Approximation properties of modified Szász–Mirakyan operators in polynomial weighted space

Abstract

We introduce certain modified Szász–Mirakyan operators in polynomial weighted spaces of functions of one variable. We studied approximation properties of these operators.

Keywords:

• Szász–Mirakyan operators
• rate of convergence
• weighted approximation
• polynomial weight

2000 Mathematics subject classifications:

• primary 41A25
• 41A30
• 41A36

Public Interest Statement

In this work, we define new sequence of operators depending on a parameter. We prove that these newly defined sequence of operators are positive and linear. Using the moments of these operators, we estimate continuous signals (functions). The admissible value of the involved parameter allows us to make appropriate choice of it, in order to have better approximation. We approximate these sequence of operators in terms of the modulus of continuity and the modulus of smoothness in polynomial weighted space.

1. Introduction

Becker (1978) studied approximation problems for functions $f\in C_p$ and Szász–Mirakyan operators(1.1) $$\begin{array}{c}\hfill S_n(f,x)=e^{-nx}\sum _{k=0}^{\infty }\frac{{(nx)}^k}{k!}f\left(\frac{k}{n}\right),\end{array}$$(1.1) $x\in {\mathbb{R}}_0=[0,\infty )$, $n\in \mathbb{N}$, where $C_p$ with fixed $p\in {\mathbb{N}}_0=\mathbb{N}\cup \{0\}$ is space generated by the weighted function$${\mathit{\omega }}_0\left(x\right)=1,{\mathit{\omega }}_p(x)={\left(1+x^p\right)}^{-1}\text{if}p\ge 1,$$

for $x\in {\mathbb{R}}_0$, and $B_p$ be the set of all functions $f:{\mathbb{R}}_0\to \mathbb{R}$ for which $f{\mathit{\omega }}_p$ is bounded on ${\mathbb{R}}_0$ and the norm is given by the following formula:$${∥f∥}_p=\underset{x\in {\mathbb{R}}_0}{sup}{\mathit{\omega }}_p(x)|f(x)|.$$

Moreover, $C_p$ be the set of all $f\in B_p$ for which $f{\mathit{\omega }}_p$ is a uniformly continuous function on ${\mathbb{R}}_0$. The spaces $B_p$ and $C_p$ are called polynomial weighted spaces.

Becker (1978) theorems on degree of approximation of $f\in C_p$ by the operators $S_n$ were proved. From these theorems, it was deduced that(1.2) $$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{S}_n(f,x)=f(x),\end{array}$$(1.2)

for every $f\in C_p$, $p\in {\mathbb{N}}_0$ and $x\in {\mathbb{R}}_0$. Moreover, the convergence (1.2) is uniform on every interval $[x_1,x_2]$, $x_1,x_2\ge 0$.

Jain (1972) introduced generalization of Szász–Mirakyan operators (1.1) with help of a Poisson type distribution, as follows:(1.3) $$\begin{array}{c}\hfill J_n^{[\mathit{\beta }]}(f,x)=\sum _{k=0}^{\infty }{\mathit{\omega }}_{\mathit{\beta }}(k,nx)f\left(\frac{k}{n}\right),\end{array}$$(1.3)

where $x\in {\mathbb{R}}_0:=[0,\infty )$, $n\in \mathbb{N}$, $0\le \mathit{\beta }<1$ and(1.4) $$\begin{array}{c}\hfill {\mathit{\omega }}_{\mathit{\beta }}(k,\mathit{\alpha })=\frac{\mathit{\alpha }}{k!}{(\mathit{\alpha }+k\mathit{\beta })}^{k-1}{e}^{-(\mathit{\alpha }+k\mathit{\beta })}\text{for}\mathit{\alpha }\in {\mathbb{R}}_0,k\in {\mathbb{N}}_0=\mathbb{N}\cup \{0\}.\end{array}$$(1.4)

The convergence properties and degree of approximation properties of $J_n^{[\mathit{\beta }]}$ were examined by Jain (1972) for $f\in C({\mathbb{R}}_0)$, the set of all real valued continuous functions f on ${\mathbb{R}}_0$. In the particular case $\mathit{\beta }=0$, $J_n^{[\mathit{\beta }]}$ turn out to well known the Szász–Mirakyan operators (Szász, 1950) which defined by (1.1). Kantorovich type extension of the operators (1.3) was discussed in Umar and Razi (1985). Various other generalization and its approximation properties of similar type of operators are studied in Agratini (2013,2014), Mishra and Patel (2013), Mishra, Khatri, Mishra, and Deepmala (2013), Örkcü (2013), Patel and Mishra (2014,2015), Rempulska and Tomczak (2009), Tarabie (2012), Bardaro and Mantellini (2006,2009). In this paper, we modify operators $J_n^{[\mathit{\beta }]}$ given by (1.3), i.e. we consider operators(1.5) $$\begin{array}{c}\hfill J_n^{[\mathit{\beta }]}(f;a_n,b_n;x)=\sum _{k=0}^{\infty }{\mathit{\omega }}_{\mathit{\beta }}(k,a_{n}x)f\left(\frac{k}{b_n}\right),x\in {\mathbb{R}}_0,n\in \mathbb{N}\end{array}$$(1.5)

for $f\in C\left([0,\infty )\right)$, where ${(a_n)}_{n=1}^{\infty }$ and ${\left(b_n\right)}_{n=1}^{\infty }$ are given increasing and unbounded numerical sequence such that $a_n\ge 1$, $b_n\ge 1$ and ${\displaystyle {\left(\frac{a_n}{b_n}\right)}_1^{\infty }}$ is non decreasing and(1.6) $$\begin{array}{c}\hfill \frac{a_n}{b_n}=1+o\left(\frac{1}{b_n}\right).\end{array}$$(1.6)

If $a_n=b_n=n$ for all $n\in \mathbb{N}$, then the operators (1.5) reduce to the operators (1.3).

