THE GENERALIZATION OF VORONOVSKAJA'S THEOREM FOR A CLASS OF LINEAR AND POSITIVE OPERATORS

MSC 2000. 41A10, 41A36.
Keywords. Bernstein operators, Bernstein-Schurer operators, BernsteinStancu operators, Kantorovich operators, Durrmeyer operators, Voronovskaja's theorem.

In this section, we recall some notions and results which we will use in this article.

Let $m$$m$mm$m$ be a nonzero natural number and $B_m:C([0,1])\to C([0,1])$$B_m:C([0,1])\to C([0,1])$B_(m):C([0,1])rarr C([0,1])B_{m}: C([0,1]) \rightarrow C([0,1])$B_m:C([0,1])\to C([0,1])$ the Bernstein operators, defined for any function $f\in C([0,1])$$f\in C([0,1])$f in C([0,1])f \in C([0,1])$f\in C([0,1])$ by

where $p_{m,k}(x)$$p_{m,k}(x)$p_(m,k)(x)p_{m, k}(x)$p_{m,k}(x)$ are the fundamental polynomials of Bernstein, defined as follows

for any $x\in [0,1]$$x\in [0,1]$x in[0,1]x \in[0,1]$x\in [0,1]$ and any $k\in \{0,1,\dots ,m\}$$k\in \{0,1,\dots ,m\}$k in{0,1,dots,m}k \in\{0,1, \ldots, m\}$k\in \{0,1,\dots ,m\}$.
In 1932, E. Voronovskaja, proved the result contained in the following theorem.

Theorem 1.1. ([13]) Let $f\in C([0,1])$$f\in C([0,1])$f in C([0,1])f \in C([0,1])$f\in C([0,1])$ be a two times derivable function at the point $x\in [0,1]$$x\in [0,1]$x in[0,1]x \in[0,1]$x\in [0,1]$. Then the equality

holds.

For the natural numbers $m$$m$mm$m$ and $p,m$$p,m$p,mp, m$p,m$ nonzero, F. Schurer (see [9]) introduced and studied in 1962, the operators ${\tilde{B}}_{m,p}:C([0,1+p])\to C([0,1])$${\tilde{B}}_{m,p}:C([0,1+p])\to C([0,1])$widetilde(B)_(m,p):C([0,1+p])rarr C([0,1])\widetilde{B}_{m, p}: C([0,1+p]) \rightarrow C([0,1])${\tilde{B}}_{m,p}:C([0,1+p])\to C([0,1])$, named Bernstein-Schurer operators, defined for any function $f\in C([0,1+p])$$f\in C([0,1+p])$f in C([0,1+p])f \in C([0,1+p])$f\in C([0,1+p])$ by

where ${\tilde{p}}_{m,k}(x)$${\tilde{p}}_{m,k}(x)$widetilde(p)_(m,k)(x)\widetilde{p}_{m, k}(x)${\tilde{p}}_{m,k}(x)$ denotes the fundamental Bernstein-Schurer polynomials, defined as follows

for any $x\in [0,1]$$x\in [0,1]$x in[0,1]x \in[0,1]$x\in [0,1]$ and any $k\in \{0,1,\dots ,m+p\}$$k\in \{0,1,\dots ,m+p\}$k in{0,1,dots,m+p}k \in\{0,1, \ldots, m+p\}$k\in \{0,1,\dots ,m+p\}$.
In 2002, D. Bărbosu proved the result contained in the following theorem.
Theorem 1.2. ([2]) Let $f\in C([0,1+p])$$f\in C([0,1+p])$f in C([0,1+p])f \in C([0,1+p])$f\in C([0,1+p])$ be a two times derivable function at the point $x\in [0,1]$$x\in [0,1]$x in[0,1]x \in[0,1]$x\in [0,1]$. Then the equality

holds.
Let $m$$m$mm$m$ be a nonzero natural number and the operators $M_n:L_1([0,1])\to C([0,1])$$M_n:L_1([0,1])\to C([0,1])$M_(n):L_(1)([0,1])rarr C([0,1])M_{n}: L_{1}([0,1]) \rightarrow C([0,1])$M_n:L_1([0,1])\to C([0,1])$ are defined for any function $f\in L_1([0,1])$$f\in L_1([0,1])$f inL_(1)([0,1])f \in L_{1}([0,1])$f\in L_1([0,1])$ by

for any $x\in [0,1]$$x\in [0,1]$x in[0,1]x \in[0,1]$x\in [0,1]$.
These operators were introduced in 1967 by J.L. Durrmeyer in [5] and were studied in 1981 by M.M. Derriennic in [4], where the following theorem can be found.

