right)=\frac{1}{R\left(1\right)\Psi \left(nxS\left(1\right)+\sigma \left(1\right)\right)}\sum _{j=0}^{\infty }{\theta }_j\left(nx\right)f\left(\frac{j}{n}\right),$$

where ${\theta }_j\left(x\right)$ is given by (2), $f\in C[0,\infty )$ and $x\ge 0$ and gave a Kantorovich-type generalization of this sequence and derived the results about the rate of convergence in [14]. Then taking advantage of this study, İçöz ane Eryiğit defined beta generalization of Stancu–Durrmeyer operators involving a generalization of Boas–Buck type polynomials via well-known Beta function in [15].

In light of this theory, we define the operator $K_n$ using a generalization of Boas–Buck type polynomials or analytic functions as

$$K_n\left(f;x\right)=\frac{1}{R\left(1\right)\Psi \left(nxS\left(1\right)+\sigma \left(1\right)\right)}\sum _{j=0}^{\infty }{\theta }_j\left(nx\right)\frac{n^{j+\lambda +1}}{\Gamma \left(j+\lambda +1\right)}\underset{0}{\overset{\infty }{\int }}{t}^{j+\lambda }{e}^{-nt}f\left(t\right)dt,$$

(1)

where $x\ge 0$ and $\lambda \ge 0.$ Here,

$$R\left(\nu \right)\Psi \left(xS\left(\nu \right)+\sigma \left(\nu \right)\right)=\sum _{j=0}^{\infty }{\theta }_j\left(x\right){\nu }^j$$

(2)

and $R,\Psi ,S$ and $\sigma$ have representations with power series at the disc $\left|v\right|<L(L>1)$ [16] as

$$\begin{array}{ccc}\hfill S\left(\nu \right)& =& \sum _{j=0}^{\infty }{s}_j{\nu }^{j+1},s_0\ne 0,\hfill \\ \hfill \sigma \left(\nu \right)& =& \sum _{j=0}^{\infty }{\sigma }_j{\nu }^{j+2}.\hfill \end{array}$$

The following conditions must be fulfilled for convergence and positivity properties of the operator $K_n$

$$\left\{\begin{array}{c}\Psi :\mathbb{R}\to \left(0,\infty \right)\\ {\theta }_j\left(x\right)\ge 0;j=0,1,2,\dots \\ R\left(1\right)>0,S^{\prime }\left(1\right)=1\end{array}\right..$$

(3)

In Section 2, we will give approximation properties of the operator with the help of the well-known Korovkin theorem. In Section 3, the rate of convergence for the operator is introduced by the meaning K-functional and the Ditzian–Totik first order modulus of smoothness. In the last section, the point-wise estimations of operators are given.

2. Approximation Properties

In this section, we will find $K_n\left(t^i;x\right)$ for $i=0,1,2$ to apply Korovkin theorem.

For $f\left(t\right)=1,$ we have

$$\begin{array}{ccc}\hfill K_n\left(1;x\right)& =& \frac{1}{R\left(1\right)\Psi \left(nxS\left(1\right)+\sigma \left(1\right)\right)}\sum _{j=0}^{\infty }{\theta }_j\left(nx\right)\frac{n^{j+\lambda +1}}{\Gamma \left(j+\lambda +1\right)}\underset{0}{\overset{\infty }{\int }}{t}^{j+\lambda }{e}^{-nt}dt\hfill \\ & =& 1.\hfill \end{array}$$

(4)

We can get the $K_n\left(t;x\right)$ by choosing $f\left(t\right)=t$,

$$\begin{array}{ccc}\hfill K_n\left(t;x\right)& =& \frac{1}{R\left(1\right)\Psi \left(nxS\left(1\right)+\sigma \left(1\right)\right)}\sum _{j=0}^{\infty }{\theta }_j\left(nx\right)\frac{n^{j+\lambda +1}}{\Gamma \left(j+\lambda +1\right)}\underset{0}{\overset{\infty }{\int }}{t}^{j+\lambda +1}{e}^{-nt}dt\hfill \\ & =& \frac{1}{R\left(1\right)\Psi \left(nxS\left(1\right)+\sigma \left(1\right)\right)}\sum _{j=0}^{\infty }{\theta }_j\left(nx\right)\frac{j}{n}+\frac{\lambda +1}{n}.\hfill \end{array}$$