The paper is organized as follows. In our manuscript, we shall study approximation properties of operators (1.5). In Section 2, we shall examine moments of the operators $J_n^{[\mathit{\beta }]}(f;a_n,b_n;x)$. We discuss approximation properties of the operators (1.5) in Section 3. We mention Kantorovich type extension of the operators $J^n(f;a_n,b_n;x)$ for further research.

2. Moments of $J_n^{[\mathit{\beta }]}(f;a_n,b_n;x)$

In order to obtain moments of $J_n^{[\mathit{\beta }]}(f;a_n,b_n;x)$, we need some background results, which are as follows:

Lemma 1

(Jain, 1972) Let $0<\mathit{\alpha }<\infty$, $0\le \mathit{\beta }<1$ and let the generalized Poisson distribution given by (1.4). Then(2.1) $$\begin{array}{c}\hfill \sum _{k=0}^{\infty }{\mathit{\omega }}_{\mathit{\beta }}(\mathit{\alpha },k)=1.\end{array}$$(2.1)

Lemma 2

(Jain, 1972) Let $0<\mathit{\alpha }<\infty$, $0\le \mathit{\beta }<1$. Suppose that$$S(r,\mathit{\alpha },\mathit{\beta }):=\sum _{k=0}^{\infty }{(\mathit{\alpha }+\mathit{\beta }k)}^{k+r-1}\frac{e^{-(\mathit{\alpha }+\mathit{\beta }k)}}{k!},r=0,1,2,\dots$$

and$$\mathit{\alpha }S(0,\mathit{\alpha },\mathit{\beta }):=1.$$

Then(2.2) $$\begin{array}{c}\hfill S(r,\mathit{\alpha },\mathit{\beta })=\mathit{\alpha }S(r-1,\mathit{\alpha },\mathit{\beta })+\mathit{\beta }S(r,\mathit{\alpha }+\mathit{\beta },\mathit{\beta }).\end{array}$$(2.2)

Also,(2.3) $$\begin{array}{c}\hfill S(r,\mathit{\alpha },\mathit{\beta })=\sum _{k=0}^{\infty }{\mathit{\beta }}^k(\mathit{\alpha }+k\mathit{\beta })S(r-1,\mathit{\alpha }+k\mathit{\beta },\mathit{\beta }).\end{array}$$(2.3)

From (2.2) and (2.3), when $0\le \mathit{\beta }<1$, we get(2.4) $$\begin{array}{cc}\hfill S(1,\mathit{\alpha },\mathit{\beta })& =\frac{1}{1-\mathit{\beta }};\hfill \\ \hfill S(2,\mathit{\alpha },\mathit{\beta })& =\frac{\mathit{\alpha }}{{(1-\mathit{\beta })}^2}+\frac{{\mathit{\beta }}^2}{{(1-\mathit{\beta })}^3};\hfill \\ \hfill S(3,\mathit{\alpha },\mathit{\beta })& =\frac{{\mathit{\alpha }}^2}{{(1-\mathit{\beta })}^3}+\frac{\mathit{\alpha }{\mathit{\beta }}^2}{{(1-\mathit{\beta })}^4}+\frac{{\mathit{\beta }}^3+2{\mathit{\beta }}^4}{{(1-\mathit{\beta })}^5};\hfill \\ \hfill S(4,\mathit{\alpha },\mathit{\beta })& =\frac{{\mathit{\alpha }}^3}{{(1-\mathit{\beta })}^4}+\frac{6{\mathit{\alpha }}^2{\mathit{\beta }}^2}{{(1-\mathit{\beta })}^5}+\frac{4\mathit{\alpha }{\mathit{\beta }}^3+11\mathit{\alpha }{\mathit{\beta }}^4}{{(1-\mathit{\beta })}^6}+\frac{{\mathit{\beta }}^4+8{\mathit{\beta }}^5+6{\mathit{\beta }}^6}{{(1-\mathit{\beta })}^7}.\hfill \end{array}$$(2.4)

In the following lemma, we have computed moments up to fourth order.

Lemma 3

Let $0\le \mathit{\beta }<1$, then the following equalities hold:

(1)	${\displaystyle J_n^{[\mathit{\beta }]}(1;a_n,b_n;x)=1;}$
(2)	${\displaystyle J_n^{[\mathit{\beta }]}(t;a_n,b_n;x)=\frac{a_{n}x}{b_n(1-\mathit{\beta })};}$
(3)	${\displaystyle J_n^{[\mathit{\beta }]}(t^2;a_n,b_n;x)=\frac{x^2{a}_n^2}{{(1-\mathit{\beta })}^2{b}_n^2}+\frac{x{a}_n}{{(1-\mathit{\beta })}^3{b}_n^2};}$
(4)	${\displaystyle J_n^{[\mathit{\beta }]}(t^3;a_n,b_n;x)=\frac{x^3{a}_n^3}{{(1-\mathit{\beta })}^3{b}_n^3}+\frac{3x^2{a}_n^2}{{(1-\mathit{\beta })}^4{b}_n^3}+\frac{x(1+2\mathit{\beta })a_n}{{(1-\mathit{\beta })}^5{b}_n^3};}$
(5)	${\displaystyle J_n^{[\mathit{\beta }]}(t^4;a_n,b_n;x)=\frac{x^4{a}_n^4}{{(1-\mathit{\beta })}^4{b}_n^4}+\frac{6x^3{a}_n^3}{{(1-\mathit{\beta })}^5{b}_n^4}+\frac{x^2(7+8\mathit{\beta })a_n^2}{{(1-\mathit{\beta })}^6{b}_n^4}+\frac{x\left(1+8\mathit{\beta }+6{\mathit{\beta }}^2\right)a_n}{{(1-\mathit{\beta })}^7{b}_n^4}}$.

