Some Properties of the Operators defined by Lupaş

AYŞEGÜL ERENÇIN$^\ast $, GÜLEN BAŞCANBAZ-TUNCA$^\ast $$^\ast $ FATMA TAŞDELEN$^{\ast \ast }$

August 14, 2014.
Published online: January 23, 2015.

$^\ast $ Abant Izzet Baysal University, Faculty of Arts and Science, Department of Mathematics, 14280, Bolu, Turkey, e-mail: erencina@hotmail.com.

$^{\ast \ast }$Ankara University, Faculty of Science, Department of Mathematics, 06100, Tandoǧan, Ankara, Turkey, e-mail: tunca@science.ankara.edu.tr, tasdelen@science.ankara.edu.tr.

In the present paper, we show that a subclass of the operators defined by Lupaş [ 12 ] preserve properties of the modulus of continuity function and Lipschitz constant and the order of a Lipschitz continuous function. We also concerned with the monotonicity of sequence of such operators for convex functions.

MSC. 41A25, 41A36

Keywords. Modulus of continuity function, Lipschitz class, monotonicity.

1 Introduction

By means of the identity

\[ \tfrac {1}{(1-a)^{\alpha }}=\sum ^{\infty }_{k=0}\tfrac {(\alpha )_{k}}{k!}a^{k},\quad |a|{\lt}1, \]

where $(\alpha )_{0}=1$, $(\alpha )_{k}=\alpha (\alpha +1)\cdots (\alpha +k-1)$, $k\geq 1$, Lupaş [ 12 ] proposed the positive linear operators

\[ {T}_{n}(f;x)=(1-a)^{nx}\sum ^{\infty }_{k=0}\tfrac {(nx)_{k}}{k!}f\left( \tfrac {k}{n}\right) a^{k},\quad x\geq 0 \]

for the functions $f:[0,\infty )\rightarrow \mathbb {R}$ and $n\in \mathbb {N}$. After that Agratini [ 3 ] , by choosing $a=\tfrac {1}{2}$ , for the operators

\begin{equation} \label{eq:1.1}L_{n}(f;x)=2^{-nx}\sum ^{\infty }_{k=0}\tfrac {(nx)_{k}}{2^{k}k!}f\left( \tfrac {k}{n}\right) \end{equation}

1.1

obtained some estimates to the order of approximation on a finite interval and proved a Voronovskaya type theorem. Furthermore, he again derived the positive linear operators $L_{n}$ via a probabilistic approach and presented the Kantorovich and Durrmeyer variants of these operators. In [ 5 ] , a better error estimation and statistical Korovkin type approximation properties of the operators $L_{n}$ were examined by Dirik. Jain and Pethe [ 9 ] , as a generalization of Szasz-Mirakjan operators, introduced the operators

\[ M_{n,\alpha }(f;x)=(1+n\alpha )^{-\tfrac {x}{\alpha }}\sum ^{\infty }_{\nu =0}\left( \alpha +\tfrac {1}{n}\right) ^{-\nu }\tfrac {x^{(\nu ,-\alpha )}}{\nu !} f\left( \tfrac {\nu }{n}\right) ,\quad x\geq 0 \]

where $x^{(0,-\alpha )}=1$, $x^{(\nu ,-\alpha )}=x(x+\alpha )\cdots (x+(\nu -1)\alpha )$, $0\leq n\alpha \leq 1$ and $n\in \mathbb {N}$. By setting $c=c_{n}=\tfrac {1}{n\alpha }$ such that $c\geq \beta $ for certain constant $\beta {\gt}0$, Abel and Ivan [ 1 ] expressed these operators in the equivalent form

\[ S_{n,c}(f;x)=\sum ^{\infty }_{\nu =0}P^{[c]}_{n,\nu }(x) f\left( \tfrac {\nu }{n}\right) ,\quad x\geq 0 \]

where $P^{[c]}_{n,\nu }(x)=\big( \tfrac {c}{1+c}\big) ^{ncx}\binom {ncx+\nu -1}{\nu }(1+c)^{-\nu }$, and studied their local approximation properties and also obtained a complete asymptotic expansion formula. We remark that when $c=1$ the operators $S_{n,c}$ reduce to the operators defined by (1.1). In [ 6 ] , Erençin and Taşdelen introduced the following generalization of the operators $L_{n}$

