Return to Article Details On some remarkable positive polynomial operators of approximation

ON SOME REMARKABLE POSITIVE POLYNOMIAL OPERATORS OF APPROXIMATION

D. D. STANCU and A. VERNESCU

Dedicated to Professor H. H. Gonska on the occasion of his 50 th 50 th  50^("th ")50^{\text {th }}50th  birthday

Abstract. In this paper there are investigated new approximation properties of a Rernstein type operator, depending on two real parameters c c ccc and d ( 0 ≤ c ≤ d ) d ( 0 ≤ c ≤ d ) d(0 <= c <= d)d(0 \leq c \leq d)d(0≤c≤d), introduced in 1969 by the first author [19].
A basic result consists in finding the maximum value (2.3) of the mean square error (2.1)-(2.2) of this operator. By using it, is constructed a best quadrature formula, which can be obtained also by means of the polynomial S m f S m f S_(m)fS_{m} fSmf, defined at (3.5).
In the last part of the paper there are established quantitative estimations of approximation in terms of first and second order moduli of smoothness.

1. INTRODUCTION

1.1. It is known that polynomial approximation represents one of the most beautiful and important part of the constructive theory of functions.
The Lagrange interpolating polynomials have a great practical interest in numerical analysis and approximation theory. Unfortunately they do not always provide uniform convergent sequences of approximation for any continuous function on a compact interval [ a , b ] [ a , b ] [a,b][a, b][a,b] of the real axis, no matter how the nodes are prescribed (see, e.g., E. W. Cheney [11]).
In 1905 E. Borel [9] proposed that for the approximation of a function f ∈ C [ 0 , 1 ] f ∈ C [ 0 , 1 ] f in C[0,1]f \in C[0,1]f∈C[0,1] to construct a polynomial having an expression similar with the Lagrange interpolating polynomial, corresponding to m + 1 m + 1 m+1m+1m+1 nodes from [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1]. Namely, in the case of the equally spaced nodes, it has to be of the form
( Q m f ) ( x ) = ∑ k = 0 m q m , k ( x ) f ( k m ) , Q m f ( x ) = ∑ k = 0 m   q m , k ( x ) f k m , (Q_(m)f)(x)=sum_(k=0)^(m)q_(m,k)(x)f((k)/(m)),\left(Q_{m} f\right)(x)=\sum_{k=0}^{m} q_{m, k}(x) f\left(\frac{k}{m}\right),(Qmf)(x)=∑k=0mqm,k(x)f(km),
where q m , k q m , k q_(m,k)q_{m, k}qm,k are appropriate polynomials of degree m m mmm, which permit that Q m f Q m f Q_(m)fQ_{m} fQmf to achieve a prescribed accuracy in the process of approximation of the function f f fff.
Following the Borel ideas, S. N. Bernstein [7] has the merit to select, in 1912, for q m , k q m , k q_(m,k)q_{m, k}qm,k the basic polynomials
(1.1) p m , k ( x ) = ( m k ) x k ( 1 − x ) m − k (1.1) p m , k ( x ) = ( m k ) x k ( 1 − x ) m − k {:(1.1)p_(m,k)(x)=((m)/(k))x^(k)(1-x)^(m-k):}\begin{equation*} p_{m, k}(x)=\binom{m}{k} x^{k}(1-x)^{m-k} \tag{1.1} \end{equation*}(1.1)pm,k(x)=(mk)xk(1−x)m−k
It should be mentioned that he was inspired by the binomial probability distribution and that he has investigated the convergence of the polynomials
(1.2) ( B m f ) ( x ) = B m ( f ( t ) ; x ) = ∑ k = 0 m p m , k ( x ) f ( k m ) (1.2) B m f ( x ) = B m ( f ( t ) ; x ) = ∑ k = 0 m   p m , k ( x ) f k m {:(1.2)(B_(m)f)(x)=B_(m)(f(t);x)=sum_(k=0)^(m)p_(m,k)(x)f((k)/(m)):}\begin{equation*} \left(B_{m} f\right)(x)=B_{m}(f(t) ; x)=\sum_{k=0}^{m} p_{m, k}(x) f\left(\frac{k}{m}\right) \tag{1.2} \end{equation*}(1.2)(Bmf)(x)=Bm(f(t);x)=∑k=0mpm,k(x)f(km)
by using the weak Bernoulli law of large numbers.
1.2. In 1969 the first author has introduced in [19] the following generalization of the Bernstein polynomials
(1.3) P m ( c , d ) ( f ( t ) ; x ) = ( P m ( c , d ) f ) ( x ) = ∑ k = 0 m p m , k ( x ) f ( k + c m + d ) , (1.3) P m ( c , d ) ( f ( t ) ; x ) = P m ( c , d ) f ( x ) = ∑ k = 0 m   p m , k ( x ) f k + c m + d , {:(1.3)P_(m)^((c,d))(f(t);x)=(P_(m)^((c,d))f)(x)=sum_(k=0)^(m)p_(m,k)(x)f((k+c)/(m+d))",":}\begin{equation*} P_{m}^{(c, d)}(f(t) ; x)=\left(P_{m}^{(c, d)} f\right)(x)=\sum_{k=0}^{m} p_{m, k}(x) f\left(\frac{k+c}{m+d}\right), \tag{1.3} \end{equation*}(1.3)Pm(c,d)(f(t);x)=(Pm(c,d)f)(x)=∑k=0mpm,k(x)f(k+cm+d),
where c c ccc and d d ddd are real parameters, independently of m m mmm, satisfying the relations: 0 ≤ c ≤ d 0 ≤ c ≤ d 0 <= c <= d0 \leq c \leq d0≤c≤d.
This polynomial is characterized by the fact that it uses equally spaced nodes, with the step h = 1 m + d h = 1 m + d h=(1)/(m+d)h=\frac{1}{m+d}h=1m+d and the starting point x 0 = c m + d x 0 = c m + d x_(0)=(c)/(m+d)x_{0}=\frac{c}{m+d}x0=cm+d. If 0 < c ≠ d 0 < c ≠ d 0 < c!=d0<c \neq d0<c≠d then it does not coincide at any node with the function f f fff, if c = 0 c = 0 c=0c=0c=0 and d ≠ c d ≠ c d!=cd \neq cd≠c then it coincides with f f fff at x 0 = 0 x 0 = 0 x_(0)=0x_{0}=0x0=0, while if 0 < c = d 0 < c = d 0 < c=d0<c=d0<c=d then it coincides with f f fff at x m = 1 x m = 1 x_(m)=1x_{m}=1xm=1. For c = d = 0 c = d = 0 c=d=0c=d=0c=d=0 we obtain the classical Bernstein polynomial B m f B m f B_(m)fB_{m} fBmf, which coincides with f f fff at x 0 = 0 x 0 = 0 x_(0)=0x_{0}=0x0=0 and x m = 1 x m = 1 x_(m)=1x_{m}=1xm=1.
In the monograph of F . Altomare and M. Campiti [3] the operator defined at (1.3) was called: ,the operator of Bernstein-Stancu" (pag. 117 and 220).
