Return to Article Details On an approximating linear positive operator of Cheney-Sharma

ON AN APPROXIMATING LINEAR POSITIVE OPERATOR OF CHENEY-SHARMA

D. D. STANCU, C. CISMAŞIU

1. INTRODUCTION

It is known that by starting from two combinatorial identities of Abel-Jensen
(1.1) ( u + v + m β ) m = ∑ k = 0 m ( m k ) u ( u + k β ) k − 1 [ v + ( m − k ) β ] m − k ( u + v + m β ) m = ∑ k = 0 m   ( m k ) u ( u + k β ) k − 1 [ v + ( m − k ) β ] m − k quad(u+v+m beta)^(m)=sum_(k=0)^(m)((m)/(k))u(u+k beta)^(k-1)[v+(m-k)beta]^(m-k)\quad(u+v+m \beta)^{m}=\sum_{k=0}^{m}\binom{m}{k} u(u+k \beta)^{k-1}[v+(m-k) \beta]^{m-k}(u+v+mβ)m=∑k=0m(mk)u(u+kβ)k−1[v+(m−k)β]m−k,
(1.2) ( u + v ) ( u + v + m β ) m − 1 = ∑ k = 0 m ( m k ) u ( u + k β ) k − 1 v [ v + ( m − k ) β ] m − 1 − k ( u + v ) ( u + v + m β ) m − 1 = ∑ k = 0 m   ( m k ) u ( u + k β ) k − 1 v [ v + ( m − k ) β ] m − 1 − k (u+v)(u+v+m beta)^(m-1)=sum_(k=0)^(m)((m)/(k))u(u+k beta)^(k-1)v[v+(m-k)beta]^(m-1-k)(u+v)(u+v+m \beta)^{m-1}=\sum_{k=0}^{m}\binom{m}{k} u(u+k \beta)^{k-1} v[v+(m-k) \beta]^{m-1-k}(u+v)(u+v+mβ)m−1=∑k=0m(mk)u(u+kβ)k−1v[v+(m−k)β]m−1−k,
Cheney and Sharma [1] have introduced and investigated two linear polynomial positive operators, of Bernstein type, P m P m P_(m)P_{m}Pm and Q m Q m Q_(m)Q_{m}Qm, defined - for any function f : [ 0 , 1 ] → R f : [ 0 , 1 ] → R f:[0,1]rarrRf:[0,1] \rightarrow \mathbf{R}f:[0,1]→R - by the following formulas
(1.3) ( P m f ) ( x ; β ) = ∑ k = 0 m p m , k ( x ; β ) f ( k m ) (1.4) ( Q m f ) ( x ; β ) = ∑ k = 0 m q m , k ( x ; β ) f ( k m ) (1.3) P m f ( x ; β ) = ∑ k = 0 m   p m , k ( x ; β ) f k m (1.4) Q m f ( x ; β ) = ∑ k = 0 m   q m , k ( x ; β ) f k m {:[(1.3)(P_(m)f)(x;beta)=sum_(k=0)^(m)p_(m,k)(x;beta)f((k)/(m))],[(1.4)(Q_(m)f)(x;beta)=sum_(k=0)^(m)q_(m,k)(x;beta)f((k)/(m))]:}\begin{align*} & \left(P_{m} f\right)(x ; \beta)=\sum_{k=0}^{m} p_{m, k}(x ; \beta) f\left(\frac{k}{m}\right) \tag{1.3}\\ & \left(Q_{m} f\right)(x ; \beta)=\sum_{k=0}^{m} q_{m, k}(x ; \beta) f\left(\frac{k}{m}\right) \tag{1.4} \end{align*}(1.3)(Pmf)(x;β)=∑k=0mpm,k(x;β)f(km)(1.4)(Qmf)(x;β)=∑k=0mqm,k(x;β)f(km)
where
(1.5) p m , k ( x ; β ) = ( m k ) x ( x + k β ) k − 1 [ 1 − x + ( m − k ) β ] m − k ( 1 + m β ) m (1.6) ( x ; β ) = ( m k ) x ( x + k β ) k − 1 ( 1 − x ) [ 1 − x + ( m − k ) β ] m − 1 − k ( 1 + m β ) m − 1 (1.5) p m , k ( x ; β ) = ( m k ) x ( x + k β ) k − 1 [ 1 − x + ( m − k ) β ] m − k ( 1 + m β ) m (1.6) ( x ; β ) = ( m k ) x ( x + k β ) k − 1 ( 1 − x ) [ 1 − x + ( m − k ) β ] m − 1 − k ( 1 + m β ) m − 1 {:[(1.5)p_(m,k)(x;beta)=((m)/(k))(x(x+k beta)^(k-1)[1-x+(m-k)beta]^(m-k))/((1+m beta)^(m))],[(1.6)(x;beta)=((m)/(k))(x(x+k beta)^(k-1)(1-x)[1-x+(m-k)beta]^(m-1-k))/((1+m beta)^(m-1))]:}\begin{align*} & p_{m, k}(x ; \beta)=\binom{m}{k} \frac{x(x+k \beta)^{k-1}[1-x+(m-k) \beta]^{m-k}}{(1+m \beta)^{m}} \tag{1.5}\\ & (x ; \beta)=\binom{m}{k} \frac{x(x+k \beta)^{k-1}(1-x)[1-x+(m-k) \beta]^{m-1-k}}{(1+m \beta)^{m-1}} \tag{1.6} \end{align*}(1.5)pm,k(x;β)=(mk)x(x+kβ)k−1[1−x+(m−k)β]m−k(1+mβ)m(1.6)(x;β)=(mk)x(x+kβ)k−1(1−x)[1−x+(m−k)β]m−1−k(1+mβ)m−1
It is obvious that for β = 0 β = 0 beta=0\beta=0β=0 these operators reduce to the classical operator B m B m B_(m)B_{m}Bm of Bernstein.