\[ L^{*}_{n}(f;x)=2^{-a_{n}x}\sum ^{\infty }_{k=0}\tfrac {(a_{n}x)_{k}}{2^{k}k!}f\big( \tfrac {k}{b_{n}}\big) ,\qquad x\geq 0 \]

where $({a_{n}})$, $({b_{n}})$ are increasing and unbounded sequences of positive numbers such that

\[ \tfrac {a_{n}}{b_{n}}=1+{\mathcal O}\big( \tfrac {1}{b_{n}}\big) ,\quad \lim _{n\rightarrow \infty }\tfrac {1}{b_{n}}=0 \]

and investigated their weighted approximation properties. Later, Erençin and Taşdelen [ 7 ] estimated the rate of convergence for the Kantorovich type version of the operators $L^{*}_{n}$ by means of the modulus of continuity, elements of local Lipschitz class and Peetre’s $K$-functional. Recently, A-statistical convergence properties of the operators $L_{n}$ and their Kantorovich type modification were studied by Tarabie in [ 14 ] .

Note that from Lemma 1 in [ 3 ] we have

\[ \begin{split} & L_{n}(1;x)=1,\\ & L_{n}(t;x)=x. \end{split} \]

In this paper, for the operators $L_{n}$ defined by (1.1), we firstly show that when $f$ is a general function of modulus of continuity, then $L_{n}(f;x):=L_{n}(f)$ is also a function of modulus of continuity with the help of the same technique of Li [ 11 ] . Later, we also show that the operators $L_{n}$ preserve the Lipschitz constant and the order of a Lipschitz continuous function. Furthermore, we discuss the monotonicity of the operators $L_{n}$ for $n$ under the convexity of $f$. We note that in the literature there are a number of papers containing preservation properties of positive linear operators. Some of them are [ 2 ] , [ 4 ] , [ 8 ] , [ 10 ] and [ 15 ] .

2 Some properties of the operators $L_{\lowercase {n}}$

In order to give some properties of the operators defined by (1.1) let us recall some definitions.

Let $f$ be a real valued continuous function defined on $[0,\infty )$. Then $f$ is said to be Lipschitz continuous of order $\gamma $ $(0{\lt}\gamma \leq 1)$ on $[0,\infty )$, if there exists $M{\gt} 0$ such that

\[ \left| f(x)-f(y)\right| \leq M\left| x-y\right| ^{\gamma } \]

for all $x,y\in [0,\infty )$. The set of Lipschitz continuous functions of order $\gamma $ with Lipschitz constant $M$ is denoted by $\operatorname {Lip}_{M}(\gamma )$.

A real valued continuous function $f$ is said to be convex on $[0,\infty )$, if

\[ f\Big( \sum ^{n}_{i=1}\alpha _{i}t_{i}\Big) \leq \sum ^{n}_{i=1}\alpha _{i}f(t_{i}) \]

for all $t_{1},t_{2},\cdots ,t_{n}\in [0,\infty )$ and for all non-negative numbers $\alpha _{1},\alpha _{2},\cdots ,\alpha _{n}$ such that $\alpha _{1}+\alpha _{2}+\cdots +\alpha _{n}=1$.

Also, a continuous and non-negative function $\omega $ defined on $[0,\infty )$ is called the modulus of continuity function, if each of the following conditions is satisfied:

$\omega (u+v)\leq \omega (u)+\omega (v)$ for $u,v\in [0,\infty )$, i.e., $\omega $ is subadditive,
$\omega (u)\geq \omega (v)$ for $u\geq v$, i.e., $\omega $ is non-decreasing,
$\lim _{u\rightarrow 0^{+}}\omega (u)=\omega (0)=0$

(see p. 106 in [ 13 ] ).

Theorem 1

If $\omega $ is a modulus of continuity function, then $L_{n}(\omega )$ is also a modulus of continuity function.

Proof â–¼

Let $x,y\in [0,\infty )$ and $x\leq y$. Then we have

\[ L_{n}(f;y)= 2^{-ny}\sum ^{\infty }_{k=0}\tfrac {(ny)_{k}}{2^{k}k!}f\left( \tfrac {k}{n}\right) = 2^{-ny}\sum ^{\infty }_{k=0}\tfrac {(n(x+(y-x)))_{k}}{2^{k}k!}f\left( \tfrac {k}{n}\right) . \]