(5)

Using the relation

$$R^{\prime }\left(1\right)\Psi \left(nxS\left(1\right)+\sigma \left(1\right)\right)+R\left(1\right){\Psi }^{\prime }\left(nxS\left(1\right)+\sigma \left(1\right)\right)\left(nx{S}^{\prime }\left(1\right)+{\sigma }^{\prime }\left(1\right)\right)=\sum _{j=0}^{\infty }j{\theta }_j\left(nx\right)$$

(6)

in [14], we utilize the above equality in (5)

$$\begin{array}{ccc}\hfill K_n\left(t;x\right)& =& \frac{1}{R\left(1\right)\Psi \left(nxS\left(1\right)+\sigma \left(1\right)\right)}\sum _{j=0}^{\infty }{\theta }_j\left(nx\right)\frac{j}{n}+\frac{\lambda +1}{n}\hfill \\ & =& \frac{{\Psi }^{\prime }\left(nxS\left(1\right)+\sigma \left(1\right)\right)}{\Psi \left(nxS\left(1\right)+\sigma \left(1\right)\right)}x+\frac{1}{n}\left[\frac{R^{\prime }\left(1\right)}{R\left(1\right)}+{\sigma }^{\prime }\left(1\right)\frac{{\Psi }^{\prime }\left(nxS\left(1\right)+\sigma \left(1\right)\right)}{\Psi \left(nxS\left(1\right)+\sigma \left(1\right)\right)}+\lambda +1\right].\hfill \end{array}$$

(7)

Finally, we find $K_n\left(t^2;x\right)$ from the Equation (1) by choosing $f\left(t\right)=t^2$,

$$\begin{array}{ccc}\hfill K_n\left(t^2;x\right)& =& \frac{1}{R\left(1\right)\Psi \left(nxS\left(1\right)+\sigma \left(1\right)\right)}\sum _{j=0}^{\infty }{\theta }_j\left(nx\right)\frac{n^{j+\lambda +1}}{\Gamma \left(j+\lambda +1\right)}\underset{0}{\overset{\infty }{\int }}{t}^{j+\lambda +2}{e}^{-nt}dt\hfill \\ & =& \frac{1}{n^2}\frac{1}{R\left(1\right)\Psi \left(nxS\left(1\right)+\sigma \left(1\right)\right)}\sum _{j=0}^{\infty }{\theta }_j\left(nx\right)j^2\hfill \\ & & +\frac{2\lambda +3}{n^2}\frac{1}{R\left(1\right)\Psi \left(nxS\left(1\right)+\sigma \left(1\right)\right)}\sum _{j=0}^{\infty }{\theta }_j\left(nx\right)j\hfill \\ & & +\frac{\left(\lambda +1\right)\left(\lambda +2\right)}{n^2}\frac{1}{R\left(1\right)\Psi \left(nxS\left(1\right)+\sigma \left(1\right)\right)}\sum _{j=0}^{\infty }{\theta }_j\left(nx\right).\hfill \end{array}$$

(8)