In this paper we prove that the second operator Q m Q m Q_(m)Q_{m}Qm preserves the linear functions and we establish several expressions for the remainder term in the corresponding approximation formula.

2. THE VALUE OF THE OPERATOR Q m Q m Q_(m)Q_{m}Qm FOR THE MONOMIAL e 1 e 1 e_(1)e_{1}e1

In [1] it was pointed out that the operator Q m Q m Q_(m)Q_{m}Qm preserves only the constant functions, after calculation of some integrals involved. But we shall prove that Q m Q m Q_(m)Q_{m}Qm preserves the linear functions.
It is easy to see that the following theorem is true.
THEOREM 2.1. The approximating polynomial Q m f Q m f Q_(m)fQ_{m} fQmf is interpolatory at both sides of the interval [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1], for any nonnegative value of the parameter β β beta\betaβ.
Proof. In order to prove this result, we have only to observe that we can write
( Q m f ) ( x ; β ) = 1 ( 1 + m β ) m − 1 { ( 1 − x ) ( 1 − x + m β ) m − 1 f ( 0 ) − − x ( x − 1 ) ∑ k = 1 m − 1 ( m k ) ( x + k β ) k − 1 [ 1 − x + ( m − k ) β ] m − 1 − k f ( k m ) + x ( x + m β ) m − 1 f ( 1 ) } Q m f ( x ; β ) = 1 ( 1 + m β ) m − 1 ( 1 − x ) ( 1 − x + m β ) m − 1 f ( 0 ) − − x ( x − 1 ) ∑ k = 1 m − 1   ( m k ) ( x + k β ) k − 1 [ 1 − x + ( m − k ) β ] m − 1 − k f k m + x ( x + m β ) m − 1 f ( 1 ) {:[(Q_(m)f)(x;beta)=(1)/((1+m beta)^(m-1)){(1-x)(1-x+m beta)^(m-1)f(0)-:}],[{:-x(x-1)sum_(k=1)^(m-1)((m)/(k))(x+k beta)^(k-1)[1-x+(m-k)beta]^(m-1-k)f((k)/(m))+x(x+m beta)^(m-1)f(1)}]:}\begin{gathered} \left(Q_{m} f\right)(x ; \beta)=\frac{1}{(1+m \beta)^{m-1}}\left\{(1-x)(1-x+m \beta)^{m-1} f(0)-\right. \\ \left.-x(x-1) \sum_{k=1}^{m-1}\binom{m}{k}(x+k \beta)^{k-1}[1-x+(m-k) \beta]^{m-1-k} f\left(\frac{k}{m}\right)+x(x+m \beta)^{m-1} f(1)\right\} \end{gathered}(Qmf)(x;β)=1(1+mβ)m−1{(1−x)(1−x+mβ)m−1f(0)−−x(x−1)∑k=1m−1(mk)(x+kβ)k−1[1−x+(m−k)β]m−1−kf(km)+x(x+mβ)m−1f(1)}
Let us consider next the monomials e j ( t ) = t j ( j ≥ 0 ) e j ( t ) = t j ( j ≥ 0 ) e_(j)(t)=t^(j)(j >= 0)e_{j}(t)=t^{j}(j \geq 0)ej(t)=tj(j≥0) for any t ∈ [ 0 , 1 ] t ∈ [ 0 , 1 ] t in[0,1]t \in[0,1]t∈[0,1]. We shall now state and prove
Theorem 2.2. The operator Q m Q m Q_(m)Q_{m}Qm reproduces the linear functions.
Proof. As it has been observed in [1], if we replace in the identity (1.2) u = x u = x u=xu=xu=x and v = 1 − x v = 1 − x v=1-xv=1-xv=1−x, we find that Q m e 0 = e 0 Q m e 0 = e 0 Q_(m)e_(0)=e_(0)Q_{m} e_{0}=e_{0}Qme0=e0, that is, the operator Q m Q m Q_(m)Q_{m}Qm reproduces the constants.
We shall prove that we also have Q m e 1 = e 1 Q m e 1 = e 1 Q_(m)e_(1)=e_(1)Q_{m} e_{1}=e_{1}Qme1=e1.