Since

\[ (n(x+(y-x)))_{k}=\sum ^{k}_{i=0}\tbinom {k}{i}(nx)_{i}(n(y-x))_{k-i} \]

one may write

\[ L_{n}(f;y)=2^{-ny}\sum ^{\infty }_{k=0}\sum ^{k}_{i=0}\tfrac {1}{2^{k}k!}\tbinom {k}{i}(nx)_{i}(n(y-x))_{k-i}f\left( \tfrac {k}{n}\right) . \]

Changing the order of the above summations and then taking $k-i=j$, we reach to

\begin{equation} \label{eq:2.1}L_{n}(f;y)= 2^{-ny}\sum ^{\infty }_{i=0}\sum ^{\infty }_{k=i}\tfrac {1}{2^{k}k!}\tbinom {k}{i}(nx)_{i}(n(y-x))_{k-i}f\left( \tfrac {k}{n}\right) = \end{equation}

2.1

\begin{equation*} = 2^{-ny}\sum ^{\infty }_{i=0}\sum ^{\infty }_{j=0}\tfrac {(nx)_{i}}{2^{i}i!}\tfrac {(n(y-x))_{j}}{2^{j}j!}f\big( \tfrac {i+j}{n}\big) . \end{equation*}

On the other hand using the identity $2^{n(y-x)}=\sum ^{\infty }_{j=0}\tfrac {(n(y-x))_{j}}{2^{j}j!}$, we get

\begin{equation} \label{eq:2.2}\begin{split} L_{n}(f;x)= & 2^{-nx}\sum ^{\infty }_{i=0}\tfrac {(nx)_{i}}{2^{i}i!}f\left( \tfrac {i}{n}\right) \\ = & 2^{-ny}2^{n(y-x)}\sum ^{\infty }_{i=0}\tfrac {(nx)_{i}}{2^{i}i!}f\left( \tfrac {i}{n}\right) \\ = & 2^{-ny}\sum ^{\infty }_{i=0}\sum ^{\infty }_{j=0}\tfrac {(nx)_{i}}{2^{i}i!}\tfrac {(n(y-x))_{j}}{2^{j}j!}f\left( \tfrac {i}{n}\right) . \end{split} \end{equation}

2.2

Hence from (2.1) and (2.2) it follows that

\begin{equation} \label{eq:2.3}L_{n}(f;y)-L_{n}(f;x)=2^{-ny}\sum ^{\infty }_{i=0}\sum ^{\infty }_{j=0}\tfrac {(nx)_{i}}{2^{i}i!}\tfrac {(n(y-x))_{j}}{2^{j}j!} \left[ f\big( \tfrac {i+j}{n}\big) -f\left( \tfrac {i}{n}\right) \right] . \end{equation}

2.6

Thus by means of the equality (2.6) and the subadditivity of $\omega $, we can write

\[ \begin{split} L_{n}(\omega ;y)-L_{n}(\omega ;x)= & 2^{-ny}\sum ^{\infty }_{i=0}\sum ^{\infty }_{j=0}\tfrac {(nx)_{i}}{2^{i}i!}\tfrac {(n(y-x))_{j}}{2^{j}j!} \left[ \omega \big( \tfrac {i+j}{n}\big) -\omega \left( \tfrac {i}{n}\right) \right] \\ \leq & 2^{-ny}\sum ^{\infty }_{i=0}\sum ^{\infty }_{j=0}\tfrac {(nx)_{i}}{2^{i}i!}\tfrac {(n(y-x))_{j}}{2^{j}j!}\omega \big( \tfrac {j}{n}\big) \\ = & 2^{-nx}\sum ^{\infty }_{i=0}\tfrac {(nx)_{i}}{2^{i}i!}2^{-n(y-x)}\sum ^{\infty }_{j=0}\tfrac {(n(y-x))_{j}}{2^{j}j!}\omega \big( \tfrac {j}{n}\big) \\ = & L_{n}(1;x)2^{-n(y-x)}\sum ^{\infty }_{j=0}\tfrac {(n(y-x))_{j}}{2^{j}j!}\omega \big( \tfrac {j}{n}\big) \\ = & 2^{-n(y-x)}\sum ^{\infty }_{j=0}\tfrac {(n(y-x))_{j}}{2^{j}j!}\omega \big( \tfrac {j}{n}\big) \\ = & L_{n}(\omega ;y-x). \end{split} \]

This shows the subadditivity of $L_{n}(\omega )$. We also infer from (2.6) that $L_{n}(\omega ;y)\geq L_{n}(\omega ;x)$ for $y\geq x$ which means that $L_{n}(\omega )$ is non-decreasing. Finally the property $L_{n}(\omega ;0)=\omega (0)=0$ is clear. Thus we may conclude that $L_{n}(\omega )$ is a modulus of continuity function.