Theorem 1.3. Let $f\in L_1([0,1])$$f\in L_1([0,1])$f inL_(1)([0,1])f \in L_{1}([0,1])$f\in L_1([0,1])$, bounded on $[0,1]$$[0,1]$[0,1][0,1]$[0,1]$. If $f$$f$ff$f$ is a two times derivable function at the point $x\in [0,1]$$x\in [0,1]$x in[0,1]x \in[0,1]$x\in [0,1]$, then

If $f$$f$ff$f$ is a two times derivable function on $[0,1]$$[0,1]$[0,1][0,1]$[0,1]$ and the function $f^{\prime \prime }$$f^{\prime \prime }$f^('')f^{\prime \prime}$f^{\prime \prime }$ is continuous on $[0,1]$$[0,1]$[0,1][0,1]$[0,1]$, then the convergence from ( 1.8 ) is uniform on $[0,1]$$[0,1]$[0,1][0,1]$[0,1]$.

For $m$$m$mm$m$ be a nonzero natural number, let the operators $K_m:L_1([0,1])\to C([0,1])$$K_m:L_1([0,1])\to C([0,1])$K_(m):L_(1)([0,1])rarr C([0,1])K_{m}: L_{1}([0,1]) \rightarrow C([0,1])$K_m:L_1([0,1])\to C([0,1])$ defined for any function $f\in L_1([0,1])$$f\in L_1([0,1])$f inL_(1)([0,1])f \in L_{1}([0,1])$f\in L_1([0,1])$ by

for any $x\in [0,1]$$x\in [0,1]$x in[0,1]x \in[0,1]$x\in [0,1]$.
The operators $K_m$$K_m$K_(m)K_{m}$K_m$, where $m$$m$mm$m$ is a nonzero natural number, are named Kantorovich operators, introduced and studied in 1930 by L.V. Kantorovich (see [10]).

For $0\le \alpha \le \beta$$0\le \alpha \le \beta$0 <= alpha <= beta0 \leq \alpha \leq \beta$0\le \alpha \le \beta$ and $m$$m$mm$m$ a nonzero natural number, define $P_m^{(\alpha ,\beta )}:C([0,1])\to C([0,1])$$P_m^{(\alpha ,\beta )}:C([0,1])\to C([0,1])$P_(m)^((alpha,beta)):C([0,1])rarr C([0,1])P_{m}^{(\alpha, \beta)}: C([0,1]) \rightarrow C([0,1])$P_m^{(\alpha ,\beta )}:C([0,1])\to C([0,1])$ for any function $f\in C([0,1])$$f\in C([0,1])$f in C([0,1])f \in C([0,1])$f\in C([0,1])$ by

for any $x\in [0,1]$$x\in [0,1]$x in[0,1]x \in[0,1]$x\in [0,1]$.
The operators $P_m^{(\alpha ,\beta )}$$P_m^{(\alpha ,\beta )}$P_(m)^((alpha,beta))P_{m}^{(\alpha, \beta)}$P_m^{(\alpha ,\beta )}$, where $m$$m$mm$m$ is a nonzero natural number, are named Bernstein-Stancu operators, introduced and studied in 1969 by D.D. Stancu in the paper [12].

In [12] is the result contained in the following theorem.
Theorem 1.4. Let $f\in C([0,1])$$f\in C([0,1])$f in C([0,1])f \in C([0,1])$f\in C([0,1])$ be a two times derivable function at the point $x\in [0,1]$$x\in [0,1]$x in[0,1]x \in[0,1]$x\in [0,1]$. Then the equality