Indeed, one can see that we can write
(2.1) ( 1 + m β ) m − 1 ( Q m e 1 ) ( x ; β ) = = ∑ k = 1 m k m ( m k ) x ( x + k β ) k − 1 ( 1 − x ) [ 1 − x + ( m − k ) β ] m − 1 − k = = ∑ k = 1 m ( m − 1 k − 1 ) x ( x + k β ) k − 1 ( 1 − x ) [ 1 − x + ( m − k ) β ] m − 1 − k (2.1) ( 1 + m β ) m − 1 Q m e 1 ( x ; β ) = = ∑ k = 1 m   k m ( m k ) x ( x + k β ) k − 1 ( 1 − x ) [ 1 − x + ( m − k ) β ] m − 1 − k = = ∑ k = 1 m   ( m − 1 k − 1 ) x ( x + k β ) k − 1 ( 1 − x ) [ 1 − x + ( m − k ) β ] m − 1 − k {:[(2.1)(1+m beta)^(m-1)(Q_(m)e_(1))(x;beta)=],[=sum_(k=1)^(m)(k)/(m)((m)/(k))x(x+k beta)^(k-1)(1-x)[1-x+(m-k)beta]^(m-1-k)=],[=sum_(k=1)^(m)((m-1)/(k-1))x(x+k beta)^(k-1)(1-x)[1-x+(m-k)beta]^(m-1-k)]:}\begin{gather*} (1+m \beta)^{m-1}\left(Q_{m} e_{1}\right)(x ; \beta)= \tag{2.1}\\ =\sum_{k=1}^{m} \frac{k}{m}\binom{m}{k} x(x+k \beta)^{k-1}(1-x)[1-x+(m-k) \beta]^{m-1-k}= \\ =\sum_{k=1}^{m}\binom{m-1}{k-1} x(x+k \beta)^{k-1}(1-x)[1-x+(m-k) \beta]^{m-1-k} \end{gather*}(2.1)(1+mβ)m−1(Qme1)(x;β)==∑k=1mkm(mk)x(x+kβ)k−1(1−x)[1−x+(m−k)β]m−1−k==∑k=1m(m−1k−1)x(x+kβ)k−1(1−x)[1−x+(m−k)β]m−1−k
because
(2.2) k m ( m k ) = ( m − 1 k − 1 ) . (2.2) k m ( m k ) = ( m − 1 k − 1 ) . {:(2.2)(k)/(m)((m)/(k))=((m-1)/(k-1)).:}\begin{equation*} \frac{k}{m}\binom{m}{k}=\binom{m-1}{k-1} . \tag{2.2} \end{equation*}(2.2)km(mk)=(m−1k−1).
If we change k − 1 = j k − 1 = j k-1=jk-1=jk−1=j and then denote again the index of summation by k k kkk, we have
(2.3) = ∑ k = 0 m − 1 ( m − 1 k ) x ( x + β + k β ) k ( 1 − x ) [ 1 − x + ( m − 1 − k ) β ] m − 2 − k = = ( 1 + m β ) ∑ k = 0 m − 1 ( m − 1 k ) x ( x + β + k β ) k − 1 ( 1 − x ) [ 1 − x + ( m − 1 − k ) β ] m − 2 − k − − ∑ k = 0 m − 1 ( m − 1 k ) x ( x + β + k β ) k − 1 ( 1 − x ) [ 1 − x + ( m − 1 − k ) β ] m − 1 − k , (2.3) = ∑ k = 0 m − 1   ( m − 1 k ) x ( x + β + k β ) k ( 1 − x ) [ 1 − x + ( m − 1 − k ) β ] m − 2 − k = = ( 1 + m β ) ∑ k = 0 m − 1   ( m − 1 k ) x ( x + β + k β ) k − 1 ( 1 − x ) [ 1 − x + ( m − 1 − k ) β ] m − 2 − k − − ∑ k = 0 m − 1   ( m − 1 k ) x ( x + β + k β ) k − 1 ( 1 − x ) [ 1 − x + ( m − 1 − k ) β ] m − 1 − k , {:[(2.3)=sum_(k=0)^(m-1)((m-1)/(k))x(x+beta+k beta)^(k)(1-x)[1-x+(m-1-k)beta]^(m-2-k)=],[=(1+m beta)sum_(k=0)^(m-1)((m-1)/(k))x(x+beta+k beta)^(k-1)(1-x)[1-x+(m-1-k)beta]^(m-2-k)-],[-sum_(k=0)^(m-1)((m-1)/(k))x(x+beta+k beta)^(k-1)(1-x)[1-x+(m-1-k)beta]^(m-1-k)","]:}\begin{align*} = & \sum_{k=0}^{m-1}\binom{m-1}{k} x(x+\beta+k \beta)^{k}(1-x)[1-x+(m-1-k) \beta]^{m-2-k}= \tag{2.3}\\ = & (1+m \beta) \sum_{k=0}^{m-1}\binom{m-1}{k} x(x+\beta+k \beta)^{k-1}(1-x)[1-x+(m-1-k) \beta]^{m-2-k}- \\ & -\sum_{k=0}^{m-1}\binom{m-1}{k} x(x+\beta+k \beta)^{k-1}(1-x)[1-x+(m-1-k) \beta]^{m-1-k}, \end{align*}(2.3)=∑k=0m−1(m−1k)x(x+β+kβ)k(1−x)[1−x+(m−1−k)β]m−2−k==(1+mβ)∑k=0m−1(m−1k)x(x+β+kβ)k−1(1−x)[1−x+(m−1−k)β]m−2−k−−∑k=0m−1(m−1k)x(x+β+kβ)k−1(1−x)[1−x+(m−1−k)β]m−1−k,
since
x + β + k β = ( 1 + m β ) − [ 1 − x + ( m − 1 − k ) β ] x + β + k β = ( 1 + m β ) − [ 1 − x + ( m − 1 − k ) β ] x+beta+k beta=(1+m beta)-[1-x+(m-1-k)beta]x+\beta+k \beta=(1+m \beta)-[1-x+(m-1-k) \beta]x+β+kβ=(1+mβ)−[1−x+(m−1−k)β]
In order to find the first sum, we replace in the identity (1.2) m m mmm by m − 1 m − 1 m-1m-1m−1 and u = x + β , v = 1 − x u = x + β , v = 1 − x u=x+beta,v=1-xu=x+\beta, v=1-xu=x+β,v=1−x.
