General Mathematics Vol. 10, No. 3–4 (2002), 57–61
Some results for modified Bernstein
polynomials
Florin Sofonea
Abstract
Our aim is to find some properties for polynomials Ln f , defined
as:
(Ln f )(x) = (Bn f )(x) −
x(1 − x)
00
(Bn f ) (x), f ∈ C[0, 1], n = 2, 3, ...
2(n − 1)
where Bn f is Bernstein polynomial corresponding to f .
2000 Mathematical Subject Classification: 41A17, 41A36.
1. Introduction
The Bernstein polynomials of f : [0, 1] → R is
(Bn f )(x) =
n µ ¶
X
n
k=0
k
k
n−k
x (1 − x)
57
µ ¶
k
f
,
n
n = 1, 2, . . .
58
Florin Sofonea
We consider the polynomials (see [2])
(1)
(Ln f )(x) = (Bn f )(x) −
x(1 − x)
00
(Bn f ) (x), n ≥ 2,
2(n − 1)
which satisfy Ln e0 = e0 , Ln e1 = e1 , Ln e2 = e2 , with ej (t) = tj , j = 0, 1, . . ..
We use the notations
K = [a, b], ∞ < a < b < +∞,
(2)
Ωj (t, x) = Ωj (t) = |t − x|j ,
ω(f ; δ) = sup |f (t) − f (x)|,
j = 0, 1, . . . , x ∈ K,
t, x ∈ K, δ ∈ [0, b − a).
|t−x|<δ
A. Lupaş [3] has proved following proposition:
Theorem 1. If L : C(K) → C(K1 ), K1 = [a1 , b1 ] ⊆ K, is a linear positive
operator, then for all f ∈ C(K) and δ > 0 we have
||f −Lf ||K1 ≤ ||f ||·||e0 −Le0 ||K1 + inf {||Le0 ||K1 +δ −m ||LΩm ||K1 }ω(f ; δ),
m=1,2,...
where || · || = max | · | and Ωm is defined in (2).
K
2. The main results
Theorem 2. If Ln f is defined by (1), then for f ∈ C[0, 1],
µ
¶
µ
¶
1
n
1
19
+ ω f;
||f − Ln f || ≤ ω f ; √
16
4
n
n
Proof. We consider m = 4 and K = [0, 1] in Theorem 1, Ω4 (t) = (t − x)4 .
Taking into account that
(Bn Ω4 )(x) =
¤
1 £
2
2
3(n
−
2)x
(1
−
x)
+
x(1
−
x)
,
n3
Some results for modified Bernstein polynomials
00
(3)
(Bn f ) (x) =
µ
¶
·
¸
n−2
2n(n − 1) X n − 2 k
n−2−k k k + 1 k + 2
,
,
;f ,
=
x (1 − x)
n2
k
n
n
n
k=0
it follows
µ
¶
19
1
||f − Bn f || ≤ ω f ; √
.
16
n
x(1 − x)
we have
2(n − 1)
¯¯
¯¯
00
00 ¯¯
¯¯
||f − Ln f || = ¯¯f − Ln f − α(Bn f ) + α(Bn f ) ¯¯ ≤
¯¯
¯¯
00 ¯¯
¯¯
≤ ||f − Bn f || + ¯¯α(Bn f ) ¯¯ .
Further, if α(x) =
But from (3) the theorem is proved because
x(1 − x)
00
(Bn f ) (x) =
2(n − 1)
¶
n−2 µ
x(1 − x) 2!n(n − 1) X n − 2 k
=
x (1 − x)n−2−k ·
2(n − 1)
n2
k
k=0
·
¸ ·
¸
k+1 k+2
k k+1
,
;f −
,
;f
n
n
n n
·
=
2
n
¶
µ
n−2
x(1 − x) X n − 2 k
x (1 − x)n−2−k ·
=
k
2
k=0
à ¡ ¢
¡ ¢
¡ k+1 ¢
¡k ¢!
k+2
f n − f k+1
−
f
f
n
n
n
·
−
≤
1
1
00
α(x)(Bn f ) (x) =
n
n−2 µ
X
n
¶
µ µ
¶
µ
¶¶
x(1 − x)
n−2 k
1
1
n−2−k
≤
n
x (1 − x)
ω f;
+ ω f;
=
2
k
n
n
k=0
µ
¶ n−2 µ
¶
1 X n−2 k
= nx(1 − x)ω f ;
x (1 − x)n−2−k =
n k=0
k
59
60
Florin Sofonea
µ
1
= nx(1 − x)ω f ;
n
¶
µ
¶
n
1
≤ ω f;
.
4
n
Corollary 1. If Ln f is defined by (1), then for f ∈ C[0, 1] we have
µ
¶
µ
¶
1
1
19
n
||f − Ln f || ≤ ω f ; √
+ ω2 f ;
16
8
n
n
Proof. We observe
¡ ¢
¡ ¢
¡ ¢
·
¸
f k+1
f k+2
f nk
k k+1 k+2
n
n
+
=
,
,
;f = 2 −
1
2
n n
n
n2
n2
n2
· µ ¶
µ
¶
µ
¶¸
µ
¶
n2
k
k+1
k+2
n2 2
k
=
f
− 2f
+f
= ∆ 1 f;
,
2
n
n
n
2 n
n
where
∆rh (f ; x)
r µ ¶
X
r
(−r)r−k f (x + kh),
=
k
k=0
r = 1, 2, . . . ,
h ∈ R.
Using the following definition
ωr (f, δ) = sup ||∆rh (f ; ·)||,
with δ > 0,
0<h≤δ
results
and
·
¸
¶
µ
k k+1 k+2
n2
1
,
,
; f ≤ ω2 f ;
,
n n
n
2
n
¶
µ
x(1 − x) 2n(n − 1) n2
1
α(x)(Bn f ) (x) ≤
ω2 f ;
=
2(n − 1)
n2
2
n
¶
¶
µ
µ
nx(1 − x)
n
1
1
=
ω2 f ;
≤ ω2 f ;
.
2
n
8
n
00
from above formula the proposition is proved.
Some results for modified Bernstein polynomials
61
References
[1] Bernstein S.N., Démonstration du théorème de Weierstrass fondée sur
la calcul des probabilités, Comm. Soc. Math. Charkow Sér. 2 t. 13,
(1912) 1-2.
[2] Franchetti C., Sull’innalzamento dell’ordine di aprossimazione dei polinomi di Bernstein, Rend. Sem. Mat. Univ. Politec. Torino, 28 (1968-69)
1-7.
[3] Lupaş A., Contribuţii la teoria aproximării prin operatori liniari, Teză
de doctorat (Cluj), (1976) (in Romanian).
[4] Sofonea F., Remainder in approximation by means of certain linear operators, Mathematical Analysis and Aproximation Theory, Burg Verlag
(Sibiu), (2002), 255-258.
University ”Lucian Blaga” of Sibiu
Department of Mathematics
Str. I. Raţiu, Nr. 5-7,
550012 - Sibiu, Romania
E-mail: florin.sofonea@ulbsibiu.ro