In the paper [19] of the author there was established the following representation of (1.3) in terms of finite differences
(1.4) ( P m ( c , d ) f ) ( x ) = ∑ j = 0 m ( m j ) ( Δ 1 m + d j f c m + d ) ( c m (1.4) P m ( c , d ) f ( x ) = ∑ j = 0 m   ( m j ) ( Δ 1 m + d j f c m + d ) c m {:(1.4)(P_(m)^((c,d))f)(x)=sum_(j=0)^(m)((m)/(j))((Delta_((1)/(m+d))^(j)f)/((c)/(m+d)))((c)/(m):}:}\begin{equation*} \left(P_{m}^{(c, d)} f\right)(x)=\sum_{j=0}^{m}\binom{m}{j}\binom{\Delta_{\frac{1}{m+d}}^{j} f}{\frac{c}{m+d}}\left(\frac{c}{m}\right. \tag{1.4} \end{equation*}(1.4)(Pm(c,d)f)(x)=∑j=0m(mj)(Δ1m+djfcm+d)(cm
as well as an expression by means of divided differences
( P m ( c , d ) f ) ( x ) = ∑ j = 0 m m | j | [ c m + d , c + 1 m + d , … , c + j m + d ; f ] ( x m + d ) j P m ( c , d ) f ( x ) = ∑ j = 0 m   m | j | c m + d , c + 1 m + d , … , c + j m + d ; f x m + d j (P_(m)^((c,d))f)(x)=sum_(j=0)^(m)m^(|j|)[(c)/(m+d),(c+1)/(m+d),dots,(c+j)/(m+d);f]((x)/(m+d))^(j)\left(P_{m}^{(c, d)} f\right)(x)=\sum_{j=0}^{m} m^{|j|}\left[\frac{c}{m+d}, \frac{c+1}{m+d}, \ldots, \frac{c+j}{m+d} ; f\right]\left(\frac{x}{m+d}\right)^{j}(Pm(c,d)f)(x)=∑j=0mm|j|[cm+d,c+1m+d,…,c+jm+d;f](xm+d)j
where m ⌊ j ⌋ = m ( m − 1 ) … ( m − j + 1 ) m ⌊ j ⌋ = m ( m − 1 ) … ( m − j + 1 ) m^(|__ j __|)=m(m-1)dots(m-j+1)m^{\lfloor j\rfloor}=m(m-1) \ldots(m-j+1)m⌊j⌋=m(m−1)…(m−j+1), the brackets representing the symbol of divided differences.
1.3. For the monomials e 0 , e 1 e 0 , e 1 e_(0),e_(1)e_{0}, e_{1}e0,e1 and e 2 e 2 e_(2)e_{2}e2, where 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), where t ∈ [ 0 , 1 ] t ∈ [ 0 , 1 ] t in[0,1]t \in[0,1]t∈[0,1], we have
(1.5) P m ( c , d ) e 0 = e 0 , ( P m ( c , d ) e 1 ) ( x ) = x + c − d x m + d , ( P m ( c , d ) e 2 ) ( x ) = x 2 + 1 ( m + d ) 2 [ m x ( 1 − x ) + ( c − d x ) ( 2 m x + d x + c ) ] (1.5) P m ( c , d ) e 0 = e 0 , P m ( c , d ) e 1 ( x ) = x + c − d x m + d , P m ( c , d ) e 2 ( x ) = x 2 + 1 ( m + d ) 2 [ m x ( 1 − x ) + ( c − d x ) ( 2 m x + d x + c ) ] {:[(1.5)P_(m)^((c,d))e_(0)=e_(0)","quad(P_(m)^((c,d))e_(1))(x)=x+(c-dx)/(m+d)","],[(P_(m)^((c,d))e_(2))(x)=x^(2)+(1)/((m+d)^(2))[mx(1-x)+(c-dx)(2mx+dx+c)]]:}\begin{gather*} P_{m}^{(c, d)} e_{0}=e_{0}, \quad\left(P_{m}^{(c, d)} e_{1}\right)(x)=x+\frac{c-d x}{m+d}, \tag{1.5}\\ \left(P_{m}^{(c, d)} e_{2}\right)(x)=x^{2}+\frac{1}{(m+d)^{2}}[m x(1-x)+(c-d x)(2 m x+d x+c)] \end{gather*}(1.5)Pm(c,d)e0=e0,(Pm(c,d)e1)(x)=x+c−dxm+d,(Pm(c,d)e2)(x)=x2+1(m+d)2[mx(1−x)+(c−dx)(2mx+dx+c)]
Because for these "test functions" we have
lim m → ∞ P m ( c , d ) e j = e j ( j = 0 , 1 , 2 ) , lim m → ∞   P m ( c , d ) e j = e j ( j = 0 , 1 , 2 ) , lim_(m rarr oo)P_(m)^((c,d))e_(j)=e_(j)quad(j=0,1,2),\lim _{m \rightarrow \infty} P_{m}^{(c, d)} e_{j}=e_{j} \quad(j=0,1,2),limm→∞Pm(c,d)ej=ej(j=0,1,2),
uniformly on the interval [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1], according to the Bohman-Korovkin convergence criterion there was possible to state the following result: if f ∈ C [ 0 , 1 ] f ∈ C [ 0 , 1 ] f in C[0,1]f \in C[0,1]f∈C[0,1] then the sequence of polynomials ( P m ( c , d ) f ) P m ( c , d ) f (P_(m)^((c,d))f)\left(P_{m}^{(c, d)} f\right)(Pm(c,d)f) converges uniformly to the function f f fff on the interval [0,1].
1.4. We mention also that for the eigenvalues of the operator P m ( c , d ) P m ( c , d ) P_(m)^((c,d))P_{m}^{(c, d)}Pm(c,d) we have obtained the following expressions
λ m , r ( P m ( c , d ) ) = m | r | ( m + d ) r = ( 1 − 1 m ) ( 1 − 2 m ) … ( 1 − r − 1 m ) ( m m + d ) r λ m , r P m ( c , d ) = m | r | ( m + d ) r = 1 − 1 m 1 − 2 m … 1 − r − 1 m m m + d r lambda_(m,r)(P_(m)^((c,d)))=(m^(|r|))/((m+d)^(r))=(1-(1)/(m))(1-(2)/(m))dots(1-(r-1)/(m))((m)/(m+d))^(r)\lambda_{m, r}\left(P_{m}^{(c, d)}\right)=\frac{m^{|r|}}{(m+d)^{r}}=\left(1-\frac{1}{m}\right)\left(1-\frac{2}{m}\right) \ldots\left(1-\frac{r-1}{m}\right)\left(\frac{m}{m+d}\right)^{r}λm,r(Pm(c,d))=m|r|(m+d)r=(1−1m)(1−2m)…(1−r−1m)(mm+d)r
where r = 0 , 1 , … , m r = 0 , 1 , … , m r=0,1,dots,mr=0,1, \ldots, mr=0,1,…,m. One can see that these quantities do not depend on the value of the parameter c c ccc.
In the case of Bernstein operator ( c = d = 0 c = d = 0 c=d=0c=d=0c=d=0 ) the point spectrum was first found in [10]. In [6] it has been given a characterization of B m B m B_(m)B_{m}Bm by using the eigenvalue λ m , 2 = λ m , 2 ( B m ) λ m , 2 = λ m , 2 B m lambda_(m,2)=lambda_(m,2)(B_(m))\lambda_{m, 2}=\lambda_{m, 2}\left(B_{m}\right)λm,2=λm,2(Bm).
Because in our case
λ m , 2 ( P m ( c , d ) ) = ( 1 − 1 m ) ( 1 + d m ) − 2 ≤ λ m , 2 ( B m ) = 1 − 1 m λ m , 2 P m ( c , d ) = 1 − 1 m 1 + d m − 2 ≤ λ m , 2 B m = 1 − 1 m lambda_(m,2)(P_(m)^((c,d)))=(1-(1)/(m))(1+(d)/(m))^(-2) <= lambda_(m,2)(B_(m))=1-(1)/(m)\lambda_{m, 2}\left(P_{m}^{(c, d)}\right)=\left(1-\frac{1}{m}\right)\left(1+\frac{d}{m}\right)^{-2} \leq \lambda_{m, 2}\left(B_{m}\right)=1-\frac{1}{m}λm,2(Pm(c,d))=(1−1m)(1+dm)−2≤λm,2(Bm)=1−1m
by using a theorem given in [6], we conclude that the best result can be achieved when c = d = 0 c = d = 0 c=d=0c=d=0c=d=0, that is in the case of Bernstein operator B m B m B_(m)B_{m}Bm.