Proof

Using equalities (2.1), (2.4–2.7) and by simple commutation, we obtain$$\begin{array}{cc}\hfill J_n^{[\mathit{\beta }]}(1;a_n,b_n;x)& =\sum _{k=0}^{\infty }{\mathit{\omega }}_{\mathit{\beta }}(k,a_{n}x)=1;\hfill \\ \hfill J_n^{[\mathit{\beta }]}(t;a_n,b_n;x)& =\frac{a_{n}x}{b_n}\sum _{k=0}^{\infty }\frac{1}{k!}{(a_{n}x+k\mathit{\beta }+\mathit{\beta })}^k{e}^{-(a_{n}x+k\mathit{\beta }+\mathit{\beta })}\hfill \\ \hfill & =\frac{a_{n}x}{b_n}S(1,a_{n}x+\mathit{\beta },\mathit{\beta })\hfill \\ \hfill & =\frac{a_{n}x}{b_n(1-\mathit{\beta })};\hfill \\ \hfill J_n^{[\mathit{\beta }]}(t^2;a_n,b_n;x)& =\sum _{k=0}^{\infty }\frac{a_{n}x}{k!}{(a_{n}x+k\mathit{\beta })}^{k-1}{e}^{-(a_{n}x+k\mathit{\beta })}\frac{k^2}{b_n^2}\hfill \\ \hfill & =\frac{a_{n}x}{b_n^2}\left[S(1,a_{n}x+\mathit{\beta },\mathit{\beta })+S(2,a_{n}x+2\mathit{\beta },\mathit{\beta })\right]\hfill \\ \hfill & =\frac{a_{n}x}{b_n^2}\left[\frac{1}{1-\mathit{\beta }}+\frac{a_{n}x+2\mathit{\beta }}{{(1-\mathit{\beta })}^2}+\frac{{\mathit{\beta }}^2}{{(1-\mathit{\beta })}^3}\right]\hfill \\ \hfill & =\frac{x^2{a}_n^2}{{(1-\mathit{\beta })}^2{b}_n^2}+\frac{x{a}_n}{{(1-\mathit{\beta })}^3{b}_n^2};\hfill \\ \hfill J_n^{[\mathit{\beta }]}(t^3;a_n,b_n;x)& =\sum _{k=0}^{\infty }\frac{a_{n}x}{k!}{(a_{n}x+k\mathit{\beta })}^{k-1}{e}^{-(a_{n}x+k\mathit{\beta })}\frac{k^3}{b_n^3}\hfill \\ \hfill & =\frac{a_{n}x}{b_n^3}\left[S(1,a_{n}x+\mathit{\beta },\mathit{\beta })+3S(2,a_{n}x+2\mathit{\beta },\mathit{\beta })+S(3,a_{n}x+3\mathit{\beta },\mathit{\beta })\right]\hfill \\ \hfill & =\frac{x^3{a}_n^3}{{(1-\mathit{\beta })}^3{b}_n^3}+\frac{3x^2{a}_n^2}{{(1-\mathit{\beta })}^4{b}_n^3}+\frac{x(1+2\mathit{\beta })a_n}{{(1-\mathit{\beta })}^5{b}_n^3};\hfill \\ \hfill J_n^{[\mathit{\beta }]}(t^4;a_n,b_n;x)& =\sum _{k=0}^{\infty }\frac{a_{n}x}{k!}{(a_{n}x+k\mathit{\beta })}^{k-1}{e}^{-(a_{n}x+k\mathit{\beta })}\frac{k^4}{b_n^4}\hfill \\ \hfill & =\frac{a_{n}x}{b_n^4}\left[S(1,a_{n}x+\mathit{\beta },\mathit{\beta })+7S(2,a_{n}x+2\mathit{\beta },\mathit{\beta })\right.\hfill \\ \hfill & \left.+6S(3,a_{n}x+3\mathit{\beta },\mathit{\beta })+S(4,a_{n}x+4\mathit{\beta },\mathit{\beta })\right]\hfill \\ \hfill & =\frac{x^4{a}_n^4}{{(1-\mathit{\beta })}^4{b}_n^4}+\frac{6x^3{a}_n^3}{{(1-\mathit{\beta })}^5{b}_n^4}+\frac{x^2(7+8\mathit{\beta })a_n^2}{{(1-\mathit{\beta })}^6{b}_n^4}+\frac{x\left(1+8\mathit{\beta }+6{\mathit{\beta }}^2\right)a_n}{{(1-\mathit{\beta })}^7{b}_n^4}.\hfill \end{array}$$

Lemma 4

Let $0\le \mathit{\beta }<1$, then the following equalities hold:

(1)	${\displaystyle J_n^{[\mathit{\beta }]}(t-x;a_n,b_n;x)=\left(\frac{a_n}{b_n(1-\mathit{\beta })}-1\right)x;}$
(2)	${\displaystyle J_n^{[\mathit{\beta }]}({(t-x)}^2;a_n,b_n;x)=x^2{\left(\frac{a_n}{(1-\mathit{\beta })b_n}-1\right)}^2+\frac{x{a}_n}{{(1-\mathit{\beta })}^3{b}_n^2};}$
(3)	${\displaystyle J_n^{[\mathit{\beta }]}({(t-x)}^3;a_n,b_n;x)=x^3{\left(\frac{a_n}{(1-\mathit{\beta })b_n}-1\right)}^3+\frac{3x^2{a}_n}{b_n^2{(1-\mathit{\beta })}^3}\left(\frac{a_n}{(1-\mathit{\beta })b_n}-1\right)+\frac{x{a}_n(1+2\mathit{\beta })}{{(1-\mathit{\beta })}^5{b}_n^3};}$
(4)	$$\begin{array}{cc}\hfill J_n^{[\mathit{\beta }]}({(t-x)}^4;a_n,b_n;x)& =x^4{\left(\frac{a_n}{(1-\mathit{\beta })b_n}-1\right)}^4+\frac{6a_n{x}^3}{{(1-\mathit{\beta })}^3{b}_n^2}{\left(\frac{a_n}{(1-\mathit{\beta })b_n}-1\right)}^2\hfill \\ \hfill & +\frac{a_n{x}^2}{{(1-\mathit{\beta })}^5{b}_n^3}\left(\frac{a_n(7+8\mathit{\beta })}{(1-\mathit{\beta })b_n}-4-8\mathit{\beta }\right)+x\left(\frac{a_n(1+8\mathit{\beta }+6{\mathit{\beta }}^2)}{{(1-\mathit{\beta })}^7{b}_n^4}\right).\hfill \end{array}$$

Proof of the above lemma, follows from the linearity of the operators $J_n^{[\mathit{\beta }]}(f;a_n,b_n;x)$.