We get
( 1 + β ) ( 1 + m β ) m − 2 = ( x + β ) ∑ k = 0 m − 1 ( m − 1 k ) ( x + β + k β ) k − 1 ( 1 − x ) ⋅ [ 1 − x + ( m − 1 − k ) β ] m − 2 − k ( 1 + β ) ( 1 + m β ) m − 2 = ( x + β ) ∑ k = 0 m − 1   ( m − 1 k ) ( x + β + k β ) k − 1 ( 1 − x ) ⋅ [ 1 − x + ( m − 1 − k ) β ] m − 2 − k {:[(1+beta)(1+m beta)^(m-2)=(x+beta)sum_(k=0)^(m-1)((m-1)/(k))(x+beta+k beta)^(k-1)(1-x)],[*[1-x+(m-1-k)beta]^(m-2-k)]:}\begin{gathered} (1+\beta)(1+m \beta)^{m-2}=(x+\beta) \sum_{k=0}^{m-1}\binom{m-1}{k}(x+\beta+k \beta)^{k-1}(1-x) \\ \cdot[1-x+(m-1-k) \beta]^{m-2-k} \end{gathered}(1+β)(1+mβ)m−2=(x+β)∑k=0m−1(m−1k)(x+β+kβ)k−1(1−x)⋅[1−x+(m−1−k)β]m−2−k
If we multiply by x x xxx and divide by x + β x + β x+betax+\betax+β, we obtain
∑ k = 0 m − 1 ( m − 1 k ) x ( x + β + k β ) k − 1 ( 1 − x ) [ 1 − x + ( m − 1 − k ) β ] m − 2 − k = = ( 1 + β ) ( 1 + m β ) m − 2 x x + β ∑ k = 0 m − 1   ( m − 1 k ) x ( x + β + k β ) k − 1 ( 1 − x ) [ 1 − x + ( m − 1 − k ) β ] m − 2 − k = = ( 1 + β ) ( 1 + m β ) m − 2 x x + β {:[sum_(k=0)^(m-1)((m-1)/(k))x(x+beta+k beta)^(k-1)(1-x)[1-x+(m-1-k)beta]^(m-2-k)=],[=(1+beta)(1+m beta)^(m-2)(x)/(x+beta)]:}\begin{gathered} \sum_{k=0}^{m-1}\binom{m-1}{k} x(x+\beta+k \beta)^{k-1}(1-x)[1-x+(m-1-k) \beta]^{m-2-k}= \\ =(1+\beta)(1+m \beta)^{m-2} \frac{x}{x+\beta} \end{gathered}∑k=0m−1(m−1k)x(x+β+kβ)k−1(1−x)[1−x+(m−1−k)β]m−2−k==(1+β)(1+mβ)m−2xx+β
which represents the first sum.
For finding the second sum we shall use the identity (1.2). We replace m m mmm by m − 1 m − 1 m-1m-1m−1 and u = x + β , v = 1 − x u = x + β , v = 1 − x u=x+beta,v=1-xu=x+\beta, v=1-xu=x+β,v=1−x and we find
( 1 + m β ) m − 1 = ( x + β ) ∑ k = 0 m − 1 ( m − 1 k ) ( x + β + k β ) k − 1 [ 1 − x + ( m − 1 − k ) β ] m − 1 − k ( 1 + m β ) m − 1 = ( x + β ) ∑ k = 0 m − 1   ( m − 1 k ) ( x + β + k β ) k − 1 [ 1 − x + ( m − 1 − k ) β ] m − 1 − k (1+m beta)^(m-1)=(x+beta)sum_(k=0)^(m-1)((m-1)/(k))(x+beta+k beta)^(k-1)[1-x+(m-1-k)beta]^(m-1-k)(1+m \beta)^{m-1}=(x+\beta) \sum_{k=0}^{m-1}\binom{m-1}{k}(x+\beta+k \beta)^{k-1}[1-x+(m-1-k) \beta]^{m-1-k}(1+mβ)m−1=(x+β)∑k=0m−1(m−1k)(x+β+kβ)k−1[1−x+(m−1−k)β]m−1−k.
It follows that we can write
∑ k = 0 m − 1 ( m − 1 k ) x ( x + β + k β ) k − 1 ( 1 − x ) [ 1 − x + ( m − 1 − k ) β ] m − 1 − k = = ( 1 + m β ) m − 1 x ( 1 − x ) x + β ∑ k = 0 m − 1   ( m − 1 k ) x ( x + β + k β ) k − 1 ( 1 − x ) [ 1 − x + ( m − 1 − k ) β ] m − 1 − k = = ( 1 + m β ) m − 1 x ( 1 − x ) x + β {:[sum_(k=0)^(m-1)((m-1)/(k))x(x+beta+k beta)^(k-1)(1-x)[1-x+(m-1-k)beta]^(m-1-k)=],[=(1+m beta)^(m-1)(x(1-x))/(x+beta)]:}\begin{gathered} \sum_{k=0}^{m-1}\binom{m-1}{k} x(x+\beta+k \beta)^{k-1}(1-x)[1-x+(m-1-k) \beta]^{m-1-k}= \\ =(1+m \beta)^{m-1} \frac{x(1-x)}{x+\beta} \end{gathered}∑k=0m−1(m−1k)x(x+β+kβ)k−1(1−x)[1−x+(m−1−k)β]m−1−k==(1+mβ)m−1x(1−x)x+β
Consequently, we have
(2.4) ( Q m e 1 ) ( x ; β ) = 1 ( 1 + m β ) m − 1 [ ( 1 + m β ) ( 1 + β ) ( 1 + m β ) m − 2 x x + β − − ( 1 + m β ) m − 1 x ( 1 − x ) x + β ] = x x + β [ 1 + β − ( 1 − x ) ] = x (2.4) Q m e 1 ( x ; β ) = 1 ( 1 + m β ) m − 1 ( 1 + m β ) ( 1 + β ) ( 1 + m β ) m − 2 x x + β − − ( 1 + m β ) m − 1 x ( 1 − x ) x + β = x x + β [ 1 + β − ( 1 − x ) ] = x {:[(2.4)(Q_(m)e_(1))(x;beta)=(1)/((1+m beta)^(m-1))[(1+m beta)(1+beta)(1+m beta)^(m-2)(x)/(x+beta)-:}],[{:-(1+m beta)^(m-1)(x(1-x))/(x+beta)]=(x)/(x+beta)[1+beta-(1-x)]=x]:}\begin{gather*} \left(Q_{m} e_{1}\right)(x ; \beta)=\frac{1}{(1+m \beta)^{m-1}}\left[(1+m \beta)(1+\beta)(1+m \beta)^{m-2} \frac{x}{x+\beta}-\right. \tag{2.4}\\ \left.-(1+m \beta)^{m-1} \frac{x(1-x)}{x+\beta}\right]=\frac{x}{x+\beta}[1+\beta-(1-x)]=x \end{gather*}(2.4)(Qme1)(x;β)=1(1+mβ)m−1[(1+mβ)(1+β)(1+mβ)m−2xx+β−−(1+mβ)m−1x(1−x)x+β]=xx+β[1+β−(1−x)]=x
Therefore we have
(2.5) Q m e 0 = e 0 , Q m e 1 = e 1 (2.5) Q m e 0 = e 0 , Q m e 1 = e 1 {:(2.5)Q_(m)e_(0)=e_(0)","Q_(m)e_(1)=e_(1):}\begin{equation*} Q_{m} e_{0}=e_{0}, Q_{m} e_{1}=e_{1} \tag{2.5} \end{equation*}(2.5)Qme0=e0,Qme1=e1
as in the case of the classical Bernstein operator B m B m B_(m)B_{m}Bm.

