On Convergence of Chlodovsky Type Durrmeyer Polynomials in Variation Seminorm

Özlem Öksüzer Yılık\(^{\ast }\), Harun Karsli\(^{\bullet }\) Fatma Tasdelen\(^{\ast }\)

August 18, 2017. Accepted: February 22, 2018. Published online: August 6, 2018.

\(^{\ast }\)Department of Mathematics, Faculty of Science, University of Ankara, Ankara, Turkey, e-mail: o.oksuzer@hotmail.com.tr; tasdelen@science.ankara.edu.tr.

\(^{\bullet }\)Abant Izzet Baysal University, Faculty of Science and Arts, Department of Mathematics, Bolu, Turkey, e-mail: karsli_h@ibu.edu.tr.

This paper deals with the variation detracting property and rate of approximation of the Chlodovsky type Durrmeyer polynomials in the space of functions of bounded variation with respect to the variation seminorm.

MSC. 41A25, 41A35, 41A36.

Keywords. Chlodovsky polynomials, Durrmeyer polynomials, convergence in variation seminorm, rate of convergence, absolutely continuous functions.

1 Introduction

Let \(X_{loc}[0,\infty )\) be the class of all complex-valued functions locally bounded on \([0,\infty )\). For \(x\in X_{loc}[0,\infty )\), the Chlodovsky polynomials \(C_{n}f\) are defined as:

\begin{equation} \left( C_{n}f\right) \left( x\right) =\sum \limits _{k=0}^{n}f\left( \tfrac {k}{n}b_{n}\right) \tbinom {n}{k}\left( \tfrac {x}{b_{n}}\right) ^{k}\left( 1-\tfrac {x}{b_{n}}\right) ^{n-k},\text{ }(0\leq x\leq b_{n}) \label{chl} \end{equation}
1

where \(n\in \mathbb {N} \) and \(\left( b_{n}\right) \) is an increasing sequence of positive numbers satisfying \(\underset {n\rightarrow \infty }{\lim }b_{n}=\infty \) and \(\underset {n\rightarrow \infty }{\lim }\frac{b_{n}}{n}=0.\)

These polynomials were introduced by I. Chlodovsky [ 1 ] in 1937 in generalization of the Bernstein polynomials, the case \(b_{n}=1,\) \(n\in \mathbb {N} \), which approximate the function \(f\) on the interval \(\left[ 0,1\right] \). Some other generalizations of the Bernstein polynomials defined on unbounded sets can be found in [ 2 ] , [ 3 ] . Works on Chlodovsky polynomials are fewer, since they are defined on an unbounded interval \([0,\infty ).\)

This generalizes Chlodovsky polynomials by incorporating Durrmeyer operators [ 4 ] , hence the name Chlodovsky-Durrmeyer operators

\begin{equation} \left( D_{n}f\right) \left( x\right) =\tfrac {n+1}{b_{n}}\sum \limits _{k=0}^{n}p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{0}^{b_{n}}f\left( t\right) p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt,\text{ \ }0\leq x\leq b_{n} \label{cd} \end{equation}
2

where \(\left( b_{n}\right) \) is a positive increasing sequence with the properties \(\underset {n\rightarrow \infty }{\lim }b_{n}=\infty \), \(\underset {n\rightarrow \infty }{\lim }\frac{b_{n}}{n}=0\) and \(\ p_{n,k}\left( x\right) :=\binom {n}{k}x^{k}\left( 1-x\right) ^{n-k}\) is the Bernstein basis. We may also mention that some articles related to Chlodovsky-Durrmeyer operators and their different generalizations are given in [ 5 ] – [ 6 ] .

The main motivation for this paper is to study the variation detracting property and rate of approximation of the Chlodovsky type Durrmeyer polynomials in the space of functions of bounded variation with respect to the variation seminorm. The first research devoted to the variation detracting property and the convergence in variation of a sequence of linear positive operators was due to Lorentz [ 7 ] . Later in [ 8 ] , authors have introduced, developed in details and studied the deep interconnections between variation detracting property and the convergence in variation for Bernstein-type polynomials and singular convolution integrals. After this fundamental study, the convergence in variation seminorm has become a new research field in the theory of approximation. For further reading on different operators, we refer to readers to [ 9 ] – [ 15 ] .

2 Notation and Auxiliary Results

For the notation; let \(I\subset \mathbb {R} \) be a bounded or unbounded interval. We denote by \(V_{\left[ I\right] }\left[ f\right] \) the total Jordan variation of the function \(f:I\rightarrow \mathbb {R} \). We deal with the class \(BV\left( I\right) \) of all the functions of bounded variation on \(I\subset \mathbb {R} \), endowed with norm \(\left\Vert .\right\Vert _{BV\left( I\right) }\), where

\begin{equation*} \left\Vert f\right\Vert _{BV\left( I\right) }:=V_{\left[ I\right] }\left[ f\right] +\left\vert f\left( a\right) \right\vert ,\text{ \ }f\in BV(I), \end{equation*}

\(a\ \)being any fixed point belonging to the interval \(I.\) If we remove the term \(\left\vert f\left( a\right) \right\vert ,\) \(V_{\left[ I\right] }\left[ f\right] \) turns into a seminorm, say \(\left\vert .\right\vert _{BV\left( I\right) }\) on the same space. So we shall say, \(TV(I)\) of all the functions of bounded variation on \(I\subset \mathbb {R} \), endowed with seminorm

\begin{equation*} \left\Vert f\right\Vert _{BV\left( I\right) }:=V_{\left[ I\right] }\left[ f\right] . \end{equation*}

Some interesting properties of the space \(TV(I)\) are presented in [ 8 ] .

In order to obtain a convergence result in the variation seminorm, it is necessary and important to state the variation detracting property. Let \(L\) be a linear operator acting on a given space \(S\) of real-valued functions defined on \(I\) such that \(BV\left( I\right) \subset S.\) The operator \(L\) possesses the variation detracting property if

\begin{equation*} V_{\left[ I\right] }\left[ Lf\right] \leq V_{\left[ I\right] }\left[ f\right] \text{ , }f\in BV(I)\text{,} \end{equation*}

holds, i.e. positive linear operators from the space of functions of bounded variation into itself do not increase the total variation of functions.