By equality (1.6) and ${\displaystyle \underset{n\to \infty }{lim}{\mathit{\beta }}_n=0}$, we obtain$$\begin{array}{cc}& \underset{n\to \infty }{lim}{b}_n{J}_n^{[{\mathit{\beta }}_n]}(t-x;a_n,b_n;x)=0;\hfill \\ \hfill & \underset{n\to \infty }{lim}{b}_n{J}_n^{[{\mathit{\beta }}_n]}({(t-x)}^2;a_n,b_n;x)=x;\hfill \\ \hfill & \underset{n\to \infty }{lim}{b}_n{J}_n^{[{\mathit{\beta }}_n]}({(t-x)}^3;a_n,b_n;x)=0;\hfill \\ \hfill & \underset{n\to \infty }{lim}{b}_n^2{J}_n^{[{\mathit{\beta }}_n]}({(t-x)}^4;a_n,b_n;x)=3x^2,\hfill \\ \hfill \end{array}$$

for every $x\in {\mathbb{R}}_0$.

3. Approximation properties

Lemma 5

Let $r\in \mathbb{N}$ be fixed number. Then there exist positive numerical coefficients ${\mathit{\lambda }}_{r,j,\mathit{\beta }}$, $1\le j\le r$, depending only on r and j such that$$J_n^{[\mathit{\beta }]}(t^r;a_n,b_n;x)=\frac{1}{b_n^r{(1-\mathit{\beta })}^r}\sum _{j=1}^r\frac{{\mathit{\lambda }}_{r,j,\mathit{\beta }}}{{(1-\mathit{\beta })}^{j-1}}{(a_{n}x)}^j,$$

for all $x\in {\mathbb{R}}_0$ and $n\in \mathbb{N}$. Moreover, we have ${\mathit{\lambda }}_{r,1,\mathit{\beta }}=1={\mathit{\lambda }}_{r,r,\mathit{\beta }}$.

The proof follows by a mathematical induction argument.

Lemma 6

For given $p\in {\mathbb{N}}_0$ and ${(a_n)}_{n=1}^{\infty }$ and ${(b_n)}_{n=1}^{\infty }$ there exists a positive constant $M_1(b_1,p,\mathit{\beta })$ such that(3.1) $$\begin{array}{c}\hfill {∥J_n^{[\mathit{\beta }]}\left(\frac{1}{{\mathit{\omega }}_p(t)};a_n,b_n;·\right)∥}_p\le M_1(b_1,p,\mathit{\beta }),n\in \mathbb{N}.\end{array}$$(3.1)

Moreover, for every $f\in C_p$, we have(3.2) $$\begin{array}{c}\hfill {∥J_n^{[\mathit{\beta }]}(f;a_n,b_n;·)∥}_p\le M_1(b_1,p,\mathit{\beta }){\Vert f\Vert }_p,n\in \mathbb{N}.\end{array}$$(3.2)

The formula (1.4), (1.5) and the inequality (3.2), show that $J_n^{[\mathit{\beta }]}$, $n\in \mathbb{N}$ is a positive linear operator from the space $C_p$ into $C_p$, $p\in {\mathbb{N}}_0$.

Proof

If $p=0$, then ${\displaystyle \Vert J_n^{[\mathit{\beta }]}\left(\frac{1}{{\mathit{\omega }}_0(t)};a_n,b_n;·\right){\Vert }_0=\underset{x\in {\mathbb{R}}_0}{sup}|J_n^{[\mathit{\beta }]}(1;a_n,b_n;x)|=1}$.

If $p\ge 1$, then by (1.5), (1.6), Lemma 3 and Lemma 5, we get$$\begin{array}{cc}\hfill {\mathit{\omega }}_p(x)J_n^{[\mathit{\beta }]}\left(\frac{1}{{\mathit{\omega }}_p(t)};a_n,b_n;x\right)& ={\mathit{\omega }}_p(x)\left\{1+J_n^{[\mathit{\beta }]}(t^p;a_n,b_n;x)\right\}\hfill \\ \hfill & =\frac{1}{1+x^p}\left\{1+\frac{1}{b_n^p{(1-\mathit{\beta })}^p}\sum _{j=1}^p\frac{{\mathit{\lambda }}_{r,j,\mathit{\beta }}}{{(1-\mathit{\beta })}^{j-1}}{(a_{n}x)}^j\right\}\hfill \\ \hfill & =\frac{1}{1+x^p}+\frac{1}{{(1-\mathit{\beta })}^p}\sum _{j=1}^p\frac{{\mathit{\lambda }}_{r,j,\mathit{\beta }}}{{(1-\mathit{\beta })}^{j-1}}\frac{1}{b_n^{p-j}}{\left(\frac{a_n}{b_n}\right)}^j\frac{x^j}{1+x^p}\hfill \\ \hfill & \le 1+\frac{1}{{(1-\mathit{\beta })}^p}\sum _{j=1}^p\frac{{\mathit{\lambda }}_{r,j,\mathit{\beta }}}{{(1-\mathit{\beta })}^{j-1}}\frac{1}{b_1^{p-j}}=M_1(b_1,p,\mathit{\beta }),\hfill \end{array}$$

for all $x\in {\mathbb{R}}_0$ and $n\in \mathbb{N}$. From this, (3.1) follows.

By (1.5) and definition of norm, we have$$\Vert J_n^{[\mathit{\beta }]}(f;a_n,b_n;·){\Vert }_p\le \Vert J_n^{[\mathit{\beta }]}(\frac{1}{{\mathit{\omega }}_p(t)};a_n,b_n;·){\Vert }_p{\Vert f\Vert }_p,$$

for every $f\in C_p$, $p\in \mathbb{N}$ and $n\in \mathbb{N}$. From (3.1), the inequalities (3.2) is achieved.