3. THE REMAINDER

Since the operator Q m Q m Q_(m)Q_{m}Qm reproduces the linear functions, it is clear that the approximation formula
(3.1) f ( x ) = ( Q m f ) ( x ; β ) + ( R m f ) ( x ; β ) (3.1) f ( x ) = Q m f ( x ; β ) + R m f ( x ; β ) {:(3.1)f(x)=(Q_(m)f)(x;beta)+(R_(m)f)(x;beta):}\begin{equation*} f(x)=\left(Q_{m} f\right)(x ; \beta)+\left(R_{m} f\right)(x ; \beta) \tag{3.1} \end{equation*}(3.1)f(x)=(Qmf)(x;β)+(Rmf)(x;β)
has the degree of exactness N = 1 N = 1 N=1N=1N=1.
First we shall give an integral representation of the remainder.
THEOREM 3.1. If the function f f fff has a continuous second derivative on the interval [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1], then we can represent the remainder of the approximation formula (3.1) under the following integral form
(3.2) ( R m f ) ( x ) = ∫ 0 1 G m ( t ; x ) f ′ ′ ( t ) d t (3.2) R m f ( x ) = ∫ 0 1   G m ( t ; x ) f ′ ′ ( t ) d t {:(3.2)(R_(m)f)(x)=int_(0)^(1)G_(m)(t;x)f^('')(t)dt:}\begin{equation*} \left(R_{m} f\right)(x)=\int_{0}^{1} G_{m}(t ; x) f^{\prime \prime}(t) \mathrm{d} t \tag{3.2} \end{equation*}(3.2)(Rmf)(x)=∫01Gm(t;x)f′′(t)dt
where
(3.3) G m ( t ; x ) = ( R m φ x ) ( t ) , φ x ( t ) = x − t + | x − t | 2 = ( x − t ) (3.3) G m ( t ; x ) = R m φ x ( t ) , φ x ( t ) = x − t + | x − t | 2 = ( x − t ) {:(3.3)G_(m)(t;x)=(R_(m)varphi_(x))(t)","quadvarphi_(x)(t)=(x-t+|x-t|)/(2)=(x-t):}\begin{equation*} G_{m}(t ; x)=\left(R_{m} \varphi_{x}\right)(t), \quad \varphi_{x}(t)=\frac{x-t+|x-t|}{2}=(x-t) \tag{3.3} \end{equation*}(3.3)Gm(t;x)=(Rmφx)(t),φx(t)=x−t+|x−t|2=(x−t)
and R m R m R_(m)R_{m}Rm operates on φ x ( t ) φ x ( t ) varphi_(x)(t)\varphi_{x}(t)φx(t) as a function of x x xxx.
Proof. The representation (3.2) can be obtained at once if we apply the wellknown theorem of Peano.
For the Peano kernel associated to the operator Q m Q m Q_(m)Q_{m}Qm we have
( R m φ x ) ( t ) = ( x − t ) + − ∑ k = 0 m q m , k ( x ; β ) ( k m − t ) + R m φ x ( t ) = ( x − t ) + − ∑ k = 0 m   q m , k ( x ; β ) k m − t + (R_(m)varphi_(x))(t)=(x-t)_(+)-sum_(k=0)^(m)q_(m,k)(x;beta)((k)/(m)-t)_(+)\left(R_{m} \varphi_{x}\right)(t)=(x-t)_{+}-\sum_{k=0}^{m} q_{m, k}(x ; \beta)\left(\frac{k}{m}-t\right)_{+}(Rmφx)(t)=(x−t)+−∑k=0mqm,k(x;β)(km−t)+
In order to find explicit expressions of this kernel, we assume that x ∈ [ s − 1 m , s m ] x ∈ s − 1 m , s m x in[(s-1)/(m),(s)/(m)]x \in\left[\frac{s-1}{m}, \frac{s}{m}\right]x∈[s−1m,sm] and we can write
(3.4) G m ( t ; x ) = x − t − ∑ k ≥ j q m , k ( x ; β ) ( k m − t ) (3.4) G m ( t ; x ) = x − t − ∑ k ≥ j   q m , k ( x ; β ) k m − t {:(3.4)G_(m)(t;x)=x-t-sum_(k >= j)q_(m,k)(x;beta)((k)/(m)-t):}\begin{equation*} G_{m}(t ; x)=x-t-\sum_{k \geq j} q_{m, k}(x ; \beta)\left(\frac{k}{m}-t\right) \tag{3.4} \end{equation*}(3.4)Gm(t;x)=x−t−∑k≥jqm,k(x;β)(km−t)
for t ∈ [ j − 1 m , j m ] t ∈ j − 1 m , j m t in[(j-1)/(m),(j)/(m)]t \in\left[\frac{j-1}{m}, \frac{j}{m}\right]t∈[j−1m,jm], where 1 ≤ j ≤ s − 1 1 ≤ j ≤ s − 1 1 <= j <= s-11 \leq j \leq s-11≤j≤s−1.