\(AC\left( I\right) \) stands for the space of all absolutely continuous real-valued functions defined on \(I\) is a closed subspace of \(TV\left( I\right) \) with respect to the convergence induced by the seminorm \(\left\Vert f\right\Vert _{TV\left( I\right) }\). Moreover, if \(\underset {n\rightarrow \infty }{\lim }V_{I}\left[ g_{n}-f\right] =0\) for a sequence \(\left( g_{n}\right) _{n\geq 1}\), \(g_{n}\in AC\left( I\right) \), \(n\in \mathbb {N} ,\) then also \(f\in AC\left[ 0,1\right] \) and

\begin{equation*} V_{I}\left[ g_{n}-f\right] =\int \limits _{I}\left\vert g_{n}^{\prime }\left( t\right) -f^{\prime }\left( t\right) \right\vert dt=\left\Vert g_{n}^{\prime }-f^{\prime }\right\Vert , \end{equation*}

where

\begin{equation*} \left\Vert f\right\Vert :=\left\Vert f\right\Vert _{L_{1}\left( I\right) }. \end{equation*}

So, convergence in variation of \(\left( g_{n}\right) _{n\geq 1}\subset AC\left( I\right) \) to \(f,\) represents the convergence of the derivatives \(\left( g_{n}^{\prime }\right) _{n\geq 1}\) to \(f^{\prime }\) in the norm \(L_{1}\left( I\right) \), the Banach space of all real-valued Lebesgue integrable functions defined on \(I.\)

Let us define the sum moments as in [ 11 ] :

\begin{equation} T_{n,m}\left( x\right) =\sum \limits _{k=0}^{n}\left[ kb_{n}-nx\right] ^{m}p_{n,k}\left( \tfrac {x}{b_{n}}\right) \label{2} \end{equation}
3

where \(m\in \mathbb {N} _{0}\) (the set of non-negative integers). Then there hold the following identities (see, e.g., [ 11 ] )

\begin{equation} T_{n,m}\left( x\right) =\left\{ \begin{tabular}{ll} $1$ & $m=0$ \\ $0$ & $m=1$ \\ $nx\left( b_{n}-x\right) $ & $m=2$ \\ $nx\left( b_{n}-x\right) \left( b_{n}-2x\right) $ & $m=3$ \\ $nx\left( b_{n}-x\right) \left( b_{n}^{2}+3\left( n-2\right) xb_{n}-3\left( n-2\right) x^{2}\right) $ & $m=4.$ \end{tabular}\right. \label{mom} \end{equation}
4

Let us define for the central moments of order \(m\in \mathbb {N} _{0},\)

\begin{equation*} T_{n,m}^{\ast }\left( x\right) =\sum \limits _{k=0}^{n}\left( \tfrac {kb_{n}}{n}-x\right) ^{m}p_{n,k}\left( \tfrac {x}{b_{n}}\right) \end{equation*}

and for any fixed \(x\in \lbrack 0,\infty )\)

\begin{equation} \left\vert T_{n,m}^{\ast }\left( x\right) \right\vert \leq A_{m}(x)\tfrac {x\left( b_{n}-x\right) }{b_{n}}\left( \tfrac {b_{n}}{n}\right) ^{\left[ \left( m+1\right) /2\right] }\text{ \ }\left( n\in \mathbb {N} ,\text{ }n>b_{n}\right) , \label{5} \end{equation}
5

where \(A_{m}(x)\) denotes a polynomial in \(x,\ \)of degree \(\left[ m/2\right] -1\), with non-negative coefficients independent of \(n\), and \(\left[ a\right] \) denotes the integral part of \(a\). For the proof see Butzer-Karsli [ 16 ] .

Since

\begin{equation*} \tfrac {d}{dx}p_{n,k}\left( \tfrac {x}{b_{n}}\right) =\tfrac {\left( kb_{n}-nx\right) }{x(b_{n}-x)}p_{n,k}\left( \tfrac {x}{b_{n}}\right) , \end{equation*}

we can write the following representations for the first derivative of \(\left( D_{n}f\right) \left( x\right) \);

\begin{equation} \left( D_{n}f\right) ^{\prime }\left( x\right) =\tfrac {\left( n+1\right) }{b_{n}x(b_{n}-x)}\sum \limits _{k=0}^{n}\left( kb_{n}-nx\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{0}^{b_{n}}f\left( t\right) p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt, \label{a} \end{equation}
6

and

\begin{equation} \left( D_{n}f\right) ^{\prime }\left( x\right) =\tfrac {n}{b_{n}}\sum \limits _{k=0}^{n-1}p_{n-1,k}\left( \tfrac {x}{b_{n}}\right) \tfrac {n+1}{b_{n}}\int \limits _{0}^{b_{n}}f\left( t\right) \left[ p_{n,k+1}\left( \tfrac {t}{b_{n}}\right) -p_{n,k}\left( \tfrac {t}{b_{n}}\right) \right] dt. \label{b} \end{equation}
7

3 Variation Detracting Property of Chlodovsky-Durrmeyer Operators

In this section, we prove the variation detracting properties of the Chlodovsky-Durrmeyer Operators.

Theorem 1

If \(f\in TV\left[ 0,b_{n}\right] \), then

\begin{equation} V_{\left[ 0,b_{n}\right] }\left[ D_{n}f\right] \leq V_{\left[ 0,b_{n}\right] }\left[ f\right] \label{14} \end{equation}
8

and

\begin{equation} \left\Vert D_{n}f\right\Vert _{BV\left[ 0,b_{n}\right] }\leq \left\Vert f\right\Vert _{BV\left[ 0,b_{n}\right] } \label{15} \end{equation}
9

hold true.

Proof â–¼
For convenience we write the Chlodovsky-Durrmeyer operators as:

\begin{equation*} \left( D_{n}f\right) \left( x\right) =\sum \limits _{k=0}^{n}p_{n,k}\left( \tfrac {x}{b_{n}}\right) F_{k,n} \end{equation*}

where

\begin{equation*} F_{k,n}:=\tfrac {n+1}{b_{n}}\int \limits _{0}^{b_{n}}f\left( t\right) p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt. \end{equation*}

As in (7), differentiating (2) and putting \(\Delta F_{k,n}=F_{k+1,n}-F_{k,n}\)

\begin{eqnarray} \left( D_{n}f\right) ^{\prime }\left( x\right) & =& \sum \limits _{k=0}^{n}p_{n,k}^{\prime }\left( \tfrac {x}{b_{n}}\right) F_{k,n}=\tfrac {1}{b_{n}}\sum \limits _{k=1}^{n}\tbinom {n}{k}k\left( \tfrac {x}{b_{n}}\right) ^{k-1}\left( 1-\tfrac {x}{b_{n}}\right) ^{n-k}F_{k,n} \notag \\ & & -\tfrac {1}{b_{n}}\sum \limits _{k=0}^{n-1}\tbinom {n}{k}\left( \tfrac {x}{b_{n}}\right) ^{k}\left( n-k\right) \left( 1-\tfrac {x}{b_{n}}\right) ^{n-k-1}F_{k,n} \notag \\ & =& \tfrac {n}{b_{n}}\sum \limits _{k=0}^{n-1}p_{n-1,k}\left( \tfrac {x}{b_{n}}\right) F_{k+1,n}-\tfrac {n}{b_{n}}\sum \limits _{k=0}^{n-1}p_{n-1,k}\left( \tfrac {x}{b_{n}}\right) F_{k,n} \notag \\ & =& \tfrac {n}{b_{n}}\sum \limits _{k=0}^{n-1}p_{n-1,k}\left( \tfrac {x}{b_{n}}\right) \left[ F_{k+1,n}-F_{k,n}\right] \notag \\ & =& \tfrac {n}{b_{n}}\sum \limits _{k=0}^{n-1}p_{n-1,k}\left( \tfrac {x}{b_{n}}\right) \Delta F_{k,n}. \label{3} \end{eqnarray}