holds.
We consider $I\subset \mathbb{R},I$$I\subset \mathbb{R},I$I subR,II \subset \mathbb{R}, I$I\subset \mathbb{R},I$ an interval and we shall use the function sets: $E(I)$$E(I)$E(I)E(I)$E(I)$, $F(I)$$F(I)$F(I)F(I)$F(I)$ which are subsets of the set of real functions defined on $I,B(I)=\{f\mid f:I\to \mathbb{R},f$$I,B(I)=\{f\mid f:I\to \mathbb{R},f$I,B(I)={f∣f:I rarrR,fI, B(I)=\{f \mid f: I \rightarrow \mathbb{R}, f$I,B(I)=\{f\mid f:I\to \mathbb{R},f$ bounded on $I\},C(I)=\{f\mid f:I\to \mathbb{R},f$$I\},C(I)=\{f\mid f:I\to \mathbb{R},f$I},C(I)={f∣f:I rarrR,fI\}, C(I)=\{f \mid f: I \rightarrow \mathbb{R}, f$I\},C(I)=\{f\mid f:I\to \mathbb{R},f$ continuous on $I\}$$I\}$I}I\}$I\}$ and $C_B(I)=B(I)\cap C(I)$$C_B(I)=B(I)\cap C(I)$C_(B)(I)=B(I)nn C(I)C_{B}(I)=B(I) \cap C(I)$C_B(I)=B(I)\cap C(I)$. For any $x\in I$$x\in I$x in Ix \in I$x\in I$, let the functions ${\phi }_x,{\psi }_x:I\to \mathbb{R}$${\phi }_x,{\psi }_x:I\to \mathbb{R}$varphi_(x),psi_(x):I rarrR\varphi_{x}, \psi_{x}: I \rightarrow \mathbb{R}${\phi }_x,{\psi }_x:I\to \mathbb{R}$, ${\phi }_x(t)=|t-x|,{\psi }_x(t)=t-x$${\phi }_x(t)=|t-x|,{\psi }_x(t)=t-x$varphi_(x)(t)=|t-x|,psi_(x)(t)=t-x\varphi_{x}(t)=|t-x|, \psi_{x}(t)=t-x${\phi }_x(t)=|t-x|,{\psi }_x(t)=t-x$, for any $t\in I$$t\in I$t in It \in I$t\in I$.

Definition 1.5. If $I\subset \mathbb{R}$$I\subset \mathbb{R}$I subRI \subset \mathbb{R}$I\subset \mathbb{R}$ is a given interval and $f\in B(I)$$f\in B(I)$f in B(I)f \in B(I)$f\in B(I)$, then the first order modulus of smoothness of $f$$f$ff$f$ is the function ${\omega }_1:[0,\mathrm{\infty })\to \mathbb{R}$${\omega }_1:[0,\mathrm{\infty })\to \mathbb{R}$omega_(1):[0,oo)rarrR\omega_{1}:[0, \infty) \rightarrow \mathbb{R}${\omega }_1:[0,\mathrm{\infty })\to \mathbb{R}$ defined for any $\delta \ge 0$$\delta \ge 0$delta >= 0\delta \geq 0$\delta \ge 0$ by

In the following, we take into account the properties of the first order modulus of smoothness and the properties of the linear positive functional.

Lemma 1.6. If $f\in C_B(I)$$f\in C_B(I)$f inC_(B)(I)f \in C_{B}(I)$f\in C_B(I)$, then ${\omega }_1(f;\cdot )$${\omega }_1(f;\cdot )$omega_(1)(f;*)\omega_{1}(f ; \cdot)${\omega }_1(f;\cdot )$ have the following properties
а) ${\omega }_1(f;0)=0$${\omega }_1(f;0)=0$omega_(1)(f;0)=0\omega_{1}(f ; 0)=0${\omega }_1(f;0)=0$;
b) ${\omega }_1(f;\cdot )$${\omega }_1(f;\cdot )$omega_(1)(f;*)\omega_{1}(f ; \cdot)${\omega }_1(f;\cdot )$ is increasing function;
c) ${\omega }_1(f;\cdot )$${\omega }_1(f;\cdot )$omega_(1)(f;*)\omega_{1}(f ; \cdot)${\omega }_1(f;\cdot )$ is uniform continuous function;
for any $\delta >0$$\delta >0$delta > 0\delta>0$\delta >0$, for any $x,t\in I$$x,t\in I$x,t in Ix, t \in I$x,t\in I$, we have
d) ${\omega }_1(f;{\phi }_x(t))\le [1+{\delta }^{-1}{\phi }_x(t)]{\omega }_1(f;\delta )$${\omega }_1\left(f;{\phi }_x(t)\right)\le \left[1+{\delta }^{-1}{\phi }_x(t)\right]{\omega }_1(f;\delta )$omega_(1)(f;varphi_(x)(t)) <= [1+delta^(-1)varphi_(x)(t)]omega_(1)(f;delta)\omega_{1}\left(f ; \varphi_{x}(t)\right) \leq\left[1+\delta^{-1} \varphi_{x}(t)\right] \omega_{1}(f ; \delta)${\omega }_1(f;{\phi }_x(t))\le [1+{\delta }^{-1}{\phi }_x(t)]{\omega }_1(f;\delta )$
and
e) $|f(t)-f(x)|\le [1+{\delta }^{-2}{\psi }_x^2(t)]{\omega }_1(f;\delta )$$|f(t)-f(x)|\le \left[1+{\delta }^{-2}{\psi }_x^2(t)\right]{\omega }_1(f;\delta )$|f(t)-f(x)| <= [1+delta^(-2)psi_(x)^(2)(t)]omega_(1)(f;delta)|f(t)-f(x)| \leq\left[1+\delta^{-2} \psi_{x}^{2}(t)\right] \omega_{1}(f ; \delta)$|f(t)-f(x)|\le [1+{\delta }^{-2}{\psi }_x^2(t)]{\omega }_1(f;\delta )$.