Theorem 1

For every $p\in {\mathbb{N}}_0$ there exists a positive constant $M_2(b_1,p,\mathit{\beta })$ such that(3.3) $$\begin{array}{c}\hfill {\mathit{\omega }}_p(x)J_n^{[\mathit{\beta }]}\left(\frac{{(t-x)}^2}{{\mathit{\omega }}_p(t)};a_n,b_n;x\right)\le M_2(b_1,p,\mathit{\beta })\left[x^2{\left(\frac{a_n}{(1-\mathit{\beta })b_n}-1\right)}^2+\frac{x}{{(1-\mathit{\beta })}^3{b}_n}\right],\end{array}$$(3.3)

for all $x\in {\mathbb{R}}_0$ and $n\in \mathbb{N}$.

Proof

If $p=0$, then (3.3) follows from values of $J_n^{[\mathit{\beta }]}\left({(t-x)}^2;a_n,b_n;x\right)$.

Let $J_n^{[\mathit{\beta }]}\left(f;x\right)=J_n^{[\mathit{\beta }]}\left(f;a_n,b_n;x\right)$. Notice that(3.4) $$\begin{array}{c}\hfill J_n^{[\mathit{\beta }]}\left(\frac{{(t-x)}^2}{{\mathit{\omega }}_p(t)};x\right)=J_n^{[\mathit{\beta }]}\left({(t-x)}^2;x\right)+J_n^{[\mathit{\beta }]}\left(t^p{(t-x)}^2;x\right).\end{array}$$(3.4)

For $p=1$, we get$$\begin{array}{cc}\hfill J_n^{[\mathit{\beta }]}\left(\frac{{(t-x)}^2}{{\mathit{\omega }}_1(t)};x\right)& =J_n^{[\mathit{\beta }]}\left({(t-x)}^2;x\right)+J_n^{[\mathit{\beta }]}\left(t{(t-x)}^2;x\right)\hfill \\ \hfill & =J_n^{[\mathit{\beta }]}\left({(t-x)}^2;x\right)+J_n^{[\mathit{\beta }]}\left({(t-x)}^3;x\right)+x{J}_n^{[\mathit{\beta }]}\left({(t-x)}^2;x\right)\hfill \\ \hfill & =(1+x)J_n^{[\mathit{\beta }]}\left({(t-x)}^2;x\right)+J_n^{[\mathit{\beta }]}\left({(t-x)}^3;x\right).\hfill \end{array}$$

Therefore,$$\begin{array}{cc}\hfill (1+x)J_n^{[\mathit{\beta }]}\left(\frac{{(t-x)}^2}{{\mathit{\omega }}_1(t)};x\right)& =x^2{\left(\frac{a_n}{(1-\mathit{\beta })b_n}-1\right)}^2+\frac{x{a}_n}{{(1-\mathit{\beta })}^3{b}_n^2}+\frac{x^3}{1+x}{\left(\frac{a_n}{(1-\mathit{\beta })b_n}-1\right)}^3\hfill \\ \hfill & +\frac{3x^2{a}_n}{(1+x)b_n^2{(1-\mathit{\beta })}^3}\left(\frac{a_n}{(1-\mathit{\beta })b_n}-1\right)+\frac{x{a}_n(1+2\mathit{\beta })}{(1+x){(1-\mathit{\beta })}^5{b}_n^3}\hfill \\ \hfill & \le M_2(b_1,p,\mathit{\beta })\left[x^2{\left(\frac{a_n}{(1-\mathit{\beta })b_n}-1\right)}^2+\frac{x}{{(1-\mathit{\beta })}^3{b}_n}\right].\hfill \end{array}$$

If $p\ge 2$, then by Lemma 5, we get$$\begin{array}{cc}\hfill {\mathit{\omega }}_p(x)J_n^{[\mathit{\beta }]}\left(t^p{(t-x)}^2;x\right)& ={\mathit{\omega }}_p(x)\left\{J_n^{[\mathit{\beta }]}\left(t^{p+2};x\right)-2x{J}_n^{[\mathit{\beta }]}\left(t^{p+1};x\right)+x^2{J}_n^{[\mathit{\beta }]}\left(t^p;x\right)\right\}\hfill \\ \hfill & =\frac{x}{b_n(1-\mathit{\beta })}\left\{\frac{1}{b_n^{p+1}{(1-\mathit{\beta })}^{p+1}}\sum _{j=1}^{p+1}\frac{{\mathit{\lambda }}_{p+2,j,\mathit{\beta }}}{{(1-\mathit{\beta })}^{j-1}}{a}_n^j\frac{x^{j-1}}{1+x^p}\right.\hfill \\ \hfill & -\frac{2}{b_n^p{(1-\mathit{\beta })}^p}\sum _{j=1}^p\frac{{\mathit{\lambda }}_{p+1,j,\mathit{\beta }}}{{(1-\mathit{\beta })}^{j-1}}{a}_n^j\frac{x^j}{1+x^p}\hfill \\ \hfill & +\left.\frac{1}{b_n^{p-1}{(1-\mathit{\beta })}^{p-1}}\sum _{j=1}^{p-1}\frac{{\mathit{\lambda }}_{p,j,\mathit{\beta }}}{{(1-\mathit{\beta })}^{j-1}}{a}_n^j\frac{x^{j+1}}{1+x^p}\right\}+\frac{1}{{(1-\mathit{\beta })}^{2p+3}}{\left(\frac{a_n}{b_n}\right)}^{p+2}\frac{x^{p+2}}{1+x^p}\hfill \\ \hfill & -\frac{2}{{(1-\mathit{\beta })}^{2p+1}}{\left(\frac{a_n}{b_n}\right)}^{p+1}\frac{x^{p+2}}{1+x^p}+\frac{1}{{(1-\mathit{\beta })}^{2p-1}}{\left(\frac{a_n}{b_n}\right)}^p\frac{x^{p+2}}{1+x^p}\hfill \\ \hfill & =\frac{x}{b_n(1-\mathit{\beta })}\left\{\frac{1}{b_n^{p+1}{(1-\mathit{\beta })}^{p+1}}\sum _{j=1}^{p+1}\frac{{\mathit{\lambda }}_{p+2,j,\mathit{\beta }}}{{(1-\mathit{\beta })}^{j-1}}{a}_n^j\frac{x^{j-1}}{1+x^p}\right.\hfill \\ \hfill & -\frac{2}{b_n^p{(1-\mathit{\beta })}^p}\sum _{j=1}^p\frac{{\mathit{\lambda }}_{p+1,j,\mathit{\beta }}}{{(1-\mathit{\beta })}^{j-1}}{a}_n^j\frac{x^j}{1+x^p}\hfill \\ \hfill & +\left.\frac{1}{b_n^{p-1}{(1-\mathit{\beta })}^{p-1}}\sum _{j=1}^{p-1}\frac{{\mathit{\lambda }}_{p,j,\mathit{\beta }}}{{(1-\mathit{\beta })}^{j-1}}{a}_n^j\frac{x^{j+1}}{1+x^p}\right\}\hfill \\ \hfill & +\frac{x^{p+2}}{1+x^p}{\left(\frac{a_n}{b_n}\right)}^p\frac{1}{{(1-\mathit{\beta })}^{2p-1}}{\left(\frac{a_n}{b_n(1-\mathit{\beta })}-1\right)}^2.\hfill \end{array}$$