1.5. The Bleimann, Butzer, Hahn (BBH) rational operator [8], given by
( L m f ) ( x ) = ∑ k = 0 m ( m k ) x k ( 1 + x ) m f ( k m + 1 − k ) ( x ≥ 0 ) , L m f ( x ) = ∑ k = 0 m   ( m k ) x k ( 1 + x ) m f k m + 1 − k ( x ≥ 0 ) , (L_(m)f)(x)=sum_(k=0)^(m)((m)/(k))(x^(k))/((1+x)^(m))f((k)/(m+1-k))quad(x >= 0),\left(L_{m} f\right)(x)=\sum_{k=0}^{m}\binom{m}{k} \frac{x^{k}}{(1+x)^{m}} f\left(\frac{k}{m+1-k}\right) \quad(x \geq 0),(Lmf)(x)=∑k=0m(mk)xk(1+x)mf(km+1−k)(x≥0),
can be obtained from the operator P m ( 0 , 1 ) P m ( 0 , 1 ) P_(m)^((0,1))P_{m}^{(0,1)}Pm(0,1) (see [15], [1], [5] and [2]) using the rational transformation t = x 1 + x ( x ≥ 0 ) t = x 1 + x ( x ≥ 0 ) t=(x)/(1+x)(x >= 0)t=\frac{x}{1+x}(x \geq 0)t=x1+x(x≥0). By means of this transformation the operator (1.3) leads to a BBH type operator having the nodes: x m . k c . d = ( k + c ) / x m . k c . d = ( k + c ) / x_(m.k)^(c.d)=(k+c)//x_{m . k}^{c . d}=(k+c) /xm.kc.d=(k+c)/. / ( m + d − k − c ) / ( m + d − k − c ) //(m+d-k-c)/(m+d-k-c)/(m+d−k−c), where 0 ≤ c ≤ d 0 ≤ c ≤ d 0 <= c <= d0 \leq c \leq d0≤c≤d. If we choose c = a c = a c=ac=ac=a and d = a + 1 ( a ≥ 0 ) d = a + 1 ( a ≥ 0 ) d=a+1(a >= 0)d=a+1(a \geq 0)d=a+1(a≥0), then we obtain a BBH type operator investigated recently by O. Agratini [5]:
( L m ( α ) f ) ( x ) = ∑ k = 0 m ( m k ) x k ( 1 + x ) m f ( k + a m + 1 − k ) . L m ( α ) f ( x ) = ∑ k = 0 m   ( m k ) x k ( 1 + x ) m f k + a m + 1 − k . (L_(m)^((alpha))f)(x)=sum_(k=0)^(m)((m)/(k))(x^(k))/((1+x)^(m))f((k+a)/(m+1-k)).\left(L_{m}^{(\alpha)} f\right)(x)=\sum_{k=0}^{m}\binom{m}{k} \frac{x^{k}}{(1+x)^{m}} f\left(\frac{k+a}{m+1-k}\right) .(Lm(α)f)(x)=∑k=0m(mk)xk(1+x)mf(k+am+1−k).
1.6. Concerning the remainder of the approximation formula
(1.6) f ( x ) = ( P m ( c , d ) f ) ( x ) + ( R m ( c , d ) ) ( x ) (1.6) f ( x ) = P m ( c , d ) f ( x ) + R m ( c , d ) ( x ) {:(1.6)f(x)=(P_(m)^((c,d))f)(x)+(R_(m)^((c,d)))(x):}\begin{equation*} f(x)=\left(P_{m}^{(c, d)} f\right)(x)+\left(R_{m}^{(c, d)}\right)(x) \tag{1.6} \end{equation*}(1.6)f(x)=(Pm(c,d)f)(x)+(Rm(c,d))(x)
in the paper [19] there was established the following representation, in terms of first and second order divided differences:
( R m ( c , d ) f ) ( x ) = d x − c m + d ∑ k = 0 m p m , k ( x ) [ x , k + c m + d ; f ] − − m x ( 1 − x ) ( m + d ) 2 ∑ k = 0 m − 1 p m − 1 , k ( x ) [ x , k + c m + d , k + c + 1 m + d ; f ] . R m ( c , d ) f ( x ) = d x − c m + d ∑ k = 0 m   p m , k ( x ) x , k + c m + d ; f − − m x ( 1 − x ) ( m + d ) 2 ∑ k = 0 m − 1   p m − 1 , k ( x ) x , k + c m + d , k + c + 1 m + d ; f . {:[(R_(m)^((c,d))f)(x)=(dx-c)/(m+d)sum_(k=0)^(m)p_(m,k)(x)[x,(k+c)/(m+d);f]-],[-(mx(1-x))/((m+d)^(2))sum_(k=0)^(m-1)p_(m-1,k)(x)[x,(k+c)/(m+d),(k+c+1)/(m+d);f].]:}\begin{aligned} & \left(R_{m}^{(c, d)} f\right)(x)=\frac{d x-c}{m+d} \sum_{k=0}^{m} p_{m, k}(x)\left[x, \frac{k+c}{m+d} ; f\right]- \\ & -\frac{m x(1-x)}{(m+d)^{2}} \sum_{k=0}^{m-1} p_{m-1, k}(x)\left[x, \frac{k+c}{m+d}, \frac{k+c+1}{m+d} ; f\right] . \end{aligned}(Rm(c,d)f)(x)=dx−cm+d∑k=0mpm,k(x)[x,k+cm+d;f]−−mx(1−x)(m+d)2∑k=0m−1pm−1,k(x)[x,k+cm+d,k+c+1m+d;f].
In the case c = d = 0 c = d = 0 c=d=0c=d=0c=d=0, when (1.3) becomes the Bernstein polynomial B m f B m f B_(m)fB_{m} fBmf, it reduces to an expression obtained already in 1964 by the first author [18]:
( R m f ) ( x ) = x ( x − 1 ) m ∑ k = 0 m p m − 1 . k ( x ) [ x , k m , k + 1 m ; f ] R m f ( x ) = x ( x − 1 ) m ∑ k = 0 m   p m − 1 . k ( x ) x , k m , k + 1 m ; f (R_(m)f)(x)=(x(x-1))/(m)sum_(k=0)^(m)p_(m-1.k)(x)[x,(k)/(m),(k+1)/(m);f]\left(R_{m} f\right)(x)=\frac{x(x-1)}{m} \sum_{k=0}^{m} p_{m-1 . k}(x)\left[x, \frac{k}{m}, \frac{k+1}{m} ; f\right](Rmf)(x)=x(x−1)m∑k=0mpm−1.k(x)[x,km,k+1m;f]
In a recent paper O. Agratini [4] studied the monotonicity properties of the sequence of polynomials (1.3).
In order to investigate the simultaneous approximation properties of the operator (1.3) there was established in [20] the following formula
( P m ( c , d ) f ) ( r ) ( x ) = m | r | ∑ j = 0 m − r p m − r . j ( x ) ( Δ 1 m + d r f ) ( j + c m + d ) P m ( c , d ) f ( r ) ( x ) = m | r | ∑ j = 0 m − r   p m − r . j ( x ) Δ 1 m + d r f j + c m + d (P_(m)^((c,d))f)^((r))(x)=m^(|r|)sum_(j=0)^(m-r)p_(m-r.j)(x)(Delta_((1)/(m+d))^(r)f)((j+c)/(m+d))\left(P_{m}^{(c, d)} f\right)^{(r)}(x)=m^{|r|} \sum_{j=0}^{m-r} p_{m-r . j}(x)\left(\Delta_{\frac{1}{m+d}}^{r} f\right)\left(\frac{j+c}{m+d}\right)(Pm(c,d)f)(r)(x)=m|r|∑j=0m−rpm−r.j(x)(Δ1m+drf)(j+cm+d)
where 0 ≤ r ≤ m 0 ≤ r ≤ m 0 <= r <= m0 \leq r \leq m0≤r≤m. By using it, there was proved the following result: if f ∈ C r [ 0 , 1 ] f ∈ C r [ 0 , 1 ] f inC^(r)[0,1]f \in C^{r}[0,1]f∈Cr[0,1] then we have
lim m → ∞ ( P m ( c ⋅ d ) f ) ( r ) = f ( r ) lim m → ∞   P m ( c ⋅ d ) f ( r ) = f ( r ) lim_(m rarr oo)(P_(m)^((c*d))f)^((r))=f^((r))\lim _{m \rightarrow \infty}\left(P_{m}^{(c \cdot d)} f\right)^{(r)}=f^{(r)}limm→∞(Pm(c⋅d)f)(r)=f(r)
uniformly on the interval [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1].