Proof. For proof see [10].
Lemma 1.7. Let $A:E(I)\to \mathbb{R}$$A:E(I)\to \mathbb{R}$A:E(I)rarrRA: E(I) \rightarrow \mathbb{R}$A:E(I)\to \mathbb{R}$ be a linear positive functional. Then
a) for any $f,g\in E(I)$$f,g\in E(I)$f,g in E(I)f, g \in E(I)$f,g\in E(I)$ with $f(x)\le g(x)$$f(x)\le g(x)$f(x) <= g(x)f(x) \leq g(x)$f(x)\le g(x)$, for any $x\in I$$x\in I$x in Ix \in I$x\in I$, we have $A(f)\le A(g)$$A(f)\le A(g)$A(f) <= A(g)A(f) \leq A(g)$A(f)\le A(g)$ and
b) $|A(f)|\le A(|f|)$$|A(f)|\le A(|f|)$|A(f)| <= A(|f|)|A(f)| \leq A(|f|)$|A(f)|\le A(|f|)$, for any $f\in E(I)$$f\in E(I)$f in E(I)f \in E(I)$f\in E(I)$.

Proof. For proof consult [10].

Theorem 2.1. Let $I\subset \mathbb{R}$$I\subset \mathbb{R}$I subRI \subset \mathbb{R}$I\subset \mathbb{R}$ be an interval, $a\in I,n\in \mathbb{N}$$a\in I,n\in \mathbb{N}$a in I,n inNa \in I, n \in \mathbb{N}$a\in I,n\in \mathbb{N}$ and the function $f:I\to \mathbb{R}$$f:I\to \mathbb{R}$f:I rarrRf: I \rightarrow \mathbb{R}$f:I\to \mathbb{R}$, $f$$f$ff$f$ is $n$$n$nn$n$ times derivable at a. According to Taylor's expansion theorem for the function $f$$f$ff$f$ around $a$$a$aa$a$, we have

where $\mu$$\mu$mu\mu$\mu$ is a bounded function and $\underset{x\to a}{lim}\mu (x-a)=0$$\underset{x\to a}{lim} \mu (x-a)=0$lim_(x rarr a)mu(x-a)=0\lim _{x \rightarrow a} \mu(x-a)=0$\underset{x\to a}{lim}\mu (x-a)=0$.
If $f^{(n)}$$f^{(n)}$f^((n))f^{(n)}$f^{(n)}$ is continuous function on $I$$I$II$I$, then for any $\delta >0$$\delta >0$delta > 0\delta>0$\delta >0$

and

for any $x\in I$$x\in I$x in Ix \in I$x\in I$.
Proof. If $n=0$$n=0$n=0n=0$n=0$, the proof is immediately. Let $n$$n$nn$n$ be a nonzero natural number. According to Taylor's expansion with the Lagrange's remainder, we have

where $\xi$$\xi$xi\xi$\xi$ is between $a$$a$aa$a$ and $x$$x$xx$x$. From (2.1) and (2.4), we obtain $\mu (x-a)=\frac{1}{n!}[f^{(n)}(\xi )-f^{(n)}(a)]$$\mu (x-a)=\frac{1}{n!}\left[f^{(n)}(\xi )-f^{(n)}(a)\right]$mu(x-a)=(1)/(n!)[f^((n))(xi)-f^((n))(a)]\mu(x-a)= \frac{1}{n!}\left[f^{(n)}(\xi)-f^{(n)}(a)\right]$\mu (x-a)=\frac{1}{n!}[f^{(n)}(\xi )-f^{(n)}(a)]$ and because $|\xi -a|\le |x-a|$$|\xi -a|\le |x-a|$|xi-a| <= |x-a||\xi-a| \leq|x-a|$|\xi -a|\le |x-a|$, we have