Since ${\displaystyle 0\le \frac{a_n}{b_n}\le 1}$ for $n\in \mathbb{N}$, ${\displaystyle {(1-\mathit{\beta })}^{-1}\le {(1-\mathit{\beta })}^{-3}}$, we have(3.5) $$\begin{array}{cc}\hfill {\mathit{\omega }}_p(x)J_n^{[\mathit{\beta }]}\left(t^p{(t-x)}^2;x\right)& \le \frac{x}{b_n{(1-\mathit{\beta })}^3}\left\{\sum _{j=1}^{p+1}\frac{{\mathit{\lambda }}_{p+2,j,\mathit{\beta }}}{b_1^{p-j+1}{(1-\mathit{\beta })}^{p+j}}\right.+2\sum _{j=1}^p\frac{{\mathit{\lambda }}_{p+1,j,\mathit{\beta }}}{b_1^{p-j}{(1-\mathit{\beta })}^{p+j-1}}\hfill \\ \hfill & +\left.\sum _{j=1}^{p-1}\frac{{\mathit{\lambda }}_{p,j,\mathit{\beta }}}{b_1^{p-j-1}{(1-\mathit{\beta })}^{p+j-2}}\right\}+\frac{x^2}{{(1-\mathit{\beta })}^{2p-1}}{\left(\frac{a_n}{b_n(1-\mathit{\beta })}-1\right)}^2.\hfill \\ \hfill & \le M_2(b_1,p,\mathit{\beta })\left\{x^2{\left(\frac{a_n}{b_n(1-\mathit{\beta })}-1\right)}^2+\frac{x}{b_n{(1-\mathit{\beta })}^3}\right\}.\hfill \end{array}$$(3.5)

for $x\in {\mathbb{R}}_0$, $n\in \mathbb{N}$. Using (3.5) in (3.4), we obtain (3.3) for $p\ge 2$.

Thus, the proof is completed.

Now, we approximate $J_n^{[\mathit{\beta }]}\left(f;a_n,b_n;x\right)$ using the modulus of continuity ${\mathit{\omega }}_1(f,C_p)$ and the modulus of smoothness ${\mathit{\omega }}_2(f,C_p)$ of function $f\in C_p$, $p\in {\mathbb{N}}_0$$${\mathit{\omega }}_1(f,C_p,t):=\underset{0\le h\le t}{sup}\Vert {▵}_{h}f(·){\Vert }_p,{\mathit{\omega }}_2(f,C_p,t):=\underset{0\le h\le t}{sup}{\Vert {▵}_h^{2}f(·)\Vert }_p,$$

for $t\ge 0$, where$${▵}_{h}f(x)=f(x+h)-f(x),{▵}_h^{2}f(x)=f(x)-2f(x+h)+f(x+2h).$$

Let(3.6) $$\begin{array}{c}\hfill {\mathit{\xi }}_{n,\mathit{\beta }}(x)=x^2{\left(\frac{a_n}{b_n(1-\mathit{\beta })}-1\right)}^2+\frac{x}{b_n{(1-\mathit{\beta })}^3},x\in {\mathbb{R}}_0,x\in \mathbb{N}.\end{array}$$(3.6)

Theorem 2

Suppose that $f\in C_p^2$ with a fixed $p\in {\mathbb{N}}_0$. Then there exists a positive constant $M_3(b_1,p,\mathit{\beta })$ such that(3.7) $$\begin{array}{c}\hfill {\mathit{\omega }}_p(x)|J_n^{[\mathit{\beta }]}\left(f;a_n,b_n;x\right)-f(x)|\le \Vert f^{\prime }{\Vert }_p|\frac{a_n}{b_n(1-\mathit{\beta })}-1|x+{\Vert f^{″}\Vert }_p{M}_3(b_1,p,\mathit{\beta }){\mathit{\xi }}_{n,\mathit{\beta }}(x),\end{array}$$(3.7)

for all $x\in {\mathbb{R}}_0$, $n\in \mathbb{N}$.

Proof

Notice that ${\displaystyle J_n^{[\mathit{\beta }]}\left(0;a_n,b_n;x\right)=f(0),n\in \mathbb{N}}$, which implies (3.7(2.1) $$\begin{array}{c}\hfill \sum _{k=0}^{\infty }{\mathit{\omega }}_{\mathit{\beta }}(\mathit{\alpha },k)=1.\end{array}$$(2.1) ) for $x=0$.