2. THE MEAN SQUARE ERROR OF THE OPERATOR p m ( c ) d ) p m ( c ) d ) p_(m)^((c)d))p_{m}^{(c) d)}pm(c)d)

2.1. Since the rate of convergence of the operators (1.3) is characterized by the value of the mean square error
(2.1) e m 2 ( x ; c , d ) = P m ( c , d ) ( ( t − x ) 2 ; x ) (2.1) e m 2 ( x ; c , d ) = P m ( c , d ) ( t − x ) 2 ; x {:(2.1)e_(m)^(2)(x;c","d)=P_(m)^((c,d))((t-x)^(2);x):}\begin{equation*} e_{m}^{2}(x ; c, d)=P_{m}^{(c, d)}\left((t-x)^{2} ; x\right) \tag{2.1} \end{equation*}(2.1)em2(x;c,d)=Pm(c,d)((t−x)2;x)
we next make an examination of it.
It is obvious that we can write
e m 2 ( x ; c , d ) = P m ( c , d ) ( t 2 ; x ) − 2 x P m ( c , d ) ( t ; x ) + x 2 e m 2 ( x ; c , d ) = P m ( c , d ) t 2 ; x − 2 x P m ( c , d ) ( t ; x ) + x 2 e_(m)^(2)(x;c,d)=P_(m)^((c,d))(t^(2);x)-2xP_(m)^((c,d))(t;x)+x^(2)e_{m}^{2}(x ; c, d)=P_{m}^{(c, d)}\left(t^{2} ; x\right)-2 x P_{m}^{(c, d)}(t ; x)+x^{2}em2(x;c,d)=Pm(c,d)(t2;x)−2xPm(c,d)(t;x)+x2
According to (1.5) we have
(2.2) e m 2 ( x ; c , d ) = m x ( 1 − x ) + ( d x − c ) 2 ( m + d ) 2 (2.2) e m 2 ( x ; c , d ) = m x ( 1 − x ) + ( d x − c ) 2 ( m + d ) 2 {:(2.2)e_(m)^(2)(x;c","d)=(mx(1-x)+(dx-c)^(2))/((m+d)^(2)):}\begin{equation*} e_{m}^{2}(x ; c, d)=\frac{m x(1-x)+(d x-c)^{2}}{(m+d)^{2}} \tag{2.2} \end{equation*}(2.2)em2(x;c,d)=mx(1−x)+(dx−c)2(m+d)2
The variance of the operator P m ( c , d ) P m ( c , d ) P_(m)^((c,d))P_{m}^{(c, d)}Pm(c,d) is defined by
v m ( x ; c , d ) = P m ( c . d ) ( e 1 − P m ( c . d ) e 1 ) 2 ( x ) = ( P m ( c . d ) e 2 ) ( x ) − [ ( P m ( c . d ) e 1 ) 2 ( x ) ] 2 . v m ( x ; c , d ) = P m ( c . d ) e 1 − P m ( c . d ) e 1 2 ( x ) = P m ( c . d ) e 2 ( x ) − P m ( c . d ) e 1 2 ( x ) 2 . v_(m)(x;c,d)=P_(m)^((c.d))(e_(1)-P_(m)^((c.d))e_(1))^(2)(x)=(P_(m)^((c.d))e_(2))(x)-[(P_(m)^((c.d))e_(1))^(2)(x)]^(2).v_{m}(x ; c, d)=P_{m}^{(c . d)}\left(e_{1}-P_{m}^{(c . d)} e_{1}\right)^{2}(x)=\left(P_{m}^{(c . d)} e_{2}\right)(x)-\left[\left(P_{m}^{(c . d)} e_{1}\right)^{2}(x)\right]^{2} .vm(x;c,d)=Pm(c.d)(e1−Pm(c.d)e1)2(x)=(Pm(c.d)e2)(x)−[(Pm(c.d)e1)2(x)]2.
If we take into consideration the identities (1.5) we obtain
v m ( x ; c , d ) = m x ( 1 − x ) ( m + d ) 2 v m ( x ; c , d ) = m x ( 1 − x ) ( m + d ) 2 v_(m)(x;c,d)=(mx(1-x))/((m+d)^(2))v_{m}(x ; c, d)=\frac{m x(1-x)}{(m+d)^{2}}vm(x;c,d)=mx(1−x)(m+d)2
and one observes that it does not depend on the parameter c c ccc.
2.2. In order to see how well a function f ∈ C [ 0 , 1 ] f ∈ C [ 0 , 1 ] f in C[0,1]f \in C[0,1]f∈C[0,1] can be approximated by the polynomial (1.3), we need to find the maximum value of (2.2) on the interval [0,1].
We shall now present a basic result of this paper.
THEOREM 2.1. If m > d 2 m > d 2 m > d^(2)m>d^{2}m>d2, then the maximum value on [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1] of the mean square error (2.2) can be represented under the form
(2.3) M m ( c , d ) = m 4 ( m + d ) 2 [ 1 + ( d − 2 c ) 2 m − d 2 ] . (2.3) M m ( c , d ) = m 4 ( m + d ) 2 1 + ( d − 2 c ) 2 m − d 2 . {:(2.3)M_(m)^((c,d))=(m)/(4(m+d)^(2))[1+((d-2c)^(2))/(m-d^(2))].:}\begin{equation*} M_{m}^{(c, d)}=\frac{m}{4(m+d)^{2}}\left[1+\frac{(d-2 c)^{2}}{m-d^{2}}\right] . \tag{2.3} \end{equation*}(2.3)Mm(c,d)=m4(m+d)2[1+(d−2c)2m−d2].
Proof. It is known that if we have a polynomial of second degree, with real coefficients: P 2 ( x ) = A x 2 + B x + C P 2 ( x ) = A x 2 + B x + C P_(2)(x)=Ax^(2)+Bx+CP_{2}(x)=A x^{2}+B x+CP2(x)=Ax2+Bx+C and A < 0 A < 0 A < 0A<0A<0, then the maximum value of this polynomial is given by P 2 ( − B 2 A ) = − Δ 4 A P 2 − B 2 A = − Δ 4 A P_(2)(-(B)/(2A))=-(Delta)/(4A)P_{2}\left(-\frac{B}{2 A}\right)=-\frac{\Delta}{4 A}P2(−B2A)=−Δ4A, where Δ = B 2 − 4 A C Δ = B 2 − 4 A C Delta=B^(2)-4AC\Delta=B^{2}-4 A CΔ=B2−4AC. In our case we have A = d 2 − m , B = m − 2 c d , C = c 2 A = d 2 − m , B = m − 2 c d , C = c 2 A=d^(2)-m,B=m-2cd,C=c^(2)A=d^{2}-m, B=m-2 c d, C=c^{2}A=d2−m,B=m−2cd,C=c2. Consequently, we find that this maximum value is given by
M m ( c , d ) = m 2 − 4 c ( d − c ) m 4 ( m + d ) 2 ( m − d 2 ) = m 4 ( m + d ) 2 [ 1 + ( d − 2 c ) 2 m − d 2 ] . M m ( c , d ) = m 2 − 4 c ( d − c ) m 4 ( m + d ) 2 m − d 2 = m 4 ( m + d ) 2 1 + ( d − 2 c ) 2 m − d 2 . M_(m)^((c,d))=(m^(2)-4c(d-c)m)/(4(m+d)^(2)(m-d^(2)))=(m)/(4(m+d)^(2))[1+((d-2c)^(2))/(m-d^(2))].M_{m}^{(c, d)}=\frac{m^{2}-4 c(d-c) m}{4(m+d)^{2}\left(m-d^{2}\right)}=\frac{m}{4(m+d)^{2}}\left[1+\frac{(d-2 c)^{2}}{m-d^{2}}\right] .Mm(c,d)=m2−4c(d−c)m4(m+d)2(m−d2)=m4(m+d)2[1+(d−2c)2m−d2].
Now it is clear that we can formulate an important consequence of this theorem.