Now we introduce the second result of this section with the following theorem.

Theorem 2

If $f \in \operatorname {Lip}_{M}(\gamma )$, then $L_{n}(f)\in \operatorname {Lip}_{M}(\gamma )$.

Proof â–¼

Suppose that $x\leq y$. By using the facts $f \in \operatorname {Lip}_{M}(\gamma )$ and $L_{n}(1,x)=1$ from (2.6) we can write

\begin{align*} \left| L_{n}(f;y)-L_{n}(f;x)\right| & \leq M 2^{-ny}\sum ^{\infty }_{i=0}\sum ^{\infty }_{j=0}\tfrac {(nx)_{i}}{2^{i}i!}\tfrac {(n(y-x))_{j}}{2^{j}j!}\big( \tfrac {j}{n}\big) ^{\gamma }\\ & =M2^{-nx}\sum ^{\infty }_{i=0}\tfrac {(nx)_{i}}{2^{i}i!}2^{-n(y-x)}\sum ^{\infty }_{j=0}\tfrac {(n(y-x))_{j}}{2^{j}j!}\big( \tfrac {j}{n}\big) ^{\gamma }\\ & =ML_{n}(1,x)2^{-n(y-x)}\sum ^{\infty }_{j=0}\tfrac {(n(y-x))_{j}}{2^{j}j!}\big( \tfrac {j}{n}\big) ^{\gamma }\\ & =M2^{-n(y-x)}\sum ^{\infty }_{j=0}\tfrac {(n(y-x))_{j}}{2^{j}j!}\big( \tfrac {j}{n}\big) ^{\gamma }. \end{align*}

Now applying Hölder’s inequality one gets

\[ \left| L_{n}(f;y)-L_{n}(f;x)\right| \leq M \Big( 2^{-n(y-x)}\sum ^{\infty }_{j=0}\tfrac {(n(y-x))_{j}}{2^{j}j!}\tfrac {j}{n}\Big) ^{\gamma } =M\left( L_{n}(t;y-x)\right) ^{\gamma }. \]

Since $L_{n}(t;x)=x$ the above inequality implies that

\[ \left| L_{n}(f;y)-L_{n}(f;x)\right| \leq M(y-x)^{\gamma }. \]

Similarly, it can be shown that when $x{\gt}y$ our claim is valid.

Now, we will study the monotonicity of the sequence of the operators $L_{n}(f;x)$ defined by (1.1) when the function $f$ is convex.

Theorem 3

If $f$ is a convex function defined on $[0,\infty )$, then $L_{n}(f;x)$ is strictly monotonically decreasing, unless $f$ is the linear function (in which case $L_{n}(f;x)=L_{n+1}(f;x)$ for all $n$).

Proof â–¼

We have

\[ \begin{split} L_{n} & (f;x)-L_{n+1}(f;x)=\\ & = 2^{-nx}\sum ^{\infty }_{k=0}\tfrac {(nx)_{k}}{2^{k}k!} f\left( \tfrac {k}{n}\right) -2^{-(n+1)x}\sum ^{\infty }_{k=0}\tfrac {((n+1)x)_{k}}{2^{k}k!} f\big( \tfrac {k}{n+1}\big) \\ & =2^{-(n+1)x}\left\{ 2^{x}\sum ^{\infty }_{k=0}\tfrac {(nx)_{k}}{2^{k}k!} f\left( \tfrac {k}{n}\right) -\sum ^{\infty }_{k=0}\tfrac {((n+1)x)_{k}}{2^{k}k!} f\big( \tfrac {k}{n+1}\big) \right\} . \end{split} \]

Using the identity $2^{x}=\sum ^{\infty }_{l=0}\tfrac {(x)_{l}}{2^{l}l !}$, one may write

\begin{align} \label{eq:2.4}& L_{n} (f;x)-L_{n+1}(f;x)= \\ & =2^{-(n+1)x}\left\{ \sum ^{\infty }_{l=0}\tfrac {(x)_{l}}{2^{l}l!}\sum ^{\infty }_{k=0}\tfrac {(nx)_{k}}{2^{k}k!} f\left( \tfrac {k}{n}\right) -\sum ^{\infty }_{k=0}\tfrac {((n+1)x)_{k}}{2^{k}k!} f\big( \tfrac {k}{n+1}\big) \right\} = \nonumber \end{align}