Let $x>0$ and let $J_n^{[\mathit{\beta }]}(f;x)=J_n^{[\mathit{\beta }]}\left(f;a_n,b_n;x\right)$. For $f\in C_p^2$ and $t\in {\mathbb{R}}_0$,(3.8) $$\begin{array}{c}\hfill f(t)=f(x)+f^{\prime }(x)(t-x)+{\int }_x^t(t-u)f^{″}(u)du.\end{array}$$(3.8)

Applying $J_n^{[\mathit{\beta }]}(f;x)$ on both sides, we obtain$$\begin{array}{c}\hfill J_n^{[\mathit{\beta }]}(f(t);x)=f(x)+f^{\prime }(x)J_n^{[\mathit{\beta }]}((t-x);x)+J_n^{[\mathit{\beta }]}\left({\int }_x^t(t-u)f^{″}(u)du;x\right).\end{array}$$

Notice that$$\left|{\int }_x^t(t-u)f^{″}(u)du\right|\le {\Vert f^{″}\Vert }_p\left(\frac{1}{{\mathit{\omega }}_p(t)}+\frac{1}{{\mathit{\omega }}_p(x)}\right){(t-x)}^2.$$

Now, using above inequality, we have$$\begin{array}{cc}\hfill {\mathit{\omega }}_p(x)|J_n^{[\mathit{\beta }]}(f(t);x)-f(x)|& \le \Vert f^{\prime }{\Vert }_p{J}_n^{[\mathit{\beta }]}((t-x);x)\hfill \\ \hfill & +\Vert f^{″}{\Vert }_p{\mathit{\omega }}_p(x)J_n^{[\mathit{\beta }]}\left(\left(\frac{1}{{\mathit{\omega }}_p(t)}+\frac{1}{{\mathit{\omega }}_p(x)}\right){(t-x)}^2;x\right)\hfill \\ \hfill & \le \Vert f^{\prime }{\Vert }_p{J}_n^{[\mathit{\beta }]}((t-x);x)\hfill \\ \hfill & +\Vert f^{″}{\Vert }_p\left({\mathit{\omega }}_p(x)J_n^{[\mathit{\beta }]}\left(\frac{{(t-x)}^2}{{\mathit{\omega }}_p(t)};x\right)+J_n^{[\mathit{\beta }]}\left({(t-x)}^2;x\right)\right).\hfill \end{array}$$

Now, using (3.3) and (3.6), we get$$\begin{array}{c}\hfill {\mathit{\omega }}_p(x)|J_n^{[\mathit{\beta }]}(f(t);x)-f(x)|\le \Vert f^{\prime }{\Vert }_p|\frac{a_n}{b_n(1-\mathit{\beta })}-1|x+{\Vert f^{″}\Vert }_p{\mathit{\xi }}_{n,\mathit{\beta }}(x)M_3(b_1,n,\mathit{\beta }).\end{array}$$

Thus, the proof is completed.

Corollary 1

Let $\mathit{\rho }(x)={(1+x^2)}^{-1}$, $x\in {\mathbb{R}}_0$. Suppose that $f\in C_p^2$ with a fixed $p=2$. Then there exists a positive constant $M_4(b_1,p,\mathit{\beta })$ such that(3.9) $$\begin{array}{cc}\hfill \Vert [J_n^{[\mathit{\beta }]}(f;a_n,b_n;x)-f(x)]{\mathit{\rho }\Vert }_2& \le \left(1-\frac{a_n}{b_n(1-\mathit{\beta })}\right){\Vert f^{\prime }\Vert }_2\hfill \\ \hfill & +M_4(b_1,p,\mathit{\beta }){\Vert f^{″}\Vert }_2{b}_n^{-1}{(1-\mathit{\beta })}^{-3},n\in \mathbb{N}\hfill \end{array}$$(3.9)

Theorem 3

Suppose that $f\in C_p$ with a fixed $p\in {\mathbb{N}}_0$. Then there exists a positive constant $M_5(b_1,p,\mathit{\beta })$ such that$$\begin{array}{cc}\hfill {\mathit{\omega }}_p|J_n^{[\mathit{\beta }]}(f;a_n,b_n;x)-f(x)|& \le |\frac{a_n}{b_n(1-\mathit{\beta })}-1|x{\left({\mathit{\xi }}_{n,\mathit{\beta }}(x)\right)}^{-1/2}{\mathit{\omega }}_1\left(f;C_p;\sqrt{{\mathit{\xi }}_{n,\mathit{\beta }}(x)}\right)\hfill \\ \hfill & +M_5(b_1,p,\mathit{\beta }){\mathit{\omega }}_2\left(f;C_p;\sqrt{{\mathit{\xi }}_{n,\mathit{\beta }}(x)}\right),\hfill \end{array}$$

for all $x>0$ and $n\in \mathbb{N}$, where ${\mathit{\xi }}_{n,\mathit{\beta }}(·)$ is defined in (3.6). For $x=0$, it follows that $J_n^{[\mathit{\beta }]}(f;a_n,b_n;0)=f(0)$.

Proof: We shall apply the Steklov function $f_h$ for $f\in C_p$:$$f_h(x)=\frac{4}{h^2}{\int }_0^{h/2}{\int }_0^{h/2}\left[f(x+s+t)-f(x+2(s+t))\right]dsdt,$$$x\in {\mathbb{R}}_0$, $h>0$, for which we have$$\begin{array}{cc}\hfill f_h^{\prime }(x)& =\frac{1}{h^2}{\int }_0^{h/2}\left[8{▵}_{h/2}f(x+s)-2{▵}_{h}f(x+2s)\right]ds,\hfill \\ \hfill f_h^{″}(x)& =\frac{1}{h^2}\left[8{▵}_{h/2}^{2}f(x)-{▵}_h^{2}f(x)\right].\hfill \end{array}$$