COROLLARY 2.1. The least maximum value (2.2) is attained for d = 2 c ( c ≥ 0 ) d = 2 c ( c ≥ 0 ) d=2c(c >= 0)d=2 c (c \geq 0)d=2c(c≥0) and it is
v m ( c ) = M m ( c , 2 c ) = m 4 ( m + 2 c ) 2 ≤ 1 4 m = v m ( 0 ) v m ( c ) = M m ( c , 2 c ) = m 4 ( m + 2 c ) 2 ≤ 1 4 m = v m ( 0 ) v_(m)(c)=M_(m)^((c,2c))=(m)/(4(m+2c)^(2)) <= (1)/(4m)=v_(m)(0)v_{m}(c)=M_{m}^{(c, 2 c)}=\frac{m}{4(m+2 c)^{2}} \leq \frac{1}{4 m}=v_{m}(0)vm(c)=Mm(c,2c)=m4(m+2c)2≤14m=vm(0)
In this case we have the approximating polynomials
(2.4) ( P m ( c , 2 c ) f ) ( x ) = ( S m ( c ) f ) ( x ) = ∑ k = 0 m p m , k ( x ) f ( k + c m + 2 c ) ( c ≥ 0 ) , (2.4) P m ( c , 2 c ) f ( x ) = S m ( c ) f ( x ) = ∑ k = 0 m   p m , k ( x ) f k + c m + 2 c ( c ≥ 0 ) , {:(2.4)(P_(m)^((c,2c))f)(x)=(S_(m)^((c))f)(x)=sum_(k=0)^(m)p_(m,k)(x)f((k+c)/(m+2c))quad(c >= 0)",":}\begin{equation*} \left(P_{m}^{(c, 2 c)} f\right)(x)=\left(S_{m}^{(c)} f\right)(x)=\sum_{k=0}^{m} p_{m, k}(x) f\left(\frac{k+c}{m+2 c}\right) \quad(c \geq 0), \tag{2.4} \end{equation*}(2.4)(Pm(c,2c)f)(x)=(Sm(c)f)(x)=∑k=0mpm,k(x)f(k+cm+2c)(c≥0),
which are important in numerical integration of functions.

3. A BEST QUADRATURE FORMULA

3.1. By using the approximation formula (1.6) we can obtain the following numerical quadrature procedure
(3.1) ∫ 0 1 f ( x ) d x = 1 m + 1 ∑ k = 0 m f ( k + c m + d ) + r m ( c , d ) ( f ) , (3.1) ∫ 0 1   f ( x ) d x = 1 m + 1 ∑ k = 0 m   f k + c m + d + r m ( c , d ) ( f ) , {:(3.1)int_(0)^(1)f(x)dx=(1)/(m+1)sum_(k=0)^(m)f((k+c)/(m+d))+r_(m)^((c,d))(f)",":}\begin{equation*} \int_{0}^{1} f(x) \mathrm{d} x=\frac{1}{m+1} \sum_{k=0}^{m} f\left(\frac{k+c}{m+d}\right)+r_{m}^{(c, d)}(f), \tag{3.1} \end{equation*}(3.1)∫01f(x)dx=1m+1∑k=0mf(k+cm+d)+rm(c,d)(f),
because
∫ 0 1 p m , k ( x ) d x = ( m k ) Γ ( k + 1 ) Γ ( m − k + 1 ) Γ ( m + 2 ) = 1 m + 1 ∫ 0 1   p m , k ( x ) d x = ( m k ) Γ ( k + 1 ) Γ ( m − k + 1 ) Γ ( m + 2 ) = 1 m + 1 int_(0)^(1)p_(m,k)(x)dx=((m)/(k))(Gamma(k+1)Gamma(m-k+1))/(Gamma(m+2))=(1)/(m+1)\int_{0}^{1} p_{m, k}(x) \mathrm{d} x=\binom{m}{k} \frac{\Gamma(k+1) \Gamma(m-k+1)}{\Gamma(m+2)}=\frac{1}{m+1}∫01pm,k(x)dx=(mk)Γ(k+1)Γ(m−k+1)Γ(m+2)=1m+1
For the monomials e 0 , e 1 , e 2 e 0 , e 1 , e 2 e_(0),e_(1),e_(2)e_{0}, e_{1}, e_{2}e0,e1,e2 the remainder of (3.1) takes the values
r m ( c , d ) ( e 0 ) = 0 , r m ( c , d ) ( e 1 ) = d − 2 c 2 ( m + d ) , r m ( c , d ) ( e 2 ) = − m + 6 c ( m + c ) − 2 d ( 2 m + d ) 6 ( m + d ) 2 . r m ( c , d ) e 0 = 0 , r m ( c , d ) e 1 = d − 2 c 2 ( m + d ) , r m ( c , d ) e 2 = − m + 6 c ( m + c ) − 2 d ( 2 m + d ) 6 ( m + d ) 2 . {:[r_(m)^((c,d))(e_(0))=0","quadr_(m)^((c,d))(e_(1))=(d-2c)/(2(m+d))","],[r_(m)^((c,d))(e_(2))=-(m+6c(m+c)-2d(2m+d))/(6(m+d)^(2)).]:}\begin{gathered} r_{m}^{(c, d)}\left(e_{0}\right)=0, \quad r_{m}^{(c, d)}\left(e_{1}\right)=\frac{d-2 c}{2(m+d)}, \\ r_{m}^{(c, d)}\left(e_{2}\right)=-\frac{m+6 c(m+c)-2 d(2 m+d)}{6(m+d)^{2}} . \end{gathered}rm(c,d)(e0)=0,rm(c,d)(e1)=d−2c2(m+d),rm(c,d)(e2)=−m+6c(m+c)−2d(2m+d)6(m+d)2.
It is easy to see that if we choose d = 2 c d = 2 c d=2cd=2 cd=2c then the degree of exactness of the corresponding quadrature formula is N = 1 N = 1 N=1N=1N=1, although the operator P m ( c .2 c ) P m ( c .2 c ) P_(m)^((c.2 c))P_{m}^{(c .2 c)}Pm(c.2c) does not reproduce the linear functions.
This can be written under the following form
(3.2) ∫ 0 1 f ( x ) d x = 1 m + 1 ∑ k = 0 m f ( k + c m + 2 c ) + K m ( c ) f ′ ′ ( ξ ) (3.2) ∫ 0 1   f ( x ) d x = 1 m + 1 ∑ k = 0 m   f k + c m + 2 c + K m ( c ) f ′ ′ ( ξ ) {:(3.2)int_(0)^(1)f(x)dx=(1)/(m+1)sum_(k=0)^(m)f((k+c)/(m+2c))+K_(m)(c)f^('')(xi):}\begin{equation*} \int_{0}^{1} f(x) \mathrm{d} x=\frac{1}{m+1} \sum_{k=0}^{m} f\left(\frac{k+c}{m+2 c}\right)+K_{m}(c) f^{\prime \prime}(\xi) \tag{3.2} \end{equation*}(3.2)∫01f(x)dx=1m+1∑k=0mf(k+cm+2c)+Km(c)f′′(ξ)
where 0 < ξ < 1 0 < ξ < 1 0 < xi < 10<\xi<10<ξ<1 and
K m ( c ) = − m − 2 c ( m + c ) 12 ( m + 2 c ) 2 = ( 2 c − 1 ) m + 2 c 2 12 ( m + 2 c ) 2 . K m ( c ) = − m − 2 c ( m + c ) 12 ( m + 2 c ) 2 = ( 2 c − 1 ) m + 2 c 2 12 ( m + 2 c ) 2 . K_(m)(c)=-(m-2c(m+c))/(12(m+2c)^(2))=((2c-1)m+2c^(2))/(12(m+2c)^(2)).K_{m}(c)=-\frac{m-2 c(m+c)}{12(m+2 c)^{2}}=\frac{(2 c-1) m+2 c^{2}}{12(m+2 c)^{2}} .Km(c)=−m−2c(m+c)12(m+2c)2=(2c−1)m+2c212(m+2c)2.