If we assume that t ∈ [ s − 1 m , x ] t ∈ s − 1 m , x t in[(s-1)/(m),x]t \in\left[\frac{s-1}{m}, x\right]t∈[s−1m,x], then we obtain
(3.5) G m ( t ; x ) = x − t − ∑ k ≥ s q m , k ( x ; β ) ( k m − t ) (3.5) G m ( t ; x ) = x − t − ∑ k ≥ s   q m , k ( x ; β ) k m − t {:(3.5)G_(m)(t;x)=x-t-sum_(k >= s)q_(m,k)(x;beta)((k)/(m)-t):}\begin{equation*} G_{m}(t ; x)=x-t-\sum_{k \geq s} q_{m, k}(x ; \beta)\left(\frac{k}{m}-t\right) \tag{3.5} \end{equation*}(3.5)Gm(t;x)=x−t−∑k≥sqm,k(x;β)(km−t)
while for t ∈ [ x , s m ] t ∈ x , s m t in[x,(s)/(m)]t \in\left[x, \frac{s}{m}\right]t∈[x,sm] we get
(3.6) G m ( t ; x ) = − ∑ k ≥ s q m , k ( x ; β ) ( k m − t ) (3.6) G m ( t ; x ) = − ∑ k ≥ s   q m , k ( x ; β ) k m − t {:(3.6)G_(m)(t;x)=-sum_(k >= s)q_(m,k)(x;beta)((k)/(m)-t):}\begin{equation*} G_{m}(t ; x)=-\sum_{k \geq s} q_{m, k}(x ; \beta)\left(\frac{k}{m}-t\right) \tag{3.6} \end{equation*}(3.6)Gm(t;x)=−∑k≥sqm,k(x;β)(km−t)
For t ∈ [ j − 1 m , j m ] ( j > s ) t ∈ j − 1 m , j m ( j > s ) t in[(j-1)/(m),(j)/(m)](j > s)t \in\left[\frac{j-1}{m}, \frac{j}{m}\right](j>s)t∈[j−1m,jm](j>s) we have
(3.7) G m ( t ; x ) = − ∑ k ≥ j q m , k ( x ; β ) ( k m − t ) (3.7) G m ( t ; x ) = − ∑ k ≥ j   q m , k ( x ; β ) k m − t {:(3.7)G_(m)(t;x)=-sum_(k >= j)q_(m,k)(x;beta)((k)/(m)-t):}\begin{equation*} G_{m}(t ; x)=-\sum_{k \geq j} q_{m, k}(x ; \beta)\left(\frac{k}{m}-t\right) \tag{3.7} \end{equation*}(3.7)Gm(t;x)=−∑k≥jqm,k(x;β)(km−t)
Because the degree of exactness of formula (3.1) is one, by replacing f ( x ) = x − t f ( x ) = x − t f(x)=x-tf(x)=x-tf(x)=x−t, the corresponding remainder vanishes and we obtain
x − t = ∑ k = 0 m q m , k ( x ; β ) ( k m − t ) = ∑ k = 0 j − 1 q m , k ( x ; β ) ( k m − t ) + ∑ k = j m q m , k ( x ; β ) ( k m − t ) x − t = ∑ k = 0 m   q m , k ( x ; β ) k m − t = ∑ k = 0 j − 1   q m , k ( x ; β ) k m − t + ∑ k = j m   q m , k ( x ; β ) k m − t x-t=sum_(k=0)^(m)q_(m,k)(x;beta)((k)/(m)-t)=sum_(k=0)^(j-1)q_(m,k)(x;beta)((k)/(m)-t)+sum_(k=j)^(m)q_(m,k)(x;beta)((k)/(m)-t)x-t=\sum_{k=0}^{m} q_{m, k}(x ; \beta)\left(\frac{k}{m}-t\right)=\sum_{k=0}^{j-1} q_{m, k}(x ; \beta)\left(\frac{k}{m}-t\right)+\sum_{k=j}^{m} q_{m, k}(x ; \beta)\left(\frac{k}{m}-t\right)x−t=∑k=0mqm,k(x;β)(km−t)=∑k=0j−1qm,k(x;β)(km−t)+∑k=jmqm,k(x;β)(km−t).