Using the relation

$$\begin{array}{ccc}\hfill \sum _{j=0}^{\infty }{\theta }_j\left(nx\right)j^2& =& {\Psi }^{″}\left(nxS\left(1\right)+\sigma \left(1\right)\right)R\left(1\right)\left(n^2{x}^2+2nx{\sigma }^{\prime }\left(1\right)+{\left({\sigma }^{\prime }\left(1\right)\right)}^2\right)\hfill \\ & & +{\Psi }^{\prime }\left(nxS\left(1\right)+\sigma \left(1\right)\right)\left[R^{\prime }\left(1\right)\left(2nx+2{\sigma }^{\prime }\left(1\right)\right)\right.\hfill \\ & & \left.+R\left(1\right)\left(nx{S}^{″}\left(1\right)+nx+{\sigma }^{\prime }\left(1\right)+{\sigma }^{″}\left(1\right)\right)\right]\hfill \\ & & +\Psi \left(nxS\left(1\right)+\sigma \left(1\right)\right)\left[R^{″}\left(1\right)+R^{\prime }\left(1\right)\right]\hfill \end{array}$$

(9)

in [14], (2), (6) on (8), then we get

$$\begin{array}{ccc}\hfill K_n\left(t^2;x\right)& =& \frac{{\Psi }^{″}\left(nxS\left(1\right)+\sigma \left(1\right)\right)}{\Psi \left(nxS\left(1\right)+\sigma \left(1\right)\right)}{x}^2+\frac{2{\sigma }^{\prime }\left(1\right)}{n}\frac{{\Psi }^{″}\left(nxS\left(1\right)+\sigma \left(1\right)\right)}{\Psi \left(nxS\left(1\right)+\sigma \left(1\right)\right)}x\hfill \\ & & +\left(\frac{2}{n}\frac{R^{\prime }\left(1\right)}{R\left(1\right)}+\frac{2\lambda +3}{n}+\frac{S^{″}\left(1\right)+1}{n}\right)\frac{{\Psi }^{\prime }\left(nxS\left(1\right)+\sigma \left(1\right)\right)}{\Psi \left(nxS\left(1\right)+\sigma \left(1\right)\right)}x\hfill \\ & & +\frac{{\left({\sigma }^{\prime }\left(1\right)\right)}^2}{n^2}\frac{{\Psi }^{″}\left(nxS\left(1\right)+\sigma \left(1\right)\right)}{\Psi \left(nxS\left(1\right)+\sigma \left(1\right)\right)}\hfill \\ & & +\left(\frac{R^{\prime }\left(1\right)}{R\left(1\right)}\frac{2{\sigma }^{\prime }\left(1\right)}{n^2}+\frac{2\lambda +3}{n^2}{\sigma }^{\prime }\left(1\right)+\frac{{\sigma }^{\prime }\left(1\right)+{\sigma }^{″}\left(1\right)}{n^2}\right)\frac{{\Psi }^{\prime }\left(nxS\left(1\right)+\sigma \left(1\right)\right)}{\Psi \left(nxS\left(1\right)+\sigma \left(1\right)\right)}\hfill \\ & & +\frac{1}{n^2}\frac{R^{″}\left(1\right)+R^{\prime }\left(1\right)}{R\left(1\right)}+\frac{2\lambda +3}{n^2}\frac{R^{\prime }\left(1\right)}{R\left(1\right)}+\frac{\left(\lambda +1\right)\left(\lambda +2\right)}{n^2}.\hfill \end{array}$$

(10)

In the next theorem, we show that the operator $K_n$ is verifying the Korovkin theorem on a compact subset of $[0,\infty )$.

Theorem 1.

If $f\in C[0,\infty )\cap \Omega$ where

$$\Omega =\left\{f:\underset{x\to \infty }{lim}\frac{f\left(x\right)}{1+x^2}\mathrm{exists}\mathrm{and}\mathrm{is}\mathrm{finite}\right\}$$

and the following conditions are satisfied:

$$\underset{\tau \to \infty }{lim}\frac{{\Psi }^{\prime }\left(\tau \right)}{\Psi \left(\tau \right)}=1,\underset{\tau \to \infty }{lim}\frac{{\Psi }^{″}\left(\tau \right)}{\Psi \left(\tau \right)}=1$$

and (3), then the sequence operator $\left(K_n\right)$ converge uniformly to f on the compact subsets of $[0,\infty )$.