In the special case c = 0 c = 0 c=0c=0c=0 formula (3.2) will be
(3.3) ∫ 0 1 f ( x ) d x = 1 m + 1 ∑ k = 0 m f ( k m ) − 1 12 m f ′ ′ ( ξ ) (3.3) ∫ 0 1   f ( x ) d x = 1 m + 1 ∑ k = 0 m   f k m − 1 12 m f ′ ′ ( ξ ) {:(3.3)int_(0)^(1)f(x)dx=(1)/(m+1)sum_(k=0)^(m)f((k)/(m))-(1)/(12 m)f^('')(xi):}\begin{equation*} \int_{0}^{1} f(x) \mathrm{d} x=\frac{1}{m+1} \sum_{k=0}^{m} f\left(\frac{k}{m}\right)-\frac{1}{12 m} f^{\prime \prime}(\xi) \tag{3.3} \end{equation*}(3.3)∫01f(x)dx=1m+1∑k=0mf(km)−112mf′′(ξ)
and it corresponds to the approximation of the function f f fff by the Bernstein polynomial B m f B m f B_(m)fB_{m} fBmf.
3.2. One observes that the least value of K m ( c ) K m ( c ) K_(m)(c)K_{m}(c)Km(c) is obtained for c = 1 2 c = 1 2 c=(1)/(2)c=\frac{1}{2}c=12, when formula (3.2) becomes
(3.4) ∫ 0 1 f ( x ) d x = 1 m + 1 ∑ k = 0 m f ( 2 k + 1 2 m + 2 ) + 1 24 ( m + 1 ) 2 f ′ ′ ( ξ ) (3.4) ∫ 0 1   f ( x ) d x = 1 m + 1 ∑ k = 0 m   f 2 k + 1 2 m + 2 + 1 24 ( m + 1 ) 2 f ′ ′ ( ξ ) {:(3.4)int_(0)^(1)f(x)dx=(1)/(m+1)sum_(k=0)^(m)f((2k+1)/(2m+2))+(1)/(24(m+1)^(2))f^('')(xi):}\begin{equation*} \int_{0}^{1} f(x) \mathrm{d} x=\frac{1}{m+1} \sum_{k=0}^{m} f\left(\frac{2 k+1}{2 m+2}\right)+\frac{1}{24(m+1)^{2}} f^{\prime \prime}(\xi) \tag{3.4} \end{equation*}(3.4)∫01f(x)dx=1m+1∑k=0mf(2k+12m+2)+124(m+1)2f′′(ξ)
which represent the well known composite midpoint or rectangular quadrature formula.
Therefore by using the Bernstein type polynomial
(3.5) ( p ( 1 2 ⋅ 1 ) f ) ( x ) = ( S m f ) ( x ) = ∑ k = 0 m p m , k ( x ) f ( 2 k + 1 2 m + 2 ) (3.5) p 1 2 ⋅ 1 f ( x ) = S m f ( x ) = ∑ k = 0 m   p m , k ( x ) f 2 k + 1 2 m + 2 {:(3.5)(p^(((1)/(2)*1))f)(x)=(S_(m)f)(x)=sum_(k=0)^(m)p_(m,k)(x)f((2k+1)/(2m+2)):}\begin{equation*} \left(p^{\left(\frac{1}{2} \cdot 1\right)} f\right)(x)=\left(S_{m} f\right)(x)=\sum_{k=0}^{m} p_{m, k}(x) f\left(\frac{2 k+1}{2 m+2}\right) \tag{3.5} \end{equation*}(3.5)(p(12⋅1)f)(x)=(Smf)(x)=∑k=0mpm,k(x)f(2k+12m+2)
we obtain the best quadrature formula having the form (3.1), namely formula (3.4).
In this case the least maximum value of the mean square error is v m ( 1 2 ) = m 4 ( m + 1 ) 2 v m 1 2 = m 4 ( m + 1 ) 2 v_(m)((1)/(2))=(m)/(4(m+1)^(2))v_{m}\left(\frac{1}{2}\right)=\frac{m}{4(m+1)^{2}}vm(12)=m4(m+1)2.

4. QUANTITATIVE ESTIMATIONS OF APPROXIMATION

4.1. Now we establish some estimates of the order of approximation of a function f ∈ C [ 0 , 1 ] f ∈ C [ 0 , 1 ] f in C[0,1]f \in C[0,1]f∈C[0,1] by means of the polynomials (1.3).
Since the constants are reproduced by the operator defined at (1.3), according to a known result (see, e.g., [17] or [12]), we can write
| f ( x ) − ( P m ( c , d ) f ) ( x ) | ≤ [ 1 + γ − 2 P m ( r , d ) ( ( t − x ) 2 : x ) ] ω 1 ( f ; γ ) , f ( x ) − P m ( c , d ) f ( x ) ≤ 1 + γ − 2 P m ( r , d ) ( t − x ) 2 : x ω 1 ( f ; γ ) , |f(x)-(P_(m)^((c,d))f)(x)| <= [1+gamma^(-2)P_(m)^((r,d))((t-x)^(2):x)]omega_(1)(f;gamma),\left|f(x)-\left(P_{m}^{(c, d)} f\right)(x)\right| \leq\left[1+\gamma^{-2} P_{m}^{(r, d)}\left((t-x)^{2}: x\right)\right] \omega_{1}(f ; \gamma),|f(x)−(Pm(c,d)f)(x)|≤[1+γ−2Pm(r,d)((t−x)2:x)]ω1(f;γ),
where ω 1 ω 1 omega_(1)\omega_{1}ω1 represents the first order modulus of continuity and γ > 0 γ > 0 gamma > 0\gamma>0γ>0.
If we take into account (2.1) and (2.2), we obtain
(4.1) | f ( x ) − ( P m ( c , d ) f ) ( x ) | ≤ [ 1 + γ − 2 e m 2 ( x ; c , d ) ] ω 1 ( f ; γ ) (4.1) f ( x ) − P m ( c , d ) f ( x ) ≤ 1 + γ − 2 e m 2 ( x ; c , d ) ω 1 ( f ; γ ) {:(4.1)|f(x)-(P_(m)^((c,d))f)(x)| <= [1+gamma^(-2)e_(m)^(2)(x;c,d)]omega_(1)(f;gamma):}\begin{equation*} \left|f(x)-\left(P_{m}^{(c, d)} f\right)(x)\right| \leq\left[1+\gamma^{-2} e_{m}^{2}(x ; c, d)\right] \omega_{1}(f ; \gamma) \tag{4.1} \end{equation*}(4.1)|f(x)−(Pm(c,d)f)(x)|≤[1+γ−2em2(x;c,d)]ω1(f;γ)
Now we can state the following important result.
THEOREM 4.1. If m > d 2 m > d 2 m > d^(2)m>d^{2}m>d2, then in the sup norm we can write the inequality
(4.2) ‖ f − P m ( c , d ) f ‖ ≤ { 1 + m 4 ( m + d ) [ 1 + ( d − 2 c ) 2 m − d 2 ] } ω 1 ( f ; 1 m + d ) (4.2) f − P m ( c , d ) f ≤ 1 + m 4 ( m + d ) 1 + ( d − 2 c ) 2 m − d 2 ω 1 f ; 1 m + d {:(4.2)||f-P_(m)^((c,d))f|| <= {1+(m)/(4(m+d))[1+((d-2c)^(2))/(m-d^(2))]}omega_(1)(f;(1)/(sqrt(m+d))):}\begin{equation*} \left\|f-P_{m}^{(c, d)} f\right\| \leq\left\{1+\frac{m}{4(m+d)}\left[1+\frac{(d-2 c)^{2}}{m-d^{2}}\right]\right\} \omega_{1}\left(f ; \frac{1}{\sqrt{m+d}}\right) \tag{4.2} \end{equation*}(4.2)‖f−Pm(c,d)f‖≤{1+m4(m+d)[1+(d−2c)2m−d2]}ω1(f;1m+d)
Proof. The idea here is to take into consideration the fact that the maximum value of (2.2) is given at (2.3) and then to choose γ = 1 / m + d γ = 1 / m + d gamma=1//sqrt(m+d)\gamma=1 / \sqrt{m+d}γ=1/m+d.