\begin{align*} & =2^{-(n+1)x}\left\{ \sum ^{\infty }_{l=0}\sum ^{\infty }_{k=l}\tfrac {(nx)_{k-l}(x)_{l}}{2^{k}l!(k-l)!}f\big( \tfrac {k-l}{n}\big) -\sum ^{\infty }_{k=0}\tfrac {((n+1)x)_{k}}{2^{k}k!} f\big( \tfrac {k}{n+1}\big) \right\} \\ & =2^{-(n+1)x}\left\{ \sum ^{\infty }_{k=0}\left[ \sum ^{k}_{l=0}\tfrac {(nx)_{l}(x)_{k-l}}{2^{k}l!(k-l)!}f\left( \tfrac {l}{n}\right) -\tfrac {((n+1)x)_{k}}{2^{k}k!} f\big( \tfrac {k}{n+1}\big) \right] \right\} . \end{align*}

Now we only need to show that for all $k=0,1,\ldots $,

\begin{equation} \label{eq:2.5}f\big( \tfrac {k}{n+1}\big) \leq \tfrac {1}{((n+1)x)_{k}}\sum ^{k}_{l=0}\tbinom {k}{l}(nx)_{l}(x)_{k-l}f\left( \tfrac {l}{n}\right) , \end{equation}

2.17

which is a direct result of convexity. In fact, set

\[ \alpha _{l}=\tbinom {k}{l}\tfrac {(nx)_{l}(x)_{k-l}}{((n+1)x)_{k}}\geq 0 \qquad {\rm and\quad } t_{l}=\tfrac {l}{n}. \]

Therefore with the help of the identity $((n+1)x)_{k}=\sum ^{k}_{l=0}\tbinom {k}{l}(nx)_{l}(x)_{k-l}$, it is clear that

\[ \sum ^{k}_{l=0}\alpha _{l}=\tfrac {1}{((n+1)x)_{k}}\sum ^{k}_{l=0}\tbinom {k}{l}(nx)_{l}(x)_{k-l}=1. \]

On the other hand, we have

\[ \begin{split} \sum ^{k}_{l=0}\alpha _{l}t_{l}= & \tfrac {1}{((n+1)x)_{k}}\sum ^{k}_{l=0}\tbinom {k}{l}(nx)_{l}(x)_{k-l}\tfrac {l}{n} \\ = & \tfrac {1}{n((n+1)x)_{k}}\sum ^{k}_{l=1}\tfrac {k!}{(l-1)!(k-l)!}(nx)_{l}(x)_{k-l}\\ = & \tfrac {k}{n((n+1)x)_{k}}\sum ^{k-1}_{l=0}\tbinom {k-1}{l}(nx)_{l+1}(x)_{k-l-1}. \end{split} \]

Since

\[ (nx)_{l+1}=nx(nx+1)_{l},\quad ((n+1)x)_{k}=(n+1)x((n+1)x+1)_{k-1} \]

and

\[ ((n+1)x+1)_{k-1}=\sum ^{k-1}_{l=0}\tbinom {k-1}{l}(nx+1)_{l}(x)_{k-l-1} \]

one may write

\[ \sum ^{k}_{l=0}\alpha _{l}t_{l}= \tfrac {k}{(n+1)((n+1)x+1)_{k-1}}\sum ^{k-1}_{l=0}\tbinom {k-1}{l}(nx+1)_{l}(x)_{k-l-1} = \tfrac {k}{n+1} \]

which, making use of the convexity of $f$, gives the inequality (2.17). Hence from (2.16) we arrive at the desired result. Clearly $L_{n}(f;x)=L_{n+1}(f;x)$ occurs only if

\[ \begin{split} f\Big( \tfrac {1}{((n+1)x)_{k}}\sum ^{k}_{l=0}\tbinom {k}{l}(nx)_{l}(x)_{k-l}\tfrac {l}{n}\Big) = & f\big( \tfrac {k}{n+1}\big) \\ = & \tfrac {1}{((n+1)x)_{k}}\sum ^{k}_{l=0}\tbinom {k}{l}(nx)_{l}(x)_{k-l}f\left( \tfrac {l}{n}\right) \\ \end{split} \]

for all $k=0,1,\ldots $ This implies that $f$ is linear in $[0,\infty )$. Thus the proof is completed.

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Some Properties of the Operators defined by Lupaş

1 Introduction

2 Some properties of the operators \(L_{\lowercase {n}}\)

Bibliography