Proof. 

Using $li{m}_{n\to \infty }{K}_n(t^i;x)=x^i$ for $i=1,2,3$ by the hypothesis of Theorem 1 and applying the Korovkin type property of (vi) of Theorem 4.1.4 from [17], we get the proof. □

Remark 1.

It is necessary to ensure the conditions

$$\underset{\tau \to \infty }{lim}\frac{{\Psi }^{\prime }\left(\tau \right)}{\Psi \left(\tau \right)}=1,\underset{\tau \to \infty }{lim}\frac{{\Psi }^{″}\left(\tau \right)}{\Psi \left(\tau \right)}=1$$

for convergence throughout the study.

Lemma 1.

Let ${\Phi }_x^k\left(t\right)={(t-x)}^k$ for $k=1,2$, we have

$$\begin{array}{ccc}\hfill K_n\left({\Phi }_x^1;x\right)& =& \left[\frac{{\Psi }^{\prime }\left(nxS\left(1\right)+\sigma \left(1\right)\right)}{\Psi \left(nxS\left(1\right)+\sigma \left(1\right)\right)}-1\right]x+\frac{1}{n}\left[\frac{R^{\prime }\left(1\right)}{R\left(1\right)}+{\sigma }^{\prime }\left(1\right)\frac{{\Psi }^{\prime }\left(nxS\left(1\right)+\sigma \left(1\right)\right)}{\Psi \left(nxS\left(1\right)+\sigma \left(1\right)\right)}+\lambda +1\right],\hfill \end{array}$$

$$\begin{array}{ccc}& & \hfill \\ \hfill \Omega \left(x\right)& =& K_n\left({\Phi }_x^2;x\right)=\left[\frac{{\Psi }^{″}\left(nxS\left(1\right)+\sigma \left(1\right)\right)}{\Psi \left(nxS\left(1\right)+\sigma \left(1\right)\right)}-2\frac{{\Psi }^{\prime }\left(nxS\left(1\right)+\sigma \left(1\right)\right)}{\Psi \left(nxS\left(1\right)+\sigma \left(1\right)\right)}+1\right]x^2\hfill \\ & & +\left(\frac{2}{n}\frac{R^{\prime }\left(1\right)}{R\left(1\right)}+\frac{2\lambda +3}{n}+\frac{S^{″}\left(1\right)+1}{n}\right)\frac{{\Psi }^{\prime }\left(nxS\left(1\right)+\sigma \left(1\right)\right)}{\Psi \left(nxS\left(1\right)+\sigma \left(1\right)\right)}x\hfill \\ & & +\frac{2}{n}\left[{\sigma }^{\prime }\left(1\right)\left(\frac{{\Psi }^{″}\left(nxS\left(1\right)+\sigma \left(1\right)\right)}{\Psi \left(nxS\left(1\right)+\sigma \left(1\right)\right)}-\frac{{\Psi }^{\prime }\left(nxS\left(1\right)+\sigma \left(1\right)\right)}{\Psi \left(nxS\left(1\right)+\sigma \left(1\right)\right)}\right)-\frac{R^{\prime }\left(1\right)}{R\left(1\right)}-\lambda -1\right]x\hfill \\ & & +\frac{{\left({\sigma }^{\prime }\left(1\right)\right)}^2}{n^2}\frac{{\Psi }^{″}\left(nxS\left(1\right)+\sigma \left(1\right)\right)}{\Psi \left(nxS\left(1\right)+\sigma \left(1\right)\right)}\hfill \\ & & +\left(\frac{R^{\prime }\left(1\right)}{R\left(1\right)}\frac{2{\sigma }^{\prime }\left(1\right)}{n^2}+\frac{2\lambda +3}{n^2}{\sigma }^{\prime }\left(1\right)+\frac{{\sigma }^{\prime }\left(1\right)+{\sigma }^{″}\left(1\right)}{n^2}\right)\frac{{\Psi }^{\prime }\left(nxS\left(1\right)+\sigma \left(1\right)\right)}{\Psi \left(nxS\left(1\right)+\sigma \left(1\right)\right)}\hfill \end{array}$$