In the special case of the operator S m ( c ) S m ( c ) S_(m)^((c))S_{m}^{(c)}Sm(c), defined at (2.4), we obtain the inequality
‖ f − S m ( c ) ‖ ≤ [ 1 + m 4 ( m + 2 c ) ] ω 1 ( f : 1 m + 2 c ) f − S m ( c ) ≤ 1 + m 4 ( m + 2 c ) ω 1 f : 1 m + 2 c ||f-S_(m)^((c))|| <= [1+(m)/(4(m+2c))]omega_(1)(f:(1)/(sqrt(m+2c)))\left\|f-S_{m}^{(c)}\right\| \leq\left[1+\frac{m}{4(m+2 c)}\right] \omega_{1}\left(f: \frac{1}{\sqrt{m+2 c}}\right)‖f−Sm(c)‖≤[1+m4(m+2c)]ω1(f:1m+2c)
It can be written also under the following form
(4.3) ‖ f − S m ( c ) f ‖ ≤ ( 5 4 − c 2 ( m + 2 c ) ) ω 1 ( f ; 1 m + 2 c ) (4.3) f − S m ( c ) f ≤ 5 4 − c 2 ( m + 2 c ) ω 1 f ; 1 m + 2 c {:(4.3)||f-S_(m)^((c))f|| <= ((5)/(4)-(c)/(2(m+2c)))omega_(1)(f;(1)/(sqrt(m+2c))):}\begin{equation*} \left\|f-S_{m}^{(c)} f\right\| \leq\left(\frac{5}{4}-\frac{c}{2(m+2 c)}\right) \omega_{1}\left(f ; \frac{1}{\sqrt{m+2 c}}\right) \tag{4.3} \end{equation*}(4.3)‖f−Sm(c)f‖≤(54−c2(m+2c))ω1(f;1m+2c)
For c = 0 c = 0 c=0c=0c=0 it reduces to the inequality of Popoviciu-Lorentz ([16]. [14]):
(4.4) ‖ f − B m f ‖ ≤ 5 4 ( ω 1 ( f ; 1 m ) . (4.4) f − B m f ≤ 5 4 ω 1 f ; 1 m . {:(4.4)||f-B_(m)f|| <= (5)/(4)(omega_(1)(f;(1)/(sqrtm)).:}:}\begin{equation*} \left\|f-B_{m} f\right\| \leq \frac{5}{4}\left(\omega_{1}\left(f ; \frac{1}{\sqrt{m}}\right) .\right. \tag{4.4} \end{equation*}(4.4)‖f−Bmf‖≤54(ω1(f;1m).
In the case of the polynomial S m f S m f S_(m)fS_{m} fSmf, defined at (3.5), we find the following inequality
‖ f − S m f ‖ ≤ C m ω 1 ( f ; 1 m + 1 ) f − S m f ≤ C m ω 1 f ; 1 m + 1 ||f-S_(m)f|| <= C_(m)omega_(1)(f;(1)/(sqrt(m+1)))\left\|f-S_{m} f\right\| \leq C_{m} \omega_{1}\left(f ; \frac{1}{\sqrt{m+1}}\right)‖f−Smf‖≤Cmω1(f;1m+1)
where C m = 1 4 ( 5 − 1 m + 1 ) C m = 1 4 5 − 1 m + 1 C_(m)=(1)/(4)(5-(1)/(m+1))C_{m}=\frac{1}{4}\left(5-\frac{1}{m+1}\right)Cm=14(5−1m+1).
If we replace in (4.1) γ = α δ ( α , δ ∈ R + ) γ = α δ α , δ ∈ R + gamma=alpha delta(alpha,delta inR_(+))\gamma=\alpha \delta\left(\alpha, \delta \in \mathbb{R}_{+}\right)γ=αδ(α,δ∈R+), we obtain
| f ( x ) − ( P m ( c , d ) f ) ( x ) | ≤ [ 1 + ( α δ ) − 2 e m 2 ( x ; c , d ) ] ω 1 ( f ; α δ ) f ( x ) − P m ( c , d ) f ( x ) ≤ 1 + ( α δ ) − 2 e m 2 ( x ; c , d ) ω 1 ( f ; α δ ) |f(x)-(P_(m)^((c,d))f)(x)| <= [1+(alpha delta)^(-2)e_(m)^(2)(x;c,d)]omega_(1)(f;alpha delta)\left|f(x)-\left(P_{m}^{(c, d)} f\right)(x)\right| \leq\left[1+(\alpha \delta)^{-2} e_{m}^{2}(x ; c, d)\right] \omega_{1}(f ; \alpha \delta)|f(x)−(Pm(c,d)f)(x)|≤[1+(αδ)−2em2(x;c,d)]ω1(f;αδ)
In the case c = d = 0 c = d = 0 c=d=0c=d=0c=d=0, by selecting δ = x ( 1 − x ) m δ = x ( 1 − x ) m delta=sqrt((x(1-x))/(m))\delta=\sqrt{\frac{x(1-x)}{m}}δ=x(1−x)m, we get
| f ( x ) − ( B m f ) ( x ) | ≤ ( 1 + 1 α 2 ) ω 1 ( f ; α x ( 1 − x m ) f ( x ) − B m f ( x ) ≤ 1 + 1 α 2 ω 1 f ; α x ( 1 − x m |f(x)-(B_(m)f)(x)| <= (1+(1)/(alpha^(2)))omega_(1)(f;alphasqrt((x(1-x)/(m)))\left|f(x)-\left(B_{m} f\right)(x)\right| \leq\left(1+\frac{1}{\alpha^{2}}\right) \omega_{1}\left(f ; \alpha \sqrt{\frac{x(1-x}{m}}\right)|f(x)−(Bmf)(x)|≤(1+1α2)ω1(f;αx(1−xm)
This inequality permits to see that, indeed, the Bernstein polynomials are interpolatory in x = 0 x = 0 x=0x=0x=0 and x = 1 x = 1 x=1x=1x=1.
Because on [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1] we have x ( 1 − x ) ≤ 1 4 x ( 1 − x ) ≤ 1 4 x(1-x) <= (1)/(4)x(1-x) \leq \frac{1}{4}x(1−x)≤14, by choosing α = 2 α = 2 alpha=2\alpha=2α=2, we arrive at the classical inequality (4.4).
4.2. Now let us use the second order modulus of smoothness
ω 2 ( f ; γ ) = sup { | f ( x − h ) − 2 f ( x ) + f ( x + h ) | : x , x ± h ∈ [ a , b ] , 0 ≤ h ≤ γ } ω 2 ( f ; γ ) = sup { | f ( x − h ) − 2 f ( x ) + f ( x + h ) | : x , x ± h ∈ [ a , b ] , 0 ≤ h ≤ γ } omega_(2)(f;gamma)=s u p{|f(x-h)-2f(x)+f(x+h)|:x,x+-h in[a,b],0 <= h <= gamma}\omega_{2}(f ; \gamma)=\sup \{|f(x-h)-2 f(x)+f(x+h)|: x, x \pm h \in[a, b], 0 \leq h \leq \gamma\}ω2(f;γ)=sup{|f(x−h)−2f(x)+f(x+h)|:x,x±h∈[a,b],0≤h≤γ}
where 0 ≤ γ ≤ 1 2 ( b − a ) 0 ≤ γ ≤ 1 2 ( b − a ) 0 <= gamma <= (1)/(2)(b-a)0 \leq \gamma \leq \frac{1}{2}(b-a)0≤γ≤12(b−a).
By using both moduli ω 1 ω 1 omega_(1)\omega_{1}ω1 and ω 2 ω 2 omega_(2)\omega_{2}ω2 one can find estimates of the approximation of the function f f fff by means of the operator (1.3).