Therefore we can write
x − t = ∑ k = j m q m , k ( x ; β ) ( k m − t ) = − ∑ k = 0 j − 1 q m , k ( x ; β ) ( t − k m ) x − t = ∑ k = j m   q m , k ( x ; β ) k m − t = − ∑ k = 0 j − 1   q m , k ( x ; β ) t − k m x-t=sum_(k=j)^(m)q_(m,k)(x;beta)((k)/(m)-t)=-sum_(k=0)^(j-1)q_(m,k)(x;beta)(t-(k)/(m))x-t=\sum_{k=j}^{m} q_{m, k}(x ; \beta)\left(\frac{k}{m}-t\right)=-\sum_{k=0}^{j-1} q_{m, k}(x ; \beta)\left(t-\frac{k}{m}\right)x−t=∑k=jmqm,k(x;β)(km−t)=−∑k=0j−1qm,k(x;β)(t−km)
Consequently, the representation (3.4) can be replaced by
(3.8) G m ( t ; x ) = − ∑ k = 0 j − 1 q m , k ( x ; β ) ( t − k m ) (3.8) G m ( t ; x ) = − ∑ k = 0 j − 1   q m , k ( x ; β ) t − k m {:(3.8)G_(m)(t;x)=-sum_(k=0)^(j-1)q_(m,k)(x;beta)(t-(k)/(m)):}\begin{equation*} G_{m}(t ; x)=-\sum_{k=0}^{j-1} q_{m, k}(x ; \beta)\left(t-\frac{k}{m}\right) \tag{3.8} \end{equation*}(3.8)Gm(t;x)=−∑k=0j−1qm,k(x;β)(t−km)
if t ∈ [ j − 1 m , j m ] t ∈ j − 1 m , j m t in[(j-1)/(m),(j)/(m)]t \in\left[\frac{j-1}{m}, \frac{j}{m}\right]t∈[j−1m,jm] and 1 ≤ j ≤ s − 1 1 ≤ j ≤ s − 1 1 <= j <= s-11 \leq j \leq s-11≤j≤s−1, while (3.5) can be replaced by
(3.9) G m ( t ; x ) = − ∑ k = 0 s − 1 q m , k ( x ; β ) ( t − k m ) , (3.9) G m ( t ; x ) = − ∑ k = 0 s − 1   q m , k ( x ; β ) t − k m , {:(3.9)G_(m)(t;x)=-sum_(k=0)^(s-1)q_(m,k)(x;beta)(t-(k)/(m))",":}\begin{equation*} G_{m}(t ; x)=-\sum_{k=0}^{s-1} q_{m, k}(x ; \beta)\left(t-\frac{k}{m}\right), \tag{3.9} \end{equation*}(3.9)Gm(t;x)=−∑k=0s−1qm,k(x;β)(t−km),
when t ∈ [ s − 1 m , x ] t ∈ s − 1 m , x t in[(s-1)/(m),x]t \in\left[\frac{s-1}{m}, x\right]t∈[s−1m,x].
THEOREM 3.2. If f ∈ C 2 [ 0 , 1 ] f ∈ C 2 [ 0 , 1 ] f inC^(2)[0,1]f \in C^{2}[0,1]f∈C2[0,1], then the remainder of the Cheney-Sharma approximation formula (3.1) can be represented under the following form
(3.10) ( R m f ) ( x ; β ) = 1 2 ( R m e 2 ) ( x ; β ) f ′ ′ ( ξ ) , 0 < ξ < 1 . (3.10) R m f ( x ; β ) = 1 2 R m e 2 ( x ; β ) f ′ ′ ( ξ ) , 0 < ξ < 1 . {:(3.10)(R_(m)f)(x;beta)=(1)/(2)(R_(m)e_(2))(x;beta)f^('')(xi)","quad0 < xi < 1.:}\begin{equation*} \left(R_{m} f\right)(x ; \beta)=\frac{1}{2}\left(R_{m} e_{2}\right)(x ; \beta) f^{\prime \prime}(\xi), \quad 0<\xi<1 . \tag{3.10} \end{equation*}(3.10)(Rmf)(x;β)=12(Rme2)(x;β)f′′(ξ),0<ξ<1.
Proof. From (3.6)-(3.9) it is easy to see that on the square D = [ 0 , 1 ] × [ 0 , 1 ] D = [ 0 , 1 ] × [ 0 , 1 ] D=[0,1]xx[0,1]D=[0,1] \times[0,1]D=[0,1]×[0,1] the function y = G m ( t ) = G m ( t ; x ) y = G m ( t ) = G m ( t ; x ) y=G_(m)(t)=G_(m)(t;x)y=G_{m}(t)=G_{m}(t ; x)y=Gm(t)=Gm(t;x) represents a polygonal continuous line situated beneath the x x xxx-axis.
By applying the first law of the mean to the integral (3.2), we get
( R m f ) ( x ; β ) = f ′ ′ ( ξ ) ∫ 0 1 G m ( t ; x ) d t R m f ( x ; β ) = f ′ ′ ( ξ ) ∫ 0 1   G m ( t ; x ) d t (R_(m)f)(x;beta)=f^('')(xi)int_(0)^(1)G_(m)(t;x)dt\left(R_{m} f\right)(x ; \beta)=f^{\prime \prime}(\xi) \int_{0}^{1} G_{m}(t ; x) \mathrm{d} t(Rmf)(x;β)=f′′(ξ)∫01Gm(t;x)dt
and formula (3.1) becomes
(3.11) f ( x ) = ( Q m f ) ( x ; β ) + f ′ ′ ( ξ ) ∫ 0 1 G m ( t ; x ) d t . (3.11) f ( x ) = Q m f ( x ; β ) + f ′ ′ ( ξ ) ∫ 0 1   G m ( t ; x ) d t . {:(3.11)f(x)=(Q_(m)f)(x;beta)+f^('')(xi)int_(0)^(1)G_(m)(t;x)dt.:}\begin{equation*} f(x)=\left(Q_{m} f\right)(x ; \beta)+f^{\prime \prime}(\xi) \int_{0}^{1} G_{m}(t ; x) \mathrm{d} t . \tag{3.11} \end{equation*}(3.11)f(x)=(Qmf)(x;β)+f′′(ξ)∫01Gm(t;x)dt.