(11)

$$\begin{array}{ccc}& & +\frac{1}{n^2}\frac{R^{″}\left(1\right)+R^{\prime }\left(1\right)}{R\left(1\right)}+\frac{2\lambda +3}{n^2}\frac{R^{\prime }\left(1\right)}{R\left(1\right)}+\frac{\left(\lambda +1\right)\left(\lambda +2\right)}{n^2}.\hfill \end{array}$$

(12)

Proof. 

Using Equations (4), (7) and (10), the proof has been obtained. □

3. Rate of Convergence

We recall the definitions of the Ditzian–Totik first order modulus of smoothness [18] and the K-functional. Let $\varphi \left(x\right)=\sqrt{x}$, $f\in C[0,\infty )$ endowed with the norm ${\left|\right|f\left|\right|}_{\infty }=\underset{x\in [0,\infty )}{sup}\left|f\left(x\right)\right|$. The first order modulus of smoothness is given by

$${\omega }_{\varphi }(f,t)=\underset{0<h\le t}{sup}\left|f\left(x+\frac{h\varphi \left(x\right)}{2}\right)-f\left(x-\frac{h\varphi \left(x\right)}{2}\right)\right|,x\pm \frac{h\varphi \left(x\right)}{2}\in [0,\infty ).$$

The corresponding K-functional to ${\omega }_{\varphi }(f;t)$ is defined by

$$K_{\varphi }(f,t)=\underset{g\in W_{\varphi }[0,\infty )}{inf}\left\{\left|\right|f-{g\left|\right|}_{\infty }+t{\left|\left|\varphi g^{{}^{\prime }}\right|\right|}_{\infty }\right\}(t>0),$$

where $W_{\varphi }[0,\infty )=\left\{g:g\in AC[0,\infty ),\left|\left|\varphi g^{{}^{\prime }}\right|\right|<\infty \right\}$ and $AC[0,\infty )$ is the class of all absolutely continuous functions on $[0,\infty )$. By [18], there exists a constant $C>0$ such that

$$K_{\varphi }(f,t)\le C{\omega }_{\varphi }(f,t).$$

Theorem 2.

Let $C_B[0,\infty )$ means the class of real valued continuous bounded functions on $[0,\infty )$. For $f\in C_B[0,\infty )$, $\varphi \left(x\right)=\sqrt{x}$, we have

$$\left|K_n(f;x)-f\left(x\right)\right|\le C{\omega }_{\varphi }\left(f,{\left(\varphi \left(x\right)\right)}^{-1}\sqrt{\Omega \left(x\right)}\right),$$

where C is a positive constant, $\Omega \left(x\right)$ is defined in (12).

Proof. 

For any $x,t\in [0,\infty )$, we have

$$g\left(t\right)=g\left(x\right)+{\int }_x^t{g}^{{}^{\prime }}\left(u\right)du=g\left(x\right)+{\int }_x^t\frac{g^{{}^{\prime }}\left(u\right)\varphi \left(u\right)}{\varphi \left(u\right)}du.$$

Thus, we obtain

$$\begin{array}{ccc}\hfill \left|g\right(t)-g(x\left)\right|& =& \left|{\int }_x^t{g}^{{}^{\prime }}\left(u\right)du\right|\le {\left|\left|\varphi g^{{}^{\prime }}\right|\right|}_{\infty }\left|{\int }_x^t\frac{1}{\varphi \left(u\right)}du\right|=2{\left|\left|\varphi g^{{}^{\prime }}\right|\right|}_{\infty }\left|\sqrt{t}-\sqrt{x}\right|\hfill \\ & =& 2{\left|\left|\varphi g^{{}^{\prime }}\right|\right|}_{\infty }\frac{|t-x|}{\sqrt{t}+\sqrt{x}}\le 2{\left|\left|\varphi g^{{}^{\prime }}\right|\right|}_{\infty }\frac{|t-x|}{\sqrt{x}}=2{\left|\left|\varphi g^{{}^{\prime }}\right|\right|}_{\infty }\frac{|t-x|}{\varphi \left(x\right)}.\hfill \\ & & \hfill \end{array}$$