For this purpose we can use an inequality of H. H. Gonska and R. K. Kovacheva [13], included in
Lemma 4.1. If K = [ a , b ] K = [ a , b ] K=[a,b]K=[a, b]K=[a,b] is a compact interval of the real axis and K ′ = [ a ′ , b ′ ] K ′ = a ′ , b ′ K^(')=[a^('),b^(')]K^{\prime}=\left[a^{\prime}, b^{\prime}\right]K′=[a′,b′] is a subinterval of it, and if we assume that L : C ( K ) → B ( K ′ ) L : C ( K ) → B K ′ L:C(K)rarr B(K^('))L: C(K) \rightarrow B\left(K^{\prime}\right)L:C(K)→B(K′) is a positive operator, such that L ( 1 ; x ) = 1 L ( 1 ; x ) = 1 L(1;x)=1L(1 ; x)=1L(1;x)=1 and 0 < γ < 1 2 ( b − a ) 0 < γ < 1 2 ( b − a ) 0 < gamma < (1)/(2)(b-a)0<\gamma<\frac{1}{2}(b-a)0<γ<12(b−a), then we have
| f ( x ) − L ( f ( t ) ; x ) | ≤ 2 γ | L ( t − x ; x ) | ω 1 ( f ; γ ) + + [ 3 2 + 3 2 γ | L ( t − x ; x ) | + 3 4 γ 2 L ( ( t − x ) 2 ; x ) ] ω 2 ( f ; γ ) | f ( x ) − L ( f ( t ) ; x ) | ≤ 2 γ | L ( t − x ; x ) | ω 1 ( f ; γ ) + + 3 2 + 3 2 γ | L ( t − x ; x ) | + 3 4 γ 2 L ( t − x ) 2 ; x ω 2 ( f ; γ ) {:[|f(x)-L(f(t);x)| <= (2)/(gamma)|L(t-x;x)|omega_(1)(f;gamma)+],[+[(3)/(2)+(3)/(2gamma)|L(t-x;x)|+(3)/(4gamma^(2))L((t-x)^(2);x)]omega_(2)(f;gamma)]:}\begin{gathered} |f(x)-L(f(t) ; x)| \leq \frac{2}{\gamma}|L(t-x ; x)| \omega_{1}(f ; \gamma)+ \\ +\left[\frac{3}{2}+\frac{3}{2 \gamma}|L(t-x ; x)|+\frac{3}{4 \gamma^{2}} L\left((t-x)^{2} ; x\right)\right] \omega_{2}(f ; \gamma) \end{gathered}|f(x)−L(f(t);x)|≤2γ|L(t−x;x)|ω1(f;γ)++[32+32γ|L(t−x;x)|+34γ2L((t−x)2;x)]ω2(f;γ)
If we take into account the relations (1.5) and (2.2) we obtain the inequality
(4.5) | f ( x ) − ( P m ( c , d ) f ) ( x ) | ≤ 2 γ ⋅ | c − d x | m + d ω 1 ( f ; γ ) + + [ 3 2 + 3 2 γ ⋅ | c − d x | m + d + 3 4 γ 2 e m 2 ( x ; c , d ) ] ω 2 ( f ; γ ) (4.5) f ( x ) − P m ( c , d ) f ( x ) ≤ 2 γ ⋅ | c − d x | m + d ω 1 ( f ; γ ) + + 3 2 + 3 2 γ ⋅ | c − d x | m + d + 3 4 γ 2 e m 2 ( x ; c , d ) ω 2 ( f ; γ ) {:[(4.5)|f(x)-(P_(m)^((c,d))f)(x)| <= (2)/(gamma)*(|c-dx|)/(m+d)omega_(1)(f;gamma)+],[+[(3)/(2)+(3)/(2gamma)*(|c-dx|)/(m+d)+(3)/(4gamma^(2))e_(m)^(2)(x;c,d)]omega_(2)(f;gamma)]:}\begin{align*} & \left|f(x)-\left(P_{m}^{(c, d)} f\right)(x)\right| \leq \frac{2}{\gamma} \cdot \frac{|c-d x|}{m+d} \omega_{1}(f ; \gamma)+ \tag{4.5}\\ & +\left[\frac{3}{2}+\frac{3}{2 \gamma} \cdot \frac{|c-d x|}{m+d}+\frac{3}{4 \gamma^{2}} e_{m}^{2}(x ; c, d)\right] \omega_{2}(f ; \gamma) \end{align*}(4.5)|f(x)−(Pm(c,d)f)(x)|≤2γ⋅|c−dx|m+dω1(f;γ)++[32+32γ⋅|c−dx|m+d+34γ2em2(x;c,d)]ω2(f;γ)
This implies the following
‖ f − P m ( c , d ) ‖ ≤ 2 γ ⋅ d m + d ω 1 ( f ; γ ) + f − P m ( c , d ) ≤ 2 γ ⋅ d m + d ω 1 ( f ; γ ) + ||f-P_(m)^((c,d))|| <= (2)/(gamma)*(d)/(m+d)omega_(1)(f;gamma)+\left\|f-P_{m}^{(c, d)}\right\| \leq \frac{2}{\gamma} \cdot \frac{d}{m+d} \omega_{1}(f ; \gamma)+‖f−Pm(c,d)‖≤2γ⋅dm+dω1(f;γ)+
+ { 3 2 + 3 d 2 δ ( m + d ) + 3 m 4 ( m + d ) 2 δ 2 [ 1 + ( d − 2 c ) 2 m − d 2 ] } ω 2 ( f ; γ ) + 3 2 + 3 d 2 δ ( m + d ) + 3 m 4 ( m + d ) 2 δ 2 1 + ( d − 2 c ) 2 m − d 2 ω 2 ( f ; γ ) +{(3)/(2)+(3d)/(2delta(m+d))+(3m)/(4(m+d)^(2)delta^(2))[1+((d-2c)^(2))/(m-d^(2))]}omega_(2)(f;gamma)+\left\{\frac{3}{2}+\frac{3 d}{2 \delta(m+d)}+\frac{3 m}{4(m+d)^{2} \delta^{2}}\left[1+\frac{(d-2 c)^{2}}{m-d^{2}}\right]\right\} \omega_{2}(f ; \gamma)+{32+3d2δ(m+d)+3m4(m+d)2δ2[1+(d−2c)2m−d2]}ω2(f;γ)
If we choose γ = 1 / m + d γ = 1 / m + d gamma=1//sqrt(m+d)\gamma=1 / \sqrt{m+d}γ=1/m+d, then we get
‖ f − P m ( c , d ) ‖ ≤ 2 d m + d ω 1 ( f ; 1 m + d ) + + { 3 2 + 3 d 2 m + d + 3 m 4 ( m + d ) [ 1 + ( d − 2 c ) 2 m − d 2 ] } ω 2 ( f ; 1 m + d ) f − P m ( c , d ) ≤ 2 d m + d ω 1 f ; 1 m + d + + 3 2 + 3 d 2 m + d + 3 m 4 ( m + d ) 1 + ( d − 2 c ) 2 m − d 2 ω 2 f ; 1 m + d {:[||f-P_(m)^((c,d))|| <= (2d)/(sqrt(m+d))omega_(1)(f;(1)/(sqrt(m+d)))+],[+{(3)/(2)+(3d)/(2sqrt(m+d))+(3m)/(4(m+d))[1+((d-2c)^(2))/(m-d^(2))]}omega_(2)(f;(1)/(sqrt(m+d)))]:}\begin{gathered} \left\|f-P_{m}^{(c, d)}\right\| \leq \frac{2 d}{\sqrt{m+d}} \omega_{1}\left(f ; \frac{1}{\sqrt{m+d}}\right)+ \\ +\left\{\frac{3}{2}+\frac{3 d}{2 \sqrt{m+d}}+\frac{3 m}{4(m+d)}\left[1+\frac{(d-2 c)^{2}}{m-d^{2}}\right]\right\} \omega_{2}\left(f ; \frac{1}{\sqrt{m+d}}\right) \end{gathered}‖f−Pm(c,d)‖≤2dm+dω1(f;1m+d)++{32+3d2m+d+3m4(m+d)[1+(d−2c)2m−d2]}ω2(f;1m+d)
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Received March 5, 1999
where C = 27 16 = 1.6875 C = 27 16 = 1.6875 C=(27)/(16)=1.6875C=\frac{27}{16}=1.6875C=2716=1.6875.
‖ f − B m f ‖ ≤ C ω 2 ( f ; 1 m ) f − B m f ≤ C ω 2 f ; 1 m ||f-B_(m)f|| <= Comega_(2)(f;(1)/(sqrtm))\left\|f-B_{m} f\right\| \leq C \omega_{2}\left(f ; \frac{1}{\sqrt{m}}\right)‖f−Bmf‖≤Cω2(f;1m)
This inequality was first given in 1994 in the work [13].

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  3. F. Altomare and M. Campiti, Korovkin-Type Approximation and Its Applications, de Gruyter, Berlin, New York, 1994.

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