Hence, for $h>0$, we have(3.10) $$\begin{array}{cc}\hfill \Vert f_h{-f\Vert }_p& \le {\mathit{\omega }}_2(f,C_p;h),\hfill \end{array}$$(3.10) (3.11) $$\begin{array}{cc}\hfill \Vert f_h^{\prime }{\Vert }_p& \le 5h^{-1}{\mathit{\omega }}_1(f,C_p;h)\frac{{\mathit{\omega }}_p(x)}{{\mathit{\omega }}_p(x+h)},\hfill \end{array}$$(3.11) (3.12) $$\begin{array}{cc}\hfill \Vert f_h^{″}{\Vert }_p& \le 9h^{-2}{\mathit{\omega }}_2(f,C_p;h),\hfill \end{array}$$(3.12)

which show that $f_h\in C_p^2$ if $f\in C_p$. By denoting $J_n^{[\mathit{\beta }]}(f;a_n,b_n;x)$ by $J_n^{[\mathit{\beta }]}(f;x)$ we can write$$\begin{array}{cc}\hfill {\mathit{\omega }}_p(x)|J_n^{[\mathit{\beta }]}(f;x)-f(x)|& \le {\mathit{\omega }}_p(x)\left\{|J_n^{[\mathit{\beta }]}(f-f_h;x)|+|J_n^{[\mathit{\beta }]}(f_h;x)-f_h(x)|\right.\hfill \\ \hfill & \left.+|f_h(x)-f(x)|\right\}:=A_1+A_2+A_3,\hfill \end{array}$$

for $x>0$, $h>0$ and $n\in \mathbb{N}$. By (3.2) and (3.9), we have$$\begin{array}{cc}\hfill A_1& \le M_1(b_1,p,\mathit{\beta }){\Vert f-f_h\Vert }_p\le M_1(b_1,p,\mathit{\beta }){\mathit{\omega }}_2(f,C_p;h),\hfill \\ \hfill A_3& \le {\mathit{\omega }}_2(f,C_p;h).\hfill \end{array}$$

Applying Theorem 2, inequalities (3.10) and (3.11), we get$$\begin{array}{cc}\hfill A_2& \le \Vert f^{\prime }{\Vert }_p|\frac{a_n}{b_n(1-\mathit{\beta })}-1|x+{\Vert f^{″}\Vert }_p{M}_3(b_1,p,\mathit{\beta }){\mathit{\xi }}_{n,\mathit{\beta }}(x)\hfill \\ \hfill & \le {\mathit{\omega }}_1(f,C_p;h)\frac{{\mathit{\omega }}_p(x)}{{\mathit{\omega }}_p(x+h)}\frac{5x}{h}|\frac{a_n}{b_n(1-\mathit{\beta })}-1|+\frac{9}{h^2}{\mathit{\omega }}_2(f,C_p;h)M_3(b_1,p,\mathit{\beta }){\mathit{\xi }}_{n,\mathit{\beta }}(x)\hfill \end{array}$$

Combining these and setting $h=\sqrt{{\mathit{\xi }}_{n,\mathit{\beta }}(x)}$, for fixed $x>0$ and $n\in \mathbb{N}$, we obtain the desired result.

Theorem 4

Let $f\in C_p$, $p\in {\mathbb{N}}_0$, and let $\mathit{\rho }(x)={(1+x^2)}^{-1}$ for $x\in {\mathbb{R}}_0$. Then there exists a positive constant $M_6(b_1,p,\mathit{\beta })$ such that$$\begin{array}{cc}\hfill \Vert \left[J_n^{[\mathit{\beta }]}(f;a_n,b_n;x)-f\right]\mathit{\rho }{\Vert }_p& \le \left(1-\frac{a_n}{b_n(1-\mathit{\beta })}\right)\sqrt{b_n}{\mathit{\omega }}_1\left(f,C_p;1/\sqrt{b_n{(1-\mathit{\beta })}^3}\right)\hfill \\ \hfill & +M_6(b_1,p,\mathit{\beta }){\mathit{\omega }}_2\left(f,C_p;1/\sqrt{b_n{(1-\mathit{\beta })}^3}\right),n\in \mathbb{N}.\hfill \end{array}$$

From Theorems 3 and 4, we derive the following corollary:

Corollary 2

Let $f\in C_p$, $p\in {\mathbb{N}}_0$, ${\mathit{\beta }}_n\to 0$ as $n\to \infty$. Then for $J_n^{[{\mathit{\beta }}_n]}$ defined by (1.5), we have(3.13) $$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{J}_n^{[{\mathit{\beta }}_n]}(f;a_n,b_n;x)=f(x),x\in {\mathbb{R}}_0.\end{array}$$(3.13)

Furthermore, the convergence of (3.12) is uniformly on every interval $[x_1,x_2]$, where $x_2>x_1\ge 0$.

Remark 1

The error of approximation of a function $f\in C_p,p\in {\mathbb{N}}_0$ by $J_n^{\left[\mathit{\beta }\right]}\left(f;a_n,b_n;.\right)$ where $a_n=n^r+\frac{1}{n}$ and $b_n=n^r,r>1$ is smaller than by the operators ().

Additional information

Funding

The authors received no direct funding for this research.

Notes on contributors

Prashantkumar Patel

Prashantkumar Patel is an assistant professor at SXCA and doing the PhD in Mathematics from NIT, Surat under VNM. His area of scientific interest includes approximation theory with positive linear operators and q-calculus which is proved by his research articles.

Vishnu Narayan Mishra

Vishnu Narayan Mishra received the PhD in Mathematics from IIT, Roorkee. His research interests are in the areas of pure and applied mathematics. He has published more than 90 research articles in reputed international journals of mathematical and engineering sciences. He is a referee and an editor of several international journals in frame of Mathematics. He guided many postgraduate and PhD students. Citations of his research contributions can be found in many books and monographs, PhD thesis and scientific journal articles.

Mediha Örkcü

Mediha Örkcü received the PhD in Mathematics from Gazi University Institute of Science Department of Mathematics during 2007–2011. Her research interest is Approximation theory. She has published many research articles in reputed international journals of Mathematics.