(13)

Next, we have

$$\begin{array}{ccc}& & \left|K_n(g;x)-g\left(x\right)\right|\hfill \\ & =& \left|K_n(g;x)-g\left(x\right)K_n(1;x)\right|\hfill \\ & \le & \frac{1}{R\left(1\right)\Psi \left(nxS\right(1)+\sigma (1\left)\right)}\sum _{j=0}^{\infty }\frac{n^{j+\lambda +1}}{\Gamma (j+\lambda +1)}{\int }_0^{\infty }{t}^{j+\lambda }{e}^{-nt}|f\left(t\right)-f\left(x\right)|dt\hfill \\ & \le & 2{\left|\left|\varphi g^{\prime }\right|\right|}_{\infty }{\left(\varphi \left(x\right)\right)}^{-1}{K}_n\left(\right|t-x|;x)\hfill \end{array}$$

(14)

by applying inequality (13). Using (14), we have

$$\begin{array}{ccc}\hfill \left|K_n(f;x)-f\left(x\right)\right|& \le & \left|K_n(f-g;x)\right|+|f-g|+\left|K_n(g;x)-g\left(x\right)\right|\hfill \\ & \le & 2\left|\right|f-{g\left|\right|}_{\infty }+2{\left|\left|\varphi g^{{}^{\prime }}\right|\right|}_{\infty }{\left(\varphi \left(x\right)\right)}^{-1}{K}_n\left(|t-x|;x\right).\hfill \end{array}$$

(15)

Taking the infimum on the right hand side over all $g\in W_{\varphi }[0,\infty )$ of (15) and using Cauchy–Schwarz inequality, we obtain

$$\begin{array}{ccc}\hfill \left|K_n(f;x)-f\left(x\right)\right|& \le & 2K_{\varphi }\left(f,{\left(\varphi \left(x\right)\right)}^{-1}{K}_n\left(|t-x|;x\right)\right)\hfill \\ & \le & 2K_{\varphi }\left(f,{\left(\varphi \left(x\right)\right)}^{-1}{\left[K_n\left({\Phi }_x^2;x\right)\right]}^{\frac{1}{2}}\right)\hfill \\ & \le & C{\omega }_{\varphi }\left(f,{\left(\varphi \left(x\right)\right)}^{-1}\sqrt{\Omega \left(x\right)}\right),\hfill \end{array}$$

where C is a positive constant, and $\Omega \left(x\right)$ is defined in (12). Theorem 2 is proved. □

4. Point-Wise Estimations

We denote that $f\in C[0,\infty )$ is in $Li{p}_M(\xi ,E)$, $\gamma \in (0,1]$, $E\subset (0,\infty )$ if it satisfies the following condition:

$$|f\left(t\right)-f\left(x\right)|\le {M|t-x|}^{\xi },t\in [0,\infty ),x\in E,$$

where M is a positive constant depending only on $\xi$ and f.

Theorem 3.

Let $\xi \in (0,1]$, E be any bounded subset on $[0,\infty )$ and $f\in C_B[0,\infty )\cap Li{p}_M(\xi ,E)$, for all $x\in [0,\infty )$, we have

$$\left|K_n(f;x)-f\left(x\right)\right|\le M\left[{\left(\Omega \left(x\right)\right)}^{\frac{\xi }{2}}+2d^{\xi }(x;E)\right],$$

where $\Omega \left(x\right)$ is defined in (12), $d(x;E)$ denotes the distance between x and E, and defined by

$$d(x;E)=inf\left\{|t-x|:t\in E\right\}.$$

Proof. 