Taking Lemma 1.6 into account, the inequalities 2.2 and 2.3 follow.
Let $a,b,a^{\prime },b^{\prime }$$a,b,a^{\prime },b^{\prime }$a,b,a^('),b^(')a, b, a^{\prime}, b^{\prime}$a,b,a^{\prime },b^{\prime }$ be real numbers, $I\subset \mathbb{R}$$I\subset \mathbb{R}$I subRI \subset \mathbb{R}$I\subset \mathbb{R}$ interval, $a<b,a^{\prime }<b^{\prime },[a,b]\subset I$$a<b,a^{\prime }<b^{\prime },[a,b]\subset I$a < b,a^(') < b^('),[a,b]sub Ia<b, a^{\prime}<b^{\prime},[a, b] \subset I$a<b,a^{\prime }<b^{\prime },[a,b]\subset I$, $[a^{\prime },b^{\prime }]\subset I$$\left[a^{\prime },b^{\prime }\right]\subset I$[a^('),b^(')]sub I\left[a^{\prime}, b^{\prime}\right] \subset I$[a^{\prime },b^{\prime }]\subset I$ and $[a,b]\cap [a^{\prime },b^{\prime }]\ne \varphi$$[a,b]\cap \left[a^{\prime },b^{\prime }\right]\ne \varphi$[a,b]nn[a^('),b^(')]!=phi[a, b] \cap\left[a^{\prime}, b^{\prime}\right] \neq \phi$[a,b]\cap [a^{\prime },b^{\prime }]\ne \varphi$. For any nonzero natural number $m$$m$mm$m$, consider the functions ${\phi }_{m,k}:I\to \mathbb{R}$${\phi }_{m,k}:I\to \mathbb{R}$varphi_(m,k):I rarrR\varphi_{m, k}: I \rightarrow \mathbb{R}${\phi }_{m,k}:I\to \mathbb{R}$ with the property that ${\phi }_{m,k}(x)\ge 0$${\phi }_{m,k}(x)\ge 0$varphi_(m,k)(x) >= 0\varphi_{m, k}(x) \geq 0${\phi }_{m,k}(x)\ge 0$ for any $x\in [a^{\prime },b^{\prime }]$$x\in \left[a^{\prime },b^{\prime }\right]$x in[a^('),b^(')]x \in\left[a^{\prime}, b^{\prime}\right]$x\in [a^{\prime },b^{\prime }]$ and any $k\in \{0,1,,\dots ,m\}$$k\in \{0,1,,\dots ,m\}$k in{0,1,,dots,m}k \in\{0,1,, \ldots, m\}$k\in \{0,1,,\dots ,m\}$ and the linear positive functionals $A_{m,k}:E([a,b])\to \mathbb{R}$$A_{m,k}:E([a,b])\to \mathbb{R}$A_(m,k):E([a,b])rarrRA_{m, k}: E([a, b]) \rightarrow \mathbb{R}$A_{m,k}:E([a,b])\to \mathbb{R}$ for any $k\in \{0,1,\dots ,m\}$$k\in \{0,1,\dots ,m\}$k in{0,1,dots,m}k \in\{0,1, \ldots, m\}$k\in \{0,1,\dots ,m\}$.

Definition 2.2. Let $m$$m$mm$m$ be a nonzero natural number. Define the operator $L_m:E([a,b])\to F(I)$$L_m:E([a,b])\to F(I)$L_(m):E([a,b])rarr F(I)L_{m}: E([a, b]) \rightarrow F(I)$L_m:E([a,b])\to F(I)$ by

for any $f\in E([a,b])$$f\in E([a,b])$f in E([a,b])f \in E([a, b])$f\in E([a,b])$ and for any $x\in \mathrm{I}$$x\in \mathrm{I}$x inIx \in \mathrm{I}$x\in \mathrm{I}$.

Proposition 2.3. For $m$$m$mm$m$ be a nonzero natural number, the $L_m$$L_m$L_(m)L_{m}$L_m$ operators are linear and positive on $E([a,b]\cap [a^{\prime },b^{\prime }])$$E\left([a,b]\cap \left[a^{\prime },b^{\prime }\right]\right)$E([a,b]nn[a^('),b^(')])E\left([a, b] \cap\left[a^{\prime}, b^{\prime}\right]\right)$E([a,b]\cap [a^{\prime },b^{\prime }])$.

Proof. The proof follows immediately.
Definition 2.4. Let $m$$m$mm$m$ be a nonzero natural number and $L_m:E([a,b])\to F(I)$$L_m:E([a,b])\to F(I)$L_(m):E([a,b])rarr F(I)L_{m}: E([a, b]) \rightarrow F(I)$L_m:E([a,b])\to F(I)$ be an operator defined in (2.5). For a natural number $i$$i$ii$i$, define $T_{m,i}^{\ast }$$T_{m,i}^{\ast }$T_(m,i)^(**)T_{m, i}^{*}$T_{m,i}^{\ast }$