Considering the representation (10) of \(\left( D_{n}f\right) ^{\prime }\), one has

\begin{eqnarray*} \left\Vert D_{n}f\right\Vert _{TV\left[ 0,b_{n}\right] } & =& V_{\left[ 0,b_{n}\right] }\left[ D_{n}f\right] =\int \limits _{0}^{b_{n}}\left\vert \left( D_{n}f\right) ^{\prime }\left( x\right) \right\vert dx \\ & \leq & \tfrac {n}{b_{n}}\sum \limits _{k=0}^{n-1}\left\vert \Delta F_{k,n}\right\vert \int \limits _{0}^{b_{n}}p_{n-1,k}\left( \tfrac {x}{b_{n}}\right) dx. \end{eqnarray*}

Since \(\tfrac {n}{b_{n}}\displaystyle \int _{0}^{b_{n}}p_{n-1,k}\left( \tfrac {x}{b_{n}}\right) dx=1\), we get

\begin{equation} \left\Vert D_{n}f\right\Vert _{TV\left[ 0,b_{n}\right] }\leq \sum \limits _{k=0}^{n-1}\left\vert \Delta F_{k,n}\right\vert . \label{*} \end{equation}
11

Now,

\begin{eqnarray*} \Delta F_{k,n} & =& \tfrac {n+1}{b_{n}}\int \limits _{0}^{b_{n}}f\left( t\right) \left[ p_{n,k+1}\left( \tfrac {t}{b_{n}}\right) -p_{n,k}\left( \tfrac {t}{b_{n}}\right) \right] dt \\ & =& \tfrac {n+1}{b_{n}}\int \limits _{0}^{b_{n}}f\left( t\right) \Delta p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt. \end{eqnarray*}

Since \(\Delta p_{n,k}=-\frac{b_{n}}{n+1}p_{n+1,k+1}^{\prime }\),

\begin{equation*} p_{n,k}^{\prime }\left( \tfrac {t}{b_{n}}\right) =-\tfrac {n}{b_{n}}\Delta p_{n-1,k-1}\left( \tfrac {t}{b_{n}}\right) , \end{equation*}

and so we get

\begin{equation} \left\vert \Delta F_{k,n}\right\vert =\left\vert \tfrac {n+1}{b_{n}}\int \limits _{0}^{b_{n}}f\left( t\right) \left[ -\tfrac {b_{n}}{n+1}p_{n+1,k+1}^{\prime }\left( \tfrac {t}{b_{n}}\right) \right] dt\right\vert . \label{**} \end{equation}
12

From (11) and (12), we obtain

\begin{eqnarray*} V_{\left[ 0,1\right] }\left[ D_{n}f\right] & =& \sum \limits _{k=0}^{n-1}\left\vert \Delta F_{k,n}\right\vert =\sum \limits _{k=0}^{n-1}\left\vert -\int \limits _{0}^{b_{n}}f\left( t\right) p_{n+1,k+1}^{\prime }\left( \tfrac {t}{b_{n}}\right) dt\right\vert \\ & =& \sum \limits _{k=0}^{n-1}\left\vert \int \limits _{0}^{b_{n}}f^{\prime }\left( t\right) p_{n+1,k+1}\left( \tfrac {t}{b_{n}}\right) dt\right\vert \\ & \leq & \sum \limits _{k=0}^{n-1}\int \limits _{0}^{b_{n}}p_{n+1,k+1}\left( \tfrac {t}{b_{n}}\right) \left\vert f^{\prime }\left( t\right) \right\vert dt \\ & =& \int \limits _{0}^{b_{n}}\sum \limits _{k=0}^{n-1}\tbinom {n+1}{k+1}\left( \tfrac {t}{b_{n}}\right) ^{k+1}\left( 1-\tfrac {t}{b_{n}}\right) ^{n-k}\left\vert f^{\prime }\left( t\right) \right\vert dt \\ & =& \int \limits _{0}^{b_{n}}\sum \limits _{k=1}^{n}\tbinom {n+1}{k}\left( \tfrac {t}{b_{n}}\right) ^{k}\left( 1-\tfrac {t}{b_{n}}\right) ^{n+1-k}\left\vert f^{\prime }\left( t\right) \right\vert dt \\ & \leq & \int \limits _{0}^{b_{n}}\sum \limits _{k=0}^{n+1}\tbinom {n+1}{k}\left( \tfrac {t}{b_{n}}\right) ^{k}\left( 1-\tfrac {t}{b_{n}}\right) ^{n+1-k}\left\vert f^{\prime }\left( t\right) \right\vert dt \end{eqnarray*}
\begin{eqnarray*} & =& \int \limits _{0}^{b_{n}}\left( \tfrac {t}{b_{n}}+1-\tfrac {t}{b_{n}}\right) ^{n+1}\left\vert f^{\prime }\left( t\right) \right\vert dt=\int \limits _{0}^{b_{n}}\left\vert f^{\prime }\left( t\right) \right\vert dt \\ & \leq & V_{\left[ 0,b_{n}\right] }\left[ f\right] . \end{eqnarray*}

The desired estimate (8) is now obvious.

Since

\begin{equation*} \left( D_{n}f\right) \left( 0\right) =\tfrac {n+1}{b_{n}}\int \limits _{0}^{b_{n}}\left( 1-\tfrac {t}{b_{n}}\right) ^{n}f\left( t\right) dt \end{equation*}

and

\begin{equation*} \left\Vert f\right\Vert _{BV\left[ I\right] }:=V_{\left[ I\right] }\left[ f\right] +\left\vert f\left( 0\right) \right\vert , \end{equation*}

relation (9) is a result of (8). Indeed,

\begin{eqnarray*} \left\Vert D_{n}f\right\Vert _{BV\left[ 0,b_{n}\right] } & =& V_{\left[ 0,b_{n}\right] }\left[ D_{n}f\right] +\left\vert \left( D_{n}f\right) \left( 0\right) \right\vert \\ & \leq & V_{\left[ 0,b_{n}\right] }\left[ f\right] +\left\vert \tfrac {n+1}{b_{n}}\int \limits _{0}^{b_{n}}\left( 1-\tfrac {t}{b_{n}}\right) ^{n}f\left( t\right) dt\right\vert . \end{eqnarray*}

Since \(f\in TV\left[ 0,b_{n}\right] \) and

\[ \left\vert \tfrac {n+1}{b_{n}}\int _{0}^{b_{n}}\left( 1-\tfrac {t}{b_{n}}\right) ^{n}f\left( t\right) dt\right\vert =\left\vert f\left( 0\right) \right\vert \leq \left\vert f\left( a\right) \right\vert \]

where \(a\) is any fixed point of \(\left[ 0,b_{n}\right] \), we get

\begin{equation*} \left\Vert D_{n}f\right\Vert _{BV\left[ 0,b_{n}\right] }\leq \left\Vert f\right\Vert _{BV\left[ 0,b_{n}\right] }. \end{equation*}

Thus, the proof of the theorem is complete.

Proof â–¼

4 Rate of Approximation in \(TV\)-norm

This section deals with the rates of approximation \(D_{n}g\) to \(g\) in the variation seminorm.

In order to obtain a convergence result in variation seminorm, we assume that \(\underset {n\rightarrow \infty }{\lim }b_{n}=\infty \) and \(\underset {n\rightarrow \infty }{\lim }\frac{b_{n}^{3}}{n}=0\)

Theorem 2

Let \(g^{\prime \prime }\in AC\left[ 0,b_{n}\right] \), then

\begin{equation*} V_{\left[ 0,b_{n}\right] }\left[ D_{n}g-g\right] \leq \tfrac {B}{\delta ^{2}}\tfrac {b_{n}^{3}}{n}\left\{ V_{\left[ 0,b_{n}\right] }\left[ g\right] +V_{\left[ 0,b_{n}\right] }\left[ g^{\prime \prime }\right] \right\} \end{equation*}

holds true, where \(\delta \) is sufficiently small positive real constant and a constant \(B{\gt}1\).

Proof â–¼
By Taylor’s formula with integral remainder term, one has

\begin{align} g\left( \tfrac {k}{n}b_{n}\right) = & \label{4} g\left( x\right) +\left( \tfrac {k}{n}b_{n}-x\right) g^{\prime }\left( x\right) \\ & +\left( \tfrac {k}{n}b_{n}-x\right) ^{2}\tfrac {g^{\prime \prime }\left( x\right) }{2}+\tfrac {1}{2}\int \limits _{x}^{\frac{k}{n}b_{n}}\left( \tfrac {k}{n}b_{n}-v\right) ^{2}g^{\prime \prime \prime }\left( v\right) dv. \nonumber \end{align}

From (13) we obtain

\begin{align*} & \left( D_{n}g\right) ^{\prime }\left( x\right) = \\ =& \tfrac {n+1}{b_{n}x(b_{n}-x)}\sum \limits _{k=0}^{n}\left( kb_{n}-nx\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{0}^{b_{n}}g\left( \tfrac {k}{n}b_{n}\right) p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt \\ =& \tfrac {\left( n+1\right) }{b_{n}x(b_{n}-x)}g\left( x\right) \sum \limits _{k=0}^{n}\left( kb_{n}-nx\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{0}^{b_{n}}p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt \\ & +\tfrac {\left( n+1\right) }{b_{n}x(b_{n}-x)}g^{\prime }\left( x\right) \sum \limits _{k=0}^{n}\left( kb_{n}-nx\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{0}^{b_{n}}\left( \tfrac {k}{n}b_{n}-x\right) p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt \\ & +\tfrac {\left( n+1\right) }{2b_{n}x(b_{n}-x)}g^{\prime \prime }\left( x\right) \sum \limits _{k=0}^{n}\left( kb_{n}-nx\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{0}^{b_{n}}\left( \tfrac {k}{n}b_{n}-x\right) ^{2}p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt \\ & +\left( R_{n}g\right) \left( x\right) . \end{align*}

where

\begin{align} & \left( R_{n}g\right) \left( x\right) = \label{6} \\ & =\tfrac {n+1}{2b_{n}x(b_{n}-x)}\sum \limits _{k=0}^{n}\left( kb_{n}\! -\! nx\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right)\! \int \limits _{0}^{b_{n}}\left[ \int \limits _{x}^{\frac{k}{n}b_{n}}\left( \! \tfrac {k}{n}b_{n}\! -\! v\right) ^{2}\! g^{\prime \prime \prime }\left( v\right) dv\right] p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt. \nonumber \end{align}

Calculating (3) and (4), we obtain

\begin{equation} \left( D_{n}g\right) ^{\prime }\left( x\right) =g^{\prime }\left( x\right) +\tfrac {b_{n}-2x}{2n}g^{\prime \prime }\left( x\right) +\left( R_{n}g\right) \left( x\right) . \label{11} \end{equation}
15

As in the proof of [ 15 ] , we choose \(\delta \) a sufficiently small positive real number, let’s divide \(\left( R_{n}g\right) \left( x\right) \) in to two parts as follows;

\begin{equation} \left( R_{n}g\right) \left( x\right) =\left( R_{n,1}g\right) \left( x\right) +\left( R_{n,2}g\right) \left( x\right) , \label{12} \end{equation}
16

where

\begin{align} & \left( R_{n,1}g\right) \left( x\right) =\\ \label{12.1} & =\tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \nonumber \\ & \quad \cdot \sum \limits _{\left\vert \frac{k}{n}-x\right\vert \leq \delta }\left( kb_{n}-nx\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{0}^{b_{n}}\left[ \int \limits _{x}^{\frac{k}{n}b_{n}}\left( \tfrac {k}{n}b_{n}-v\right) ^{2}g^{\prime \prime \prime }\left( v\right) dv\right] p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt\nonumber \end{align}

and

\begin{align} & \left( R_{n,2}g\right) \left( x\right)=\\ \label{12.2} & =\tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \nonumber \\ & \quad \cdot \sum \limits _{\left\vert \frac{k}{n}-x\right\vert {\gt}\delta }\left( kb_{n}-nx\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{0}^{b_{n}}\left[ \int \limits _{x}^{\frac{k}{n}b_{n}}\left( \tfrac {k}{n}b_{n}-v\right) ^{2}g^{\prime \prime \prime }\left( v\right) dv\right] p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt. \nonumber \end{align}

In order to estimate the integration domain of the double integral in the remainder term (14), we divide the summation into different sums as following;

\begin{equation*} \left( R_{n,1}g\right) \left( x\right) =A_{2,n}g+A_{5,n}g\text{ ~ and }\left( R_{n,2}g\right) \left( x\right) =A_{1,n}g+A_{3,n}g+A_{4,n}g+A_{6,n}g. \end{equation*}

Here \(A_{i,n}g\) for \(i=1,\ldots ,6,\)

\begin{align*} & A_{1,n}g =\\ & =\tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \\ & \quad \cdot \sum \limits _{\delta {\lt}x-\frac{k}{n}b_{n}\leq x}\! \! \left( kb_{n}-nx\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \! \int \limits _{0}^{\frac{k}{n}b_{n}}\! \left[ \int \limits _{x}^{\frac{k}{n}b_{n}}\! \left( \tfrac {k}{n}b_{n}-v\right) ^{2}g^{\prime \prime \prime }\left( v\right) dv\right] p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt, \\ & A_{2,n}g =\\ & =\tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \\ & \quad \cdot \sum \limits _{0\leq x-\frac{k}{n}b_{n}\leq \delta }\! \! \left( kb_{n}-nx\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{\frac{k}{n}b_{n}}^{x}\! \! \left[ \int \limits _{x}^{\frac{k}{n}b_{n}}\! \left( \tfrac {k}{n}b_{n}-v\right) ^{2}g^{\prime \prime \prime }\left( v\right) dv\right] p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt, \end{align*}
\begin{align*} & A_{3,n}g =\\ & =\tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \\ & \quad \cdot \sum \limits _{\delta {\lt}x-\frac{k}{n}b_{n}\leq x}\! \! \left( kb_{n}-nx\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{x}^{b_{n}}\left[ \int \limits _{x}^{\frac{k}{n}b_{n}}\left( \tfrac {k}{n}b_{n}-v\right) ^{2}g^{\prime \prime \prime }\left( v\right) dv\right] p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt, \end{align*}
\begin{align*} & A_{4,n}g =\\ & =\tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \\ & \quad \cdot \sum \limits _{\delta {\lt}\frac{k}{n}b_{n}-x\leq 1-x}\! \! \left( kb_{n}-nx\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{0}^{x}\! \! \left[ \int \limits _{x}^{\frac{k}{n}b_{n}}\! \! \left( \tfrac {k}{n}b_{n}-v\right) ^{2}g^{\prime \prime \prime }\left( v\right) dv\right] p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt, \\ & A_{5,n}g =\\ & =\tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \\ & \quad \cdot \sum \limits _{0{\lt}\frac{k}{n}b_{n}-x\leq \delta }\! \! \left( kb_{n}-nx\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{x}^{\frac{k}{n}b_{n}}\left[ \int \limits _{x}^{\frac{k}{n}b_{n}}\left( \tfrac {k}{n}b_{n}-v\right) ^{2}g^{\prime \prime \prime }\left( v\right) dv\right] p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt, \end{align*}

and

\begin{align*} & A_{6,n}g=\\ & =\frac{n+1}{2b_{n}x(b_{n}-x)}\cdot \\ & \quad \cdot \sum \limits _{\delta {\lt}\frac{k}{n}b_{n}-x\leq 1-x}\! \! \! \! \! \left( kb_{n}-nx\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \! \! \int \limits _{\tfrac {k}{n}b_{n}}^{b_{n}}\! \! \left[ \int \limits _{x}^{\frac{k}{n}b_{n}}\! \! \left( \tfrac {k}{n}b_{n}\! -\! v\right) ^{2}\! \! g^{\prime \prime \prime }\left( v\right) dv\right]\! \! p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt. \end{align*}

It is easy to see that, \(A_{1,n}g+A_{2,n}g=-A_{4,n}g\) and \(A_{5,n}g+A_{6,n}g=-A_{3,n}g.\) So one has

\begin{equation*} \left\vert A_{1,n}g+A_{2,n}g\right\vert =\left\vert -A_{4,n}g\right\vert \leq \left\vert A_{1,n}g\right\vert +\left\vert A_{2,n}g\right\vert \end{equation*}

and

\begin{equation*} \left\vert A_{5,n}g+A_{6,n}g\right\vert =\left\vert -A_{3,n}g\right\vert \leq \left\vert A_{5,n}g\right\vert +\left\vert A_{6,n}g\right\vert . \end{equation*}

So, we get

\begin{equation*} \left\vert \left( R_{n}g\right) \left( x\right) \right\vert \leq 2\left( \left\vert A_{1,n}g\right\vert +\left\vert A_{2,n}g\right\vert +\left\vert A_{5,n}g\right\vert +\left\vert A_{6,n}g\right\vert \right) \end{equation*}

or

\begin{equation*} \left\vert \left( R_{n}g\right) \left( x\right) \right\vert \leq 2\left( \left\vert -A_{3,n}g\right\vert +\left\vert -A_{4,n}g\right\vert \right) . \end{equation*}

Now we only estimate \(A_{i,n}g\) for \(i=1,2,5\), and \(6\) respectively. Firstly, let us estimate \(A_{1,n}g\) as follows;

\begin{align*} & \left\vert A_{1,n}g\right\vert \leq \\ & \leq \tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \\ & \quad \cdot \sum \limits _{\delta {\lt}x-\frac{k}{n}b_{n}\leq x}\left\vert kb_{n}-nx\right\vert p_{n,k}\left( \tfrac {x}{b_{n}}\right) \! \! \int \limits _{0}^{\frac{k}{n}b_{n}}\left\vert \int \limits _{x}^{\frac{k}{n}b_{n}}\! \! \left( \tfrac {k}{n}b_{n}-v\right) ^{2}\! g^{\prime \prime \prime }\left( v\right) dv\right\vert p_{n,k}\left(\tfrac {t}{b_{n}}\right)\! \! \ dt \\ & \leq \tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \\ & \quad \cdot \sum \limits _{\delta {\lt}x-\frac{k}{n}b_{n}\leq x}\left\vert kb_{n}-nx\right\vert p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{0}^{\frac{k}{n}b_{n}}\left\vert \int \limits _{x}^{\frac{k}{n}b_{n}}\left( \tfrac {k}{n}b_{n}-v\right) ^{2}\left\vert g^{\prime \prime \prime }\left( v\right) \right\vert dv\right\vert p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt \\ & =\tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \\ & \quad \cdot \sum \limits _{\delta {\lt}x-\frac{k}{n}b_{n}}\left( nx-kb_{n}\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{0}^{\frac{k}{n}b_{n}}\left[ \int \limits _{x}^{\frac{k}{n}b_{n}}\left( \tfrac {k}{n}b_{n}-v\right) ^{2}\left\vert g^{\prime \prime \prime }\left( v\right) \right\vert dv\right] p_{n,k}\left(\tfrac {t}{b_{n}}\right)\! \! \ dt \end{align*}
\begin{align*} & \leq \tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \\ & \quad \cdot \sum \limits _{\delta {\lt}x-\frac{k}{n}b_{n}}\left( nx-kb_{n}\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{0}^{\frac{k}{n}b_{n}}\left( \tfrac {k}{n}b_{n}-x\right) ^{2}\left[ \int \limits _{\frac{k}{n}b_{n}}^{x}\left\vert g^{\prime \prime \prime }\left( v\right) \right\vert dv\right] p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt \\ & \leq \tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \\ & \quad \cdot \sum \limits _{\delta {\lt}x-\frac{k}{n}b_{n}}\left( nx-kb_{n}\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) x^{2}\int \limits _{0}^{\frac{k}{n}b_{n}}\left[ \int \limits _{\frac{k}{n}b_{n}}^{x}\left\vert g^{\prime \prime \prime }\left( v\right) \right\vert dv\right] p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt\\ & \leq \tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \\ & \quad \cdot \sum \limits _{\delta {\lt}x-\frac{k}{n}b_{n}}\left( nx-kb_{n}\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) x^{2}\int \limits _{0}^{\frac{k}{n}b_{n}}\left[ \int \limits _{0}^{b_{n}}\left\vert g^{\prime \prime \prime }\left( v\right) \right\vert dv\right] p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt \\ & \leq \tfrac {\left( n+1\right) x^{2}}{2b_{n}x(b_{n}-x)}\left\Vert g^{\prime \prime \prime }\right\Vert \sum \limits _{\delta {\lt}x-\frac{k}{n}b_{n}}\left( nx-kb_{n}\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{0}^{b_{n}}p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt \\ & =\tfrac {x^{2}}{2x(b_{n}-x)}\left\Vert g^{\prime \prime \prime }\right\Vert \sum \limits _{\delta {\lt}x-\frac{k}{n}b_{n}}\left( nx-kb_{n}\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) . \end{align*}

Since

\begin{equation*} \sum \limits _{\delta {\lt}x-\frac{k}{n}b_{n}}\left( nx-kb_{n}\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) =\sum \limits _{\frac{k}{n}b_{n}-x{\lt}-\delta }\left( kb_{n}-nx\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \text{,} \end{equation*}

we get

\begin{equation} \left\vert A_{1,n}g\right\vert \leq \tfrac {x^{2}}{2x(b_{n}-x)}\left\Vert g^{\prime \prime \prime }\right\Vert \sum \limits _{\frac{k}{n}b_{n}-x<-\delta }\left( kb_{n}-nx\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) . \label{7} \end{equation}
19

Analogously, \(A_{2,n}g\) can be estimated by

\begin{align*} & \left\vert A_{2,n}g\right\vert \leq \\ & \leq \tfrac {n+1}{2b_{n}x(b_{n}-x)}\\ & \quad \cdot \sum \limits _{0\leq x-\frac{k}{n}b_{n}\leq \delta }\left\vert kb_{n}-nx\right\vert p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{\frac{k}{n}b_{n}}^{x}\left\vert \int \limits _{x}^{\frac{k}{n}b_{n}}\left( \tfrac {k}{n}b_{n}-v\right) ^{2}g^{\prime \prime \prime }\left( v\right) dv\right\vert p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt\\ & \leq \tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \\ & \quad \cdot \sum \limits _{0\leq x-\frac{k}{n}b_{n}\leq \delta }\left\vert kb_{n}-nx\right\vert p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{\frac{k}{n}b_{n}}^{x}\left\vert \int \limits _{x}^{\frac{k}{n}b_{n}}\left( \tfrac {k}{n}b_{n}-v\right) ^{2}\left\vert g^{\prime \prime \prime }\left( v\right) \right\vert dv\right\vert p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt\\ & \leq \tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \\ & \quad \cdot \sum \limits _{x-\frac{k}{n}b_{n}\leq \delta }\left( nx-kb_{n}\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{\frac{k}{n}b_{n}}^{x}\left[ \int \limits _{\frac{k}{n}b_{n}}^{x}\left( \tfrac {k}{n}b_{n}-v\right) ^{2}\left\vert g^{\prime \prime \prime }\left( v\right) \right\vert dv\right] p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt\\ & \leq \tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \\ & \quad \cdot \sum \limits _{x-\frac{k}{n}b_{n}\leq \delta }\left( nx-kb_{n}\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{\frac{k}{n}b_{n}}^{x}\left( \tfrac {k}{n}b_{n}-x\right) ^{2}\left[ \int \limits _{\frac{k}{n}b_{n}}^{x}\left\vert g^{\prime \prime \prime }\left( v\right) \right\vert dv\right] p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt \\ & \leq \tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \\ & \quad \cdot \sum \limits _{x-\frac{k}{n}b_{n}\leq \delta }\left( nx-kb_{n}\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \left( \tfrac {k}{n}b_{n}-x\right) ^{2}\int \limits _{\frac{k}{n}b_{n}}^{x}\left[ \int \limits _{\frac{k}{n}b_{n}}^{x}\left\vert g^{\prime \prime \prime }\left( v\right) \right\vert dv\right] p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt \end{align*}
\begin{align*} & \leq \tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \\ & \quad \cdot \sum \limits _{x-\frac{k}{n}b_{n}\leq \delta }\left( nx-kb_{n}\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \left( \tfrac {k}{n}b_{n}-x\right) ^{2}\int \limits _{\frac{k}{n}b_{n}}^{x}\left[ \int \limits _{0}^{b_{n}}\left\vert g^{\prime \prime \prime }\left( v\right) \right\vert dv\right] p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt\\ & \leq \tfrac {n+1}{2b_{n}x(b_{n}-x)}\left\Vert g^{\prime \prime \prime }\right\Vert \cdot \\ & \quad \cdot \sum \limits _{x-\frac{k}{n}b_{n}\leq \delta }\left( nx-kb_{n}\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \left( \tfrac {k}{n}b_{n}-x\right) ^{2}\int \limits _{\frac{k}{n}b_{n}}^{x}p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt \\ & \leq \tfrac {\left\Vert g^{\prime \prime \prime }\right\Vert }{2n^{2}x(b_{n}-x)}\cdot \\ & \quad \cdot \sum \limits _{x-\frac{k}{n}b_{n}\leq \delta }\left\vert kb_{n}-nx\right\vert p_{n,k}^{1/2}\left( \tfrac {x}{b_{n}}\right) \left( kb_{n}-nx\right) ^{2}p_{n,k}^{1/2}\left( \tfrac {x}{b_{n}}\right) . \end{align*}

In view of Hölder inequality, (4) and (5), we get

\begin{align*} & \left\vert A_{2,n}g\right\vert \leq \\ & \leq \tfrac {\left\Vert g^{\prime \prime \prime }\right\Vert }{2n^{2}x(b_{n}-x)}\left( \sum \limits _{k=0}^{n}\left( kb_{n}-nx\right) ^{2}p_{n,k}\left( \tfrac {x}{b_{n}}\right) \right) ^\frac {1}{2}\left( \sum \limits _{k=0}^{n}\left( kb_{n}-nx\right) ^{4}p_{n,k}\left( \tfrac {x}{b_{n}}\right) \right) ^\frac {1}{2} \\ & \leq \tfrac {\sqrt{b_{n}}}{n}\left\Vert g^{\prime \prime \prime }\right\Vert \end{align*}

which implies that

\begin{equation} \left\Vert A_{2,n}g\right\Vert \leq \tfrac {\sqrt{b_{n}}}{n}\left\Vert g^{\prime \prime \prime }\right\Vert . \label{8} \end{equation}
20

As to the term \(A_{5,n}g\), noting (4) and (5), we have

\begin{align*} & \left\vert A_{5,n}g\right\vert \leq \\ & \leq \tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \\ & \quad \cdot \sum \limits _{0{\lt}\frac{k}{n}b_{n}-x\leq \delta }\left\vert kb_{n}-nx\right\vert p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{x}^{\frac{k}{n}b_{n}}\left\vert \int \limits _{x}^{\frac{k}{n}b_{n}}\left( \tfrac {k}{n}b_{n}-v\right) ^{2}g^{\prime \prime \prime }\left( v\right) dv\right\vert p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt \\ & \leq \tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \\ & \quad \cdot \sum \limits _{\frac{k}{n}b_{n}-x\leq \delta }\left( kb_{n}-nx\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{x}^{\frac{k}{n}b_{n}}\left[ \int \limits _{x}^{\frac{k}{n}b_{n}}\left( \tfrac {k}{n}b_{n}-v\right) ^{2}\left\vert g^{\prime \prime \prime }\left( v\right) \right\vert dv\right] p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt \end{align*}
\begin{align*} & \leq \tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \\ & \quad \cdot \sum \limits _{\frac{k}{n}b_{n}-x\leq \delta }\left( kb_{n}-nx\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \left( \tfrac {k}{n}b_{n}-x\right) ^{2}\int \limits _{x}^{\frac{k}{n}b_{n}}\left[ \int \limits _{x}^{\frac{k}{n}b_{n}}\left\vert g^{\prime \prime \prime }\left( v\right) \right\vert dv\right] p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt \\ & \leq \tfrac {\left\Vert g^{\prime \prime \prime }\right\Vert }{2n^{2}x(b_{n}-x)}\sum \limits _{k=0}^{n}\left\vert kb_{n}-nx\right\vert ^{3}p_{n,k}\left( \tfrac {x}{b_{n}}\right) \end{align*}

As in the proof of \(A_{2,n}g\), one has

\begin{equation*} \left\vert A_{5,n}g\right\vert \leq \tfrac {\sqrt{b_{n}}}{n}\left\Vert g^{\prime \prime \prime }\right\Vert , \end{equation*}

which yields

\begin{equation} \left\Vert A_{5,n}g\right\Vert \leq \tfrac {\sqrt{b_{n}}}{n}\left\Vert g^{\prime \prime \prime }\right\Vert . \label{9} \end{equation}
21

Finally to the next term \(A_{6,n}g\), we have

\begin{align*} & \left\vert A_{6,n}g\right\vert \leq \\ & \leq \tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \\ & \quad \cdot \sum \limits _{\delta {\lt}\frac{k}{n}b_{n}-x\leq 1-x}\left\vert kb_{n}-nx\right\vert p_{n,k}\left( \tfrac {x}{b_{n}}\right)\! \! \! \int \limits _{\frac{k}{n}b_{n}}^{b_{n}}\left\vert \int \limits _{x}^{\frac{k}{n}b_{n}}\left( \tfrac {k}{n}b_{n}-v\right) ^{2}g^{\prime \prime \prime }\left( v\right) dv\right\vert p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt\notag \end{align*}
\begin{align} & \leq \tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \notag \\ & \quad \cdot \! \! \sum \limits _{\delta {\lt}\frac{k}{n}b_{n}-x\leq 1-x}\! \! \left\vert kb_{n}-nx\right\vert p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{\frac{k}{n}b_{n}}^{b_{n}}\left\vert \! \! \int \limits _{x}^{\frac{k}{n}b_{n}}\left( \tfrac {k}{n}b_{n}-v\right) ^{2}\! \! \left\vert g^{\prime \prime \prime }\left( v\right) \right\vert dv\right\vert p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt \notag \\ & \leq \tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \notag \\ & \quad \cdot \sum \limits _{\delta {\lt}\frac{k}{n}b_{n}-x}\left( kb_{n}-nx\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \int \limits _{\frac{k}{n}b_{n}}^{b_{n}}\left( \tfrac {k}{n}b_{n}-x\right) ^{2}\left[ \int \limits _{x}^{\frac{k}{n}b_{n}}\left\vert g^{\prime \prime \prime }\left( v\right) \right\vert dv\right] p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt \notag \end{align}
\begin{align} & \leq \tfrac {n+1}{2b_{n}x(b_{n}-x)}\cdot \notag \\ & \quad \cdot \sum \limits _{\delta {\lt}\frac{k}{n}b_{n}-x}\left( kb_{n}-nx\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \left( b_{n}-x\right) ^{2}\int \limits _{\frac{k}{n}b_{n}}^{b_{n}}\left[ \int \limits _{0}^{b_{n}}\left\vert g^{\prime \prime \prime }\left( v\right) \right\vert dv\right] p_{n,k}\left( \tfrac {t}{b_{n}}\right) dt \notag \\ & \leq \tfrac {\left( b_{n}-x\right) ^{2}}{2x(b_{n}-x)}\left\Vert g^{\prime \prime \prime }\right\Vert \cdot \sum \limits _{\delta {\lt}\frac{k}{n}b_{n}-x}\left( kb_{n}-nx\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) . \label{10} \end{align}

Collecting (19) and (21), we obtain

\begin{eqnarray} \left\vert A_{1,n}g\right\vert +\left\vert A_{6,n}g\right\vert & \leq & \tfrac {x^{2}+\left( b_{n}-x\right) ^{2}}{2x(b_{n}-x)}\left\Vert g^{\prime \prime \prime }\right\Vert \sum \limits _{\left\vert \frac{k}{n}b_{n}-x\right\vert {\gt}\delta }\left( kb_{n}-nx\right) p_{n,k}\left( \tfrac {x}{b_{n}}\right) \notag \\ & \leq & \tfrac {x^{2}+\left( b_{n}-x\right) ^{2}}{2x(b_{n}-x)}\left\Vert g^{\prime \prime \prime }\right\Vert n\! \! \sum \limits _{\left\vert \frac{k}{n}b_{n}-x\right\vert {\gt}\delta }\left( \tfrac {k}{n}b_{n}\! -\! x\right) \! \tfrac {\left( \frac{k}{n}b_{n}\! -x\right) ^{2}}{\delta ^{2}}p_{n,k}\left( \tfrac {x}{b_{n}}\right) \notag \\ & =& \tfrac {x^{2}+\left( b_{n}-x\right) ^{2}}{2x(b_{n}-x)}\left\Vert g^{\prime \prime \prime }\right\Vert \frac{n}{\delta ^{2}}\sum \limits _{\left\vert \frac{k}{n}b_{n}-x\right\vert {\gt}\delta }\left( \tfrac {k}{n}b_{n}-x\right) ^{3}p_{n,k}\left( \tfrac {x}{b_{n}}\right) \\ & \leq & \tfrac {b_{n}^{3}}{n\delta ^{2}}\left\Vert g^{\prime \prime \prime }\right\Vert . \label{16} \end{eqnarray}

In view of (23), one obtains

\begin{equation} \left\Vert A_{1,n}g\right\Vert +\left\Vert A_{6,n}g\right\Vert \leq \tfrac {b_{n}^{3}}{n\delta ^{2}}\left\Vert g^{\prime \prime \prime }\right\Vert . \label{17} \end{equation}
24

Thus, altogether with the results in (20), (21) and (24), we have for (16)

\begin{equation*} \left\Vert R_{n}g\right\Vert \leq 2\left( \tfrac {\sqrt{b_{n}}}{n}+\tfrac {\sqrt{b_{n}}}{n}+\tfrac {b_{n}^{3}}{n\delta ^{2}}\right) \left\Vert g^{\prime \prime \prime }\right\Vert \leq \tfrac {6b_{n}^{3}}{\delta ^{2}n}\left\Vert g^{\prime \prime \prime }\right\Vert . \end{equation*}

Finally we obtain by using (15)

\begin{equation*} \left\Vert \left( D_{n}g\right) ^{\prime }-g^{\prime }\right\Vert \leq \tfrac {b_{n}}{2n}\left\Vert g^{\prime \prime }\right\Vert +\tfrac {6b_{n}^{3}}{\delta ^{2}n}\left\Vert g^{\prime \prime \prime }\right\Vert . \end{equation*}

According to Stein’s inequality (see, e.g., [ 17 , Th. A10.1 ] ) one has

\begin{eqnarray*} \left\Vert g^{\prime \prime }\left( x\right) \right\Vert _{L_{1}\left( 0,1\right) } & \leq & C\sqrt{\left\Vert g^{\prime }\left( x\right) \right\Vert _{L_{1}\left( 0,1\right) }\left\Vert g^{\prime \prime \prime }\right\Vert _{L_{1}\left( 0,1\right) }} \\ & \leq & C\left( \left\Vert g^{\prime }\right\Vert _{L_{1}\left( 0,1\right) }+\left\Vert g^{\prime \prime \prime }\right\Vert _{L_{1}\left( 0,1\right) }\right) , \end{eqnarray*}

where \(C{\gt}1\) is a constant. So, we have

\begin{equation} \left\Vert \left( D_{n}g\right) ^{\prime }-g^{\prime }\right\Vert \leq \tfrac {B}{\delta ^{2}}\tfrac {b_{n}^{3}}{n}\left( \left\Vert g^{\prime }\right\Vert +\left\Vert g^{\prime \prime \prime }\right\Vert \right) \label{***} \end{equation}
25

where \(B{\gt}1\) is a constant. This finally establishes the theorem.

Proof â–¼

Bibliography

1

I. Chlodovsky, Sur le développement des fonctions définies dans un intervalle infini en séries de polynomes de M. S. Bernstein, Compositio Math., 4 (1937), 380–393.

2

T. Hermann, Approximation of unbounded functions on unbounded interval, Acta. Math. Hungar., 29 (1977), 393–398. \includegraphics[scale=0.1]{ext-link.png}

3

G. Bleimann, P.L. Butzer, L. Hahn, A Bernstein-type operator approximating continuous functions on the semi-axis, Indag. Math., 42 (1980), 255–262. \includegraphics[scale=0.1]{ext-link.png}

4

J.L. Durrmeyer, Une formule d’inversion de la transformée de Laplace: Applications à la théorie des moments, Thèse de 3e Cycle, Faculté des Sciences de l’Université de Paris, 1967.

5

H. Karsli, P. Pych-Taberska, On the rates of convergence of Chlodovsky–Durrmeyer operators and their Bézier variant, Georgian Math. J., 16 (2009) no. 4, 693–704.

6

H. Karsli, Order of convergence of Chlodowsky type Durrmeyer operators for functions with derivatives of bounded variation, Indian J. Pure Appl. Math., 38 (2007) no. 5, 353.

7

G.G. Lorentz, Bernstein polynomials, University of Toronto Press, Toronto (1953).

8

C. Bardaro, P.L. Butzer, R.L. Stens, G. Vinti, Convergence in variation and rates of approximation for Bernstein-type polynomials and singular convolution integrals, Analysis (Munich), 23 (2003) no. 4, 299–346. \includegraphics[scale=0.1]{ext-link.png}

9

O. Agratini, On the variation detracting property of a class of operators, Appl. Math. Lett., 19 (2006), 1261–1264. \includegraphics[scale=0.1]{ext-link.png}

10

A. Kivinukk, T. Metsmagi, Approximation in variation by the Meyer-König and Zeller operators, Proc. Estonian Acad. Sci., 60 (2011) no. 2, 88–97. \includegraphics[scale=0.1]{ext-link.png}

11

H. Karsli, On convergence of Chlodovsky and Chlodovsky-Kantorovich polynomials in the variation seminorm, Mediterr. J. Math., 10 (2013), 41–56. \includegraphics[scale=0.1]{ext-link.png}

12

Ö. Öksüzer, H. Karsli, F. Tasdelen, On convergence of Bernstein-Stancu polynomials in the variation seminorm, Numer. Funct. Anal. Optim., 37 (2016) no. 4, 1–20.

13

H.Gül  Ince  Ilarslan, G. Başcanbaz-Tunca, Convergence in variation for Bernstein-type operators, Mediterr. J. Math., 13 (2015) no. 5, pp. 2577–2592. \includegraphics[scale=0.1]{ext-link.png}

14

U. Abel, O. Agratini, On the variation detracting property of operators of Balázs and Szabados, Acta Math. Hungarica, 150 (2016) no. 2, 383–395. \includegraphics[scale=0.1]{ext-link.png}

15

H. Karsli, Ö. Öksüzer Yılık, F. Tasdelen, Convergence of the Bernstein–Durrmeyer operators in variation seminorm, Results Math., 72 (2017), 1257–1270. \includegraphics[scale=0.1]{ext-link.png}

16

P.L. Butzer, H. Karsli, Voronovskaya-type theorems for derivatives of the Bernstein-Chlodovsky polynomials and the Száasz-Mirakyan operator, Comment. Math., 49 (2009) no. 1, 33–57.

17

R.M. Trigub, E.S. Belinsky, Fourier Analysis and Approximation of Functions. Kluwer Academic Publishers, Dordrecht (2004).