Let $\overline{E}$ be the closure of E. Then, there is at least a point $t_0$ such that $d(x;E)=\left|x-t_0\right|$. By the triangle inequality

$$|f\left(t\right)-f\left(x\right)|\le \left|f\left(t\right)-f\left(t_0\right)\right|+\left|f\left(x\right)-f\left(t_0\right)\right|,$$

we have

$$\begin{array}{ccc}\hfill \left|K_n(f;x)-f\left(x\right)\right|& \le & K_n\left(\left|f\left(t\right)-f\left(t_0\right)\right|;x\right)+K_n\left(\left|f\left(x\right)-f\left(t_0\right)\right|;x\right)\hfill \\ & \le & M\left[K_n\left({\left|t-t_0\right|}^{\xi };x\right)+{\left|x-t_0\right|}^{\xi }\right]\hfill \\ & \le & M\left[K_n\left({|t-x|}^{\xi }+{\left|x-t_0\right|}^{\xi };x\right)+{\left|x-t_0\right|}^{\xi }\right]\hfill \\ & =& M\left[K_n\left({|t-x|}^{\xi };x\right)+2{\left|x-t_0\right|}^{\xi }\right].\hfill \end{array}$$

Letting $p=\frac{2}{\xi }$ and $q=\frac{2}{2-\xi }$, using Hölder inequality, we obtain

$$\begin{array}{ccc}\hfill \left|K_n(f;x)-f\left(x\right)\right|& \le & M\left\{{\left[K_n\left({|t-x|}^{p\xi };x\right)\right]}^{\frac{1}{p}}{\left[K_n\left(1^q;x\right)\right]}^{\frac{1}{q}}+2d^{\xi }(x;E)\right\}\hfill \\ & \le & M\left\{{\left[K_n\left({\Phi }_x^2\left(x\right);x\right)\right]}^{\frac{\xi }{2}}+2d^{\xi }(x;E)\right\}\hfill \\ & \le & M\left[{\left(\Omega \left(x\right)\right)}^{\frac{\xi }{2}}+2d^{\xi }(x;E)\right].\hfill \end{array}$$

Theorem 3 is proved. □

Finally, we give the local direct estimate by using the Lipschitz-type maximal function of the order $\xi$, which is defined as

$${\tilde{\omega }}_{\xi }(f,x)=\underset{x,t\in {\mathbb{R}}_{+},x\ne t}{sup}\frac{\left|f\right(t)-f(x\left)\right|}{{|t-x|}^{\xi }},\xi \in (0,1].$$

(16)

Theorem 4.

Let $f\in C_B[0,\infty )$ and $\xi \in (0,1]$, for all $x\in [0,\infty )$, we have

$$\left|K_n(f;x)-f\left(x\right)\right|\le {\tilde{\omega }}_{\xi }(f,x){\left(\Omega \left(x\right)\right)}^{\frac{\xi }{2}},$$

where $\Omega \left(x\right)$ is defined in (12).

Proof. 

By (16), we have

$$\left|K_n(f;x)-f\left(x\right)\right|\le {\tilde{\omega }}_{\xi }(f,x)K_n\left({|t-x|}^{\xi };x\right).$$

Applying Hölder inequality, we obtain

$$\begin{array}{ccc}\hfill \left|K_n(f;x)-f\left(x\right)\right|& \le & {\tilde{\omega }}_{\xi }(f,x){\left[K_n\left({\Phi }_x^2\left(x\right);x\right)\right]}^{\frac{\xi }{2}}\hfill \\ & \le & {\tilde{\omega }}_{\xi }(f,x){\left(\Omega \left(x\right)\right)}^{\frac{\xi }{2}}.\hfill \end{array}$$

Theorem 4 is proved. □

5. Conclusions

With the help of this paper, we shed light on other authors by taking a general family of polynomials using in the definition of the operators. The new studies can be done using this family used in this study.