If we replace here f ( x ) = e 2 ( x ) = x 2 f ( x ) = e 2 ( x ) = x 2 f(x)=e_(2)(x)=x^(2)f(x)=e_{2}(x)=x^{2}f(x)=e2(x)=x2, we obtain
x 2 = ( Q m e 2 ) ( x ; β ) + 2 ∫ 0 1 ( t ; x ) d t x 2 = Q m e 2 ( x ; β ) + 2 ∫ 0 1   ( t ; x ) d t x^(2)=(Q_(m)e_(2))(x;beta)+2int_(0)^(1)(t;x)dtx^{2}=\left(Q_{m} e_{2}\right)(x ; \beta)+2 \int_{0}^{1}(t ; x) \mathrm{d} tx2=(Qme2)(x;β)+2∫01(t;x)dt
Consequently, we can write
(3.12) ∫ 0 1 G m ( t ; x ) d t = 1 2 [ x 2 − ( Q m e 2 ) ( x ) ] = 1 2 ( R m e 2 ) ( x ; β ) . (3.12) ∫ 0 1   G m ( t ; x ) d t = 1 2 x 2 − Q m e 2 ( x ) = 1 2 R m e 2 ( x ; β ) . {:(3.12)int_(0)^(1)G_(m)(t;x)dt=(1)/(2)[x^(2)-(Q_(m)e_(2))(x)]=(1)/(2)(R_(m)e_(2))(x;beta).:}\begin{equation*} \int_{0}^{1} G_{m}(t ; x) \mathrm{d} t=\frac{1}{2}\left[x^{2}-\left(Q_{m} e_{2}\right)(x)\right]=\frac{1}{2}\left(R_{m} e_{2}\right)(x ; \beta) . \tag{3.12} \end{equation*}(3.12)∫01Gm(t;x)dt=12[x2−(Qme2)(x)]=12(Rme2)(x;β).
Formulas (3.11) and (3.12) lead us to the desired approximation formula with the expression (3.10) for the remainder term.
In the special case β = 0 β = 0 beta=0\beta=0β=0, when Q m = B m Q m = B m Q_(m)=B_(m)Q_{m}=B_{m}Qm=Bm, the formulas corresponding to (3.2)-(3.3) and (3.10) were first established in our old paper [3].
Remark. Since the polynomial Q m f Q m f Q_(m)fQ_{m} fQmf is interpolatory at both sides of the interval [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1], it is clear that ( R m e 2 ) ( x ; β ) R m e 2 ( x ; β ) (R_(m)e_(2))(x;beta)\left(R_{m} e_{2}\right)(x ; \beta)(Rme2)(x;β) contains the factor x ( x − 1 ) x ( x − 1 ) x(x-1)x(x-1)x(x−1).
Since R m e 0 = 0 , R m e 1 = 0 R m e 0 = 0 , R m e 1 = 0 R_(m)e_(0)=0,R_(m)e_(1)=0R_{m} e_{0}=0, R_{m} e_{1}=0Rme0=0,Rme1=0 and R m f ≠ 0 R m f ≠ 0 R_(m)f!=0R_{m} f \neq 0Rmf≠0 for any convex function of the first order, we can apply a criterion of T . Popoviciu [2] and we can find that the remainder is of a simple form.
Consequently, we can state
THEOREM 3.3. If the second-order divided differences of the function f f fff are bounded on the interval [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1], then there exist three points t m , 1 , t m , 2 t m , 1 , t m , 2 t_(m,1),t_(m,2)t_{m, 1}, t_{m, 2}tm,1,tm,2 and t m , 3 t m , 3 t_(m,3)t_{m, 3}tm,3 from [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1] which might depend on f f fff, such that the remainder of the approximation formula (3.1) can be represented under the form
(3.13) ( R m f ) ( x ; β ) = ( R m e 2 ) ( x ; β ) [ t m , 1 , t m , 2 , t m , 3 ; f ] . (3.13) R m f ( x ; β ) = R m e 2 ( x ; β ) t m , 1 , t m , 2 , t m , 3 ; f . {:(3.13)(R_(m)f)(x;beta)=(R_(m)e_(2))(x;beta)[t_(m,1),t_(m,2),t_(m,3);f].:}\begin{equation*} \left(R_{m} f\right)(x ; \beta)=\left(R_{m} e_{2}\right)(x ; \beta)\left[t_{m, 1}, t_{m, 2}, t_{m, 3} ; f\right] . \tag{3.13} \end{equation*}(3.13)(Rmf)(x;β)=(Rme2)(x;β)[tm,1,tm,2,tm,3;f].
It is clear that, if f ∈ C 2 [ 0 , 1 ] f ∈ C 2 [ 0 , 1 ] f inC^(2)[0,1]f \in C^{2}[0,1]f∈C2[0,1] and we apply the mean-value theorem of divided differences, we can obtain formula (3.10) from formula (3.13).

REFERENCES

  1. E. W. Cheney and A. Sharma, On a generalization of Bernstein polynomials, Riv. Mat. Univ. Parma 5 (1964), 77-84.
  2. T. Popoviciu, Sur le reste dans certaines formules linéaires d'approximation de l'analyse, Mathematica 1 (24) (1959), 95-142.
  3. D. D. Stancu, Evaluation of the remainder term in approximation formulas by Bernstein polynomials, Math. Comput. 17 (1963), 270-278.
Received May 15, 1996
D. D. Stancu
Faculty of Mathematics and Computer Science "Babes-Bolyai" University 3400 Cluj-Napoca
Romania
C. Cismaşiu
Department of Mathematics
Transylvania University
2200 Braşov
Romania

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