MATEMATIQKI VESNIK
UDK 517.984
originalni nauqni rad
research paper
61 (2009), 53–60
ASYMPTOTIC BEHAVIOUR OF DIFFERENTIATED
BERNSTEIN POLYNOMIALS
Heiner Gonska and Ioan Raşa
Abstract. In the present note we give a full quantitative version of a theorem of Floater
dealing with the asymptotic behaviour of differentiated Bernstein polynomials. While Floater’s
result is a generalization of the classical Voronovskaya theorem, ours generalizes a hardly known
quantitative version of this theorem due to Videnskiı̆, among others.
1. Introduction
In a recent article Floater [2] proved the following
Theorem 1. If f ∈ C k+2 [0, 1] for some k ≥ 0, then
n
o 1 dk
lim n (Bn f )(k) (x) − f (k) (x) = · k {x(1 − x)f 00 (x)},
n→∞
2 dx
uniformly for x ∈ [0, 1].
Here Bn is the Bernstein operator defined for a function f : [0, 1] → R and
x ∈ [0, 1] by
µ ¶
n
P
i
Bn f (x) =
pn,i (x),
f
n
i=0
where
pn,i (x) =
µ ¶
n i
x (1 − x)n−i , i = 0, . . . , n.
i
In the sequel we will also use the abbreviations X = x(1 − x) and ei (x) = xi for
i = 0, 1, 2, . . . . Floater’s result is a generalization of the classical Voronovskaya
theorem (see [10]) which is obtained for k = 0. In a recent paper [3] the latter
AMS Subject Classification: 41A10, 41A17, 41A25, 41A28, 41A36.
Keywords and phrases: Voronovskaya theorem; Bernstein operators; simultaneous approximation; k-th Kantorovich modification; convex operators.
Presented at the international conference Analysis, Topology and Applications 2008
(ATA2008), held in Vrnjačka Banja, Serbia, from May 30 to June 4, 2008.
53
54
H. Gonska, I. Raşa
theorem was given in quantitative form as follows, improving an earlier estimate
by Videnskiı̆ (see [9]).
Theorem 2. For f ∈ C 2 [0, 1], x ∈ [0, 1] and n ∈ N one has
Ã
!
r
¯
¯
¯
¯
1
x(1 − x)
¯n · [Bn (f ; x) − f (x)] − x(1 − x) · f 00 (x)¯ ≤ x(1 − x) · ω̃ f 00 ;
.
+
¯
¯
2
2
n2
n
Here ω̃ is the least concave majorant of ω, the first order modulus of continuity,
satisfying
ω(f ; ²) ≤ ω̃(f ; ²) ≤ 2ω(f ; ²),
² ≥ 0.
The above inequality follows from a more general asymptotic statement which was
inspired by results of Bernstein [1] and Mamedov [7]. This is given in
Theorem 3. Let q ∈ N0 , f ∈ C q [0, 1] and L : C[0, 1] → C[0, 1] be a positive
linear operator. Then
¯
¯
q
¯
P
f (r) (x) ¯¯
r
¯L(f ; x) −
L((e1 − x) ; x) ·
¯
r! ¯
r=0
µ
¶
L(|e1 − x|q ; x)
L(|e1 − x|q+1 ; x)
(q)
≤
ω̃ f ;
.
q!
(q + 1)L(|e1 − x|q ; x)
It is the aim of this note to prove a quantitative version of Floater’s result. In
doing so we will make essential use of a corollary of Theorem 3 for the case q = 2.
Corollary 1. Under the assumptions of Theorem 3 one has, for q = 2, the
inequality
¯
¯
¯
¯
¯L(f ; x) − f (x) · L(e0 ; x) − f 0 (x) · L(e1 − x; x) − 1 f 00 (x) · L((e1 − x)2 ; x)¯
¯
¯
2
s
!
Ã
L((e1 − x)4 ; x)
1
2
00 1
.
≤ · L((e1 − x) ; x) · ω̃ f ; ·
2
3
L((e1 − x)2 ; x)
The square root is obtained by using the Cauchy-Schwarz inequality for positive
linear functionals.
2. An auxiliary result
An operator L : C[0, 1] → C k [0, 1] is said to be convex of order k − 1 if it
preserves convexity of order k − 1, k ∈ N ∪ {0}. This means that any function f
with divided differences
[x0 , . . . , xk ; f ] ≥ 0 for any x0 < · · · < xk ∈ [0, 1]
is mapped to a function Lf having the same property.
Asymptotic behaviour of differentiated Bernstein polynomials
55
The Bernstein operator is an example of a mapping which is convex of all
orders k ∈ N ∪ {0}.
For an operator L being convex of order k − 1 and satisfying L(Πk−1 ) ⊆ Πk−1
consider
Z x
(x − t)k−1
Ik : C[0, 1] → C[0, 1] given by (Ik f )(x) =
f (t) dt.
(k − 1)!
0
Let Qk := Dk ◦ L ◦ Ik where Dk =
dk
.
dxk
Qk may be considered as a k-th order Kantorovich modification of L. Since
L is convex of order k − 1, it follows that Qk is a linear and positive (convex of
order −1) operator. Since Ik Dk f − f ∈ Πk−1 and L(Πk−1 ) ⊆ Πk−1 , we have
L(Ik Dk f − f ) ∈ Πk−1 . It follows Dk LIk Dk f = Dk Lf , hence Qk Dk f = Dk Lf , for
all f ∈ C k [0, 1].
To our knowledge the latter construction is due to Sendov and Popov [8].
3. Main result
In this section we will prove the main result of this note by providing the
following quantitative version of Floater’s convergence result.
Theorem 4. If f ∈ C k+2 [0, 1] for some k ≥ 0, then
¯
¯
k
¯
¯
¯n[(Bn f )(k) (x) − f (k) (x)] − 1 · d {x(1 − x)f 00 (x)}¯
¯
¯
k
2 dx
¶
µ
µ ¶
1
1
.
· max {|f (κ) (x)|} + O(1) · ω̃ f (k+2) ; √
≤O
k≤κ≤k+2
n
n
¡ ¢
¡ ¢
Here O n1 and O(1) represent sequences of order O n1 and O(1), respectively,
which depend on the fixed k.
Proof. Put Qkn := Dk Bn Ik . For this positive linear operator we apply Corollary 1 and write the left hand side of the inequality for f ∈ C k+2 [0, 1] as
|Qkn (f (k) ; x) − f (k) (x) · Qkn (e0 ; x) − f (k+1) (x) · Qkn (e1 − x; x)
1
− · f (k+2) (x) · Qkn ((e1 − x)2 ; x)|
2
= |Dk Bn (f ; x) − f (k) (x) + f (k) (x)(1 − Qkn (e0 ; x)) − f (k+1) (x) · Qkn (e1 − x; x)
1
1
dk
− f (k+2) (x) · Qkn ((e1 − x)2 ; x) −
· k {x(1 − x) · f 00 (x)}
2
2n dx
1
dk
+
·
{x(1 − x) · f 00 (x)}|
2n dxk
1
dk
= |Dk Bn (f ; x) − f (k) (x) −
· k {x(1 − x)f 00 (x)}
2n dx
− {(Qkn (e0 ; x) − 1) · f (k) (x) + f (k+1) (x) · Qkn (e1 − x; x)
1
dk
1
· k {x(1 − x) · f 00 (x)}|.
+ f (k+2) (x) · Qkn ((e1 − x)2 ; x) −
2
2n dx
56
H. Gonska, I. Raşa
Multiplying the inequality of Corollary 1 by n and using the (second) triangular
inequality yields
¯
¯
1 dk
¯
¯
¯n · {Dk Bn (f ; x) − f (k) (x)} − · k {x(1 − x) · f 00 (x)}¯
2
dx
¯
¯
≤ ¯n · (Qkn (e0 ; x) − 1) · f (k) (x) + f (k+1) (x) · n · Qkn (e1 − x; x)
¯
1
1 dk
¯
+ f (k+2) (x) · n · Qkn ((e1 − x)2 ; x) − · k {x(1 − x) · f 00 (x)}¯
2
2
dx
s
Ã
!
n
Qkn ((e1 − x)4 ; x)
k
2
(k+2) 1
+ · Qn ((e1 − x) ; x) · ω̃ f
;
.
2
3 Qkn ((e1 − x)2 ; x)
Both summands of the r.h.s. will now be inspected separately. In order to do so,
first observe that by Leibniz’ rule one has
½
¾
1
k(k − 1) (k)
1 dk
00
(k+2)
(k+1)
0
·
{Xf (x)} =
f
(x) · X + k · f
(x) · X +
f (x)(−2) .
2 dxk
2
2
Note that this is correct also for k ∈ {0, 1}. So the first summand can be estimated
from above by
¯
¯
¯
¯
¯
¯
¯
¯
¯n · (Qkn (e0 ; x) − 1) + k(k − 1) ¯ ·|f (k) (x)|+ ¯n · Qkn (e1 − x; x) − k X 0 ¯ ·|f (k+1) (x)|
¯
¯
¯
2
2 ¯
¯
¯
¯n
1 ¯¯
k
2
¯
+ ¯ · Qn ((e1 − x) ; x) − X ¯ · |f (k+2) (x)|
2
2
:= Akn · |f (k) (x)| + Bnk · |f (k+1) (x)| + Cnk · |f (k+2) (x)|.
Now for n ≥ 1 and k ≥ 0, also noting that (n)0 = 1, we have
¯
¯ µ
¶
¯
k(k − 1) ¯¯
(n)k
k
¯
−1 +
An = ¯n ·
¯
nk
2
¯
¯
¯
¯
k terms
¯
¯ z
}|
{
¯ n(n − 1) . . . (n − k + 1) −nk
k(k − 1) ¯¯
¯
+
= ¯n ·
¯
¯
¯
nk
2
¯
¯
¯
¯
¯
¯
¯
¯
µ ¶
¯ (n)k − nk
k(k − 1) ¯¯ ¯¯ k(k − 1) k(k − 1) ¯¯
1
¯
+
+
=¯
¯ ≤ ¯−
¯+O n .
nk−1
2
2
2
In order to verify the last inequality it is only necessary to observe that
½
¾
1
k(k − 1) k−1
1
k
k
k
{(n)
−
n
}
=
n
−
n
+
(lower
order
terms
in
n)
−
n
.
k
nk−1
nk−1
2
Note that Akn = 0 for k ∈ {0, 1}.
Asymptotic behaviour of differentiated Bernstein polynomials
57
Moreover (see [5], pp. 44–45), for n ≥ 1 and k ≥ 0 we get
¯
¯¯
¯
¯
¯ ¯ (n)k k 0 k
¯
k
k
k
0¯ ¯
0¯
¯
Bn = ¯n · Qn (e1 − x; x) − · X ¯ ¯n · k ·
X − ·X ¯
2
n
2n
2
¯
¯
µ ¶
¯ k 0 k(k − 1)
k 0 ¯¯ (n)k
1
= |X | · ¯ k − 1¯¯ ≤ |X | ·
=O
.
2
n
2
2k
n
We also have Bnk = 0 for k ∈ {0, 1}. For the last factor we have (see [6], p. 26) for
n≥k+2
¯
¯
¯n
1 ¯
Cnk = ¯¯ · Qkn ((e1 − x)2 ; x) − X ¯¯
2
2
¯
¯
¸
·
¯ n (n)k
1 ¯
1
= ¯¯ · k+2 (n − k(k + 1)) · X + k(3k + 1) − X ¯¯
2 n
12
2
¯
¯
½
¾
¯1
¯
(n)k
1 (n)k
¯
= ¯¯ X
·
(n
−
k(k
+
1))
−
1
+
k(3k
+
1)
¯
k+1
k+1
2
n
24 n
¯
¯
µ ¶
¯
1 ¯ (n)k
1
.
≤ X ¯¯ k+1 (n − k(k + 1)) − 1¯¯ + O
2
n
n
It remains to consider the quantity inside | . . . |. The latter is equal to
µ
¶
1
k(k − 1) k−1
k
k−2
n −
n
+ O(n
)(n − k(k + 1) − 1
nk+1
2
µ ¶
µ ¶
µ ¶
µ ¶
µ ¶
k(k − 1)
1
1
1
1
1
=1−
+O
−O
+O
−O
−1=O
.
2
2
3
2n
n
n
n
n
n
Note that Cnk = 0 for k = 0. So we know that
¯
¯
k
¯
¯
¯n · {Dk Bn (f ; x) − f (k) (x)} − 1 · d {x(1 − x) · f 00 (x)}¯
¯
¯
k
2 dx
µ ¶
1
· max{|f (k) (x)|, |f (k+1) (x)|, |f (k+2) (x)|}
≤O
n
s
!
Ã
n
1
Qkn ((e1 − x)4 ; x)
k
2
(k+2)
+ · Qn ((e1 − x) ; x) · ω̃ f
;
,
2
3 Qkn ((e1 − x)2 ; x)
where O depends only on k.
For the factor in front of ω̃(f (k+2) ; . . . ) we have already observed that
·
¸
n (n)k
n
1
2
k
· Qn ((e1 − x) ; x) = · k+2 (n − k(k + 1)) · X + k(3k + 1) = O(1).
2
2 n
12
Hence it remains to consider the square root in ω̃(f (k+2) ; . . . ). This is done in the
following lemmas dealing with the moments of Qkn .
Lemma 1. Suppose that Ln : Π → Π, n ≥ 1, is a linear operator mapping
polynomials into polynomials and such that Ln (Πj ) ⊂ Πj , n ≥ 1, j ≥ 0. If we
58
H. Gonska, I. Raşa
define
1
Ln ((e1 − x)m ; x), n ≥ 1, m ≥ 0,
m!
1
k
Rn,p
(x) := Qkn ((e1 − x)p ; x), n ≥ 1, k ≥ 0, p ≥ 0,
p!
Mn,m (x) :=
then
k
Rn,p
(x) =
µ
p+k
P
k
p+k−j
j=p
Proof. We will use the notation
f, f ∈ C[0, 1].
First observe that
(k)
¶
(j−p)
· Mn,j
(x).
f = Ik f to denote a k-th antiderivative of
Mn,m ∈ Πm for n ≥ 1 and m ≥ 0.
Now let k ≥ 0, p ≥ 0 be fixed, f ∈ Πp , x ∈ [0, 1]. Then
(k)
f (t) =
p+k
P
j=0
and hence
Ln ((k) f )(x) =
Thus
Qkn f
=
p+k
P
((
(k)
f)
(j)
j=0
1
((k) f )(j) (x) (t − x)j ,
j!
p+k
P
j=0
(k)
Mn,j )
((k) f )(j) (x)Mn,j (x).
µ ¶
k (k) (j+i)
(k−i)
=
( f)
· Mn,j .
j=0 i=0 i
p+k
k
P P
(k−i)
Noting that Mn,j
≡ 0 if 0 ≤ i < k − j, we may write
µ ¶
p+k
k
P
P
k (i+j−k) (k−i)
k
Qn f =
f
Mn,j .
j=0 i=max{0,k−j} i
Substituting i + j − j = `, i = ` + k − j, the latter becomes
µ
¶
p+k
j
P
P
k
(j−`)
=
f (`) Mn,j
j=0 `=max{j−k,0} ` + k − j
µ
¶
p+k
P min{`+k,p+k}
P
k
(j−`)
=
f (`) Mn,j , f ∈ Πp .
`+k−j
`=0
j=`
This is correct in view of (aj,` ∈ R)
p+k
P
j
P
j=0 `=max{j−k,0}
aj,`
 k+`

P




a
,
if
k
+
`
≤
k
+
p,
j,`



p+k
P  j=`
=
 k+p

P
`=0 


aj,` , if k + ` > k + p.


j=`
=
p+k
P min{`+k,p+k}
P
`=0
j=`
aj,` .
Asymptotic behaviour of differentiated Bernstein polynomials
59
Note that the l.h.s. corresponds to “horizontal summation first, then vertical”, while
the r.h.s. corresponds to the opposite.
f
(`)
1
k
We obtain Rn,p
(x) if we put f = p!
(e1 − x)p in Qkn f , also observing that
(x) = 0 for ` ∈ {0, . . