Durrmeyer-Stancu type operators

Ovidiu T. Pop\(^\ast \) Dan Bărbosu\(^\S \)

August 05, 2008

\(^\ast \)National College “Mihai Eminescu”, 5 Mihai Eminescu Street, 440014 Satu Mare,
Romania, e-mail: ovidiutiberiu@yahoo.com

\(^\S \)North University of Baia Mare, Dept. of Math. and Comp. Science, Victoriei 76, 43012 Baia Mare, Romania, e-mail: danbarbosu@yahoo.com

Considering two given real parameters \(\alpha , \beta \) satisfying the conditions \(0\leq \alpha \leq \beta \). D. D. Stancu [7] constructed and studied the linear positive operators \(P^{(\alpha ,\beta )}_m:C([0,1])\to C([0,1])\), defined for any \(f\in C([0,1])\) and any positive integer \(m\) by (1). In this paper we are dealing with the Durrmeyer form of Stancu’s operators. Some approximation properties of these Durrmeyer-Stancu operators are established. As a particular case, we retrieve approximation properties for the classical Durrmeyer operators [5].

MSC. 41A10, 41A25, 41A36, 41A63.

Keywords. Linear positive operators, Durrmeyer operators, first order modulus of smoothness, Shisha-Mond theorem.

1 Preliminaries

Let \(\alpha , \beta \) be real parameters satisfying the conditions \(0\leq \alpha \leq \beta \).
In 1969 D. D. Stancu [7] constructed and studied the linear positive operators \(P^{(\alpha ,\beta )}_m:C([0,1])\to C([0,1])\) defined for any \(f\in C([0,1])\), any \(x\in [0,1]\) and any positive integer \(m\) by:

\begin{equation} \label{1.1} \left(p^{(\alpha ,\beta )}_m f\right)(x)=\sum ^m_{k=0} p_{m,k}(x) f\left(\tfrac {k+\alpha }{m+\beta }\right) \end{equation}

where

\begin{equation} \label{1.2} p_{m,k}(x)=\tbinom {m}{k} x^k(1-x)^{m-k} \end{equation}

are the fundamental Bernstein polynomials [4].

In the monograph by F. Altomare and M. Campiti [2], the operators (1) are called “the Bernstein-Stancu operators”.

In 1967, J. L. Durrmeyer [5] introduced the operators \(D_m:L_1([0,1])\to C([0,1])\), defined for any \(f\in C([0,1])\), any \(x\in [0,1]\) and any positive integer \(m\) by:

\begin{equation} \label{1.3} (D_m f)(x)=(m+1)\sum ^m_{k=0}p_{m,k}(x)\int ^1_0 p_{mk}(t) f(t){\rm d}t. \end{equation}

Note that the operators (3) are known in the mathematical literature as the “Durrmeyer operators”. Following the Durrmeyer ideas, the second author introduced and studied Durrmeyer-Schurer operators [3].

The aim of the present paper is to construct the Durrmeyer type operators associated to the operators (1), denoted Durrmeyer-Stancu operators and to establish some of them approximation properties.

In Section 2 we recall the well-known theorem due to O. Shisha and B. Mond [6] and some properties of Durrmeyer operators [5]. We also introduce here the Durrmeyer-Stancu operators, denoted \(D_m^{(\alpha ,\beta )}\) and we compute the images of test functions by these operators.

Section 3 contains the main results of the paper which are: a convergence theorem for the sequence \(\big\{ D_m^{(\alpha ,\beta )} f\big\} _{m\in \mathbb N}\); the approximation order of \(f\) by \(D_m^{(\alpha ,\beta )} f\), under different assumption on the approximated function \(f\).
As particular case, we get the classical Durrmeyer operator (3).

2 Auxiliary results

As usually, \(\omega _1(f;\delta )\) denotes the first order modulus of smoothness for the function \(f\) and \(e_j(x)=x^j\) \((j=0,1,\dots )\) are the test functions.

First, we recall the following result, due to O. Shisha and B. Mond [6].

Theorem 1

Let \(I\) be a non-empty interval of real axis, let \(L:C_B(I)\to B(I)\) be a linear and positive operator and, for \(x\in I\), let \(\varphi _x:I\to \mathbb R\) be defined by \(\varphi _x(t)=|t-x|\), for any \(t\in I\).

Let \(f\in C_B(I)\) be given; for any \(\delta {\gt}0\) and any \(x\in I\) the following

\begin{align} \label{2.1} |(Lf)(x)-f(x)|& \leq |f(x)||(Le_0)(x)-1|+\\ & \quad +\left\{ (Le_0)(x)+\delta ^{-1}\sqrt{(Le_0)(x)(L\varphi ^2_x)(x)} \right\} \omega _1(f;\delta )\nonumber \end{align}

holds.
Let \(f\) be a given differentiable function such that \(f'\in C_B(I)\); for any \(\delta {\gt}0\) and any \(x\in I\), the following

\begin{align} \label{2.2} |(Lf)(x)-f(x)|& \leq |f(x)||(Le_0)(x)-1|+\\ & \quad +|f’(x)||(Le_1)(x)-x(Le_0)(x)|+\nonumber \\ & \quad +\sqrt{(L\varphi ^2_x)(x)}\left\{ \sqrt{Le_0)(x)}+ \delta ^{-1}\sqrt{(L\varphi ^2_x)(x)}\right\} \omega _1(f’;\delta )\nonumber \end{align}

holds.

Next, let us to recall some properties of Durrmeyer operators.

Lemma 2

[5]. For any positive integer \(m\) and any \(x\in [0,1]\), the Durrmeyer operators \((\ref{1.3})\) verify:

\begin{align} \label{2.3} & (D_m e_0)(x)=1;\\ & (D_m e_1)(x)=\tfrac {mx+1}{m+2}\, ;\label{2.4}\\ & (D_m e_2)(x)=\tfrac {m(m-1)x^2+4mx+2}{(m+2)(m+3)}\, ;\label{2.5}\\ & (D_m\varphi ^2_x)(x)=\tfrac {2(m-3)x(1-x)+2}{(m+2)(m+3)}\, .\label{2.6} \end{align}

Definition 3

Let \(\alpha , \beta \) be real parameters satisfying the conditions \(0\leq \alpha \leq \beta \). The Durrmeyer-Stancu operators \(D_m^{(\alpha ,\beta )}:L_1([0,1])\to C([0,1])\) are defined for any \(f\in L_1([0,1])\), any \(x\in [0,1]\) and any positive integer \(m\) by:

\begin{equation} \label{2.7} \left(D_m^{(\alpha ,\beta )} f\right)(x)=(m+1)\sum ^m_{k=0} p_{m,k}(x)\int ^1_0 p_{m,k}(t) f\left(\tfrac {mt+\alpha }{m+\beta }\right){\rm d}t. \end{equation}

Lemma 4

The operators \((\ref{2.7})\) are linear and positive.

Proof â–¼

Directly, from Definition 3.

Lemma 5

For any \(x\in [0,1]\) and any positive integer \(m\) the Durrmeyer-Stancu operators \((\ref{2.7})\) verify:

\begin{equation} \label{2.8} \left(D_m^{(\alpha ,\beta )}e_0\right)(x)=1; \end{equation}

\begin{equation} \label{2.9} \left(D_m^{(\alpha ,\beta )} e_1\right)(x)=\tfrac {m^2}{(m+\beta )(m+2)}\, x+ \tfrac {(\alpha +1)m+2\alpha }{(m+\beta )(m+2)}\, ; \end{equation}

\begin{align} \label{2.10} \left(D_m^{(\alpha ,\beta )} e_2\right)(x)& = \tfrac {m^3(m-1)}{(m+\beta )^2(m+2)(m+3)}\, x^2\\ & \quad +\tfrac {4m^3+2\alpha m^2(m+3)}{(m+\beta )^2(m+2)(m+3)}\, x\nonumber \\ & \quad +\tfrac {2m^2+2\alpha m(m+3)+\alpha ^2(m+2)(m+3)} {(m+\beta )^2(m+2)(m+1)}\, ;\nonumber \end{align}

\begin{align} \label{2.11} \left(D_m^{(\alpha ,\beta )}\varphi ^2_x\right)(x)& = \left(\tfrac {m}{m+\beta }\right)^2 \tfrac {2(m-3)x(1-x)+2}{(m+2)(m+3)}+\\ & \quad +\! \tfrac {\! \left\{ \! \beta ^2(m\! +\! 2)\! +\! 4\beta m\! \right\} x^2\! -\! 2 \left\{ \! \alpha \beta (m\! +\! 2)\! +\! \beta m\! +\! 2\alpha m\! \right\} x\! + \! \alpha ^2(m\! +\! 2)\! +\! 2m\alpha } {(m+\beta )^2 (m+2)}.\nonumber \end{align}

Proof â–¼

It is easy to observe the following identities:

\begin{align*} \left(D_m^{(\alpha ,\beta )}e_0\right)(x)& =\left(D_m e_0\right)(x)\\ \left(D_m^{(\alpha ,\beta )}e_1\right)(x)& =\tfrac {m}{m+\beta }\, (D_m e_1)(x)+\tfrac {\alpha }{m+\beta }\, (D_m e_0)(x)\\ \left(D_m^{(\alpha ,\beta )}e_2\right)(x)& = \left(\tfrac {m}{m+\beta }\right)^2(D_m e_2)(x)+ \tfrac {2\alpha m}{(m+\beta )^2}\, (D_m e_1)(x)+\\ & \quad +\tfrac {\alpha ^2}{(m+\beta )^2}\, (D_m e_0)(x)\end{align*}

\begin{align*} \left(D_m^{(\alpha ,\beta )}\varphi _x^2\right)(x)& = \left(D_m^{(\alpha ,\beta )}e_2\right)(x)-2x\left(D_m^{(\alpha ,\beta )}e_1\right)(x) +x^2\left(D_m^{(\alpha ,\beta )}e_0\right)(x). \end{align*}

Next, one applies Lemma 2.

Lemma 6

For any positive integer \(m\) and any \(x\in [0,1]\) the following

\begin{equation} \label{2.12} \delta _m^{(\alpha ,\beta )}(x)\leq \delta ^{(\alpha ,\beta )}_m \end{equation}

holds, where

\begin{equation} \label{2.13} \delta _m^{(\alpha ,\beta )}(x)=\sqrt{\left(D_m^{(\alpha ,\beta )} \varphi ^2_m\right)(x)} \end{equation}

and

\begin{equation} \label{2.14} \delta _m^{(\alpha ,\beta )}= \sqrt{\! \left(\! \tfrac {m}{m\! +\! \beta }\! \right)^2 \tfrac {m\! +\! 1}{2(m\! +\! 2)(m\! +\! 3)}\! +\! \max \left\{ \! \tfrac {\alpha ^2(m\! +\! 2)\! +\! 2m\alpha } {(m\! +\! \beta )^2(m\! +\! 2)}\, ,\tfrac {(m\! +\! 2)(\alpha \! -\! \beta )^2\! -\! 2m(\alpha \! -\! \beta )} {(m+\beta )^2(m+2)}\right\} \! } .\end{equation}

Proof â–¼

Because \(x(1-x)\leq \tfrac {1}{4}\) for any \(x\in [0,1]\), we get

\begin{equation} \label{2.15} \tfrac {2(m-3)x(1-x)+2}{(m+2)(m+3)}\leq \tfrac {m+1}{2(m+2)(m+3)} \end{equation}

for any positive integer \(m\).

For any positive integer \(m\), let us to introduce the following notations:

\begin{align*} \gamma ^{(\alpha ,\beta )}_m & =\max \left\{ \tfrac {\alpha ^2(m+2)+2m\alpha } {(m+\beta )^2(m+2)}\, ,\tfrac {(m+2)(\alpha -\beta )^2-2m(\alpha -\beta )} {(m+\beta )^2(m+2)}\right\} ;\\ a_m & =\tfrac {\beta ^2(m+2)+4\beta m}{(m+\beta )^2(m+2)}\, ;\\ b_m& =-2\tfrac {\alpha \beta (m+2)+\beta m+2\alpha m} {(m+\beta )^2(m+2)}\, ;\\ c_m & =\tfrac {\alpha ^2(m+2)+2m\alpha }{(m+\beta )^2(m+2)}\, . \end{align*}

Let \(f_m[0,1]\to \mathbb R\) be defined for any \(x\in [0,1]\) by:

\[ f_m(x)=a_m x^2+b_m x+c_m. \]

If \(\beta =0\), follows \(\alpha =0\) and \(f_m(x)=0\), for any \(x\in [0,1]\).

If \(\beta \neq 0\) the function \(f\) is a polynomial function of second degree and because \(a_m{\gt}0\) follows that \(f_m\) has a maximum value in the point

\[ x_m=-\tfrac {b_m}{2a_m}= \tfrac {\alpha \beta (m+2)+\beta m+2\alpha m} {\beta ^2(m+2)+4\beta m}\, . \]

Because \(\alpha \leq \beta \), we get

\[ x_m\leq \tfrac {\beta ^2(m+2)+3\beta m} {\beta ^2(m+2)+4\beta m}{\lt}1. \]

Taking the above inequality into account, yields:

\[ f_m(x)\leq \max \left\{ f_m(0), f_m(1)\right\} , \]

for any \(x\in [0,1]\). Follows that for any \(x\in [0,1]\) the following

\begin{equation} \label{2.16} f_m(x)\leq \gamma ^{(\alpha ,\beta )}_m. \end{equation}

Applying (18) and (19) we get (15).

3 Main results

Theorem 7

For any \(f\in L_1([0,1])\) the sequence \(\big\{ D^{(\alpha ,\beta )}_m f\big\} \) converges to \(f\), uniformly on \([0,1]\).

Proof â–¼

Lemma 5 (identity (14)) follows \(\lim \limits _{m\to \infty }\big(D_m^{(\alpha ,\beta )}\varphi ^2_x\big)(x)=0\), uniformly on \([0,1]\). Then one applies the well known Bohman-Korovkin theorem ([1] or [8]).

Theorem 8

For any \(f\in L_1([0,1])\), any \(x\in [0,1]\), any \(\delta {\gt}0\) and any positive integer \(m\), the following

\begin{equation} \label{3.1} \left|\left(D_m^{(\alpha ,\beta )} f\right)(x)-f(x)\right|\leq \left(1+\tfrac {1}{\delta }\, \delta _m^{(\alpha ,\beta )}(x)\right) \omega _1 (f;\delta ) \end{equation}
20

holds.
For any \(f\in C^1([0,1])\), any \(x\in [0,1]\), any \(\delta {\gt}0\) and any positive integer \(m\), the following

\begin{align} \label{3.2} \left|\left(D_m^{(\alpha ,\beta )} f\right)(x)-f(x)\right|& \leq |f’(x)|\bigg|-\tfrac {(\beta +2)m+2\beta }{(m+\beta )(m+2)}\, x +\tfrac {(\alpha +1)m+2\alpha }{(m+\beta )(m+2)}\bigg|+\\ & \quad +\delta _m^{(\alpha ,\beta )}(x)\left(1+\tfrac {1}{\delta }\, \delta _m^{(\alpha ,\beta )}(x)\right)\omega _1(f’;\delta )\nonumber \end{align}

holds.

In \((\ref{3.1})\) and \((\ref{3.2})\) \(\delta _m^{(\alpha ,\beta )}(x)\) is defined at \((\ref{2.13})\).

Proof â–¼

One applies Theorem 1 and Lemma 5.

Theorem 9

For any \(f\in L_1([0,1])\), any \(x\in [0,1]\) and any positive integer \(m\), the following

\begin{equation} \label{3.3} \left|\left(D_m^{(\alpha ,\beta )} f\right)(x)-f(x)\right| \leq 2\omega _1\left(f; \delta _m^{(\alpha ,\beta )}(x)\right) \end{equation}
22

holds.
For any \(f\in C^1([0,1])\), any \(x\in [0,1]\) and any positive integer \(m\) the following

\begin{align} \label{3.4} \left|\left(D_m^{(\alpha ,\beta )} f\right)(x)-f(x)\right|& \leq |f’(x)|\, \bigg|\tfrac {(\beta +2)m+2\beta }{(m+\beta )(m+2)}\, x+ \tfrac {(\alpha +1)m+2\alpha }{(m+\beta )(m+2)}\bigg|+\\ & \quad +2\delta _m^{(\alpha ,\beta )}(x)\omega _1\left(f’; \delta _m^{(\alpha ,\beta )} (x)\right).\nonumber \end{align}

Proof â–¼

In Theorem 8 we choose \(\delta =\delta _{m,p}^{(\alpha ,\beta )}(x)\).

Remark 10

For any positive integer \(m\), let \(g_m:[0,1]\to \mathbb R\) be defined by

\[ g_m(x)=-\tfrac {(\beta +2)m+2\beta }{(m+\beta )(m+2)}\, x+ \tfrac {(\alpha +1)m+2\alpha }{(m+\beta )(m+2)} \]

for any \(x\in [0,1]\). It is immediately that

\begin{equation} \label{3.5} |g_m(x)|\leq \eta _m^{(\alpha ,\beta )} \end{equation}

for any \(x\in [0,1]\), where

\begin{align} \label{3.6} \eta _m^{(\alpha ,\beta )} & =\max \left\{ |g_m(0)|, g_m(1)\right\} =\\ & =\max \left\{ \tfrac {(\alpha +1)m+2\alpha }{(m+\beta )(m+2)}\, , \tfrac {|m(\alpha -\beta -1)+2(\alpha -\beta )|} {(m+\beta )(m+2)}\right\} .\nonumber \end{align}

â–¡

Applying then Theorem 9, Lemma 6 and (24), we get:

Corollary 11

\((i)\) For any \(f\in L_1([0,1])\), any \(x\in [0,1]\) and any positive integer \(m\), the following

\begin{equation} \label{3.7} \left|\left(D_m^{(\alpha ,\beta )} f\right)(x)-f(x)\right) \leq 2\omega _1\left(f; \delta _m^{(\alpha ,\beta )}\right) \end{equation}

holds.

\((ii)\) For any \(f\in C^1([0,1])\), any \(x\in [0,1]\) and any positive integer \(m\), the following

\begin{equation} \label{3.8} \left|\left(D_m^{(\alpha ,\beta )} f\right)(x)-f(x)\right|\leq M_1\eta _m^{(\alpha ,\beta )}+2\delta _m^{(\alpha ,\beta )} \omega _1\left(f'; \delta _m^{(\alpha ,\beta )}\right) \end{equation}

where \(M_1=\max \limits _{x\in [0,1]}|f'(x)|\).

Remark 12

For any \(\alpha =\beta =0\), the operators \(D_m^{(0,0)}\) are the Durrmeyer operators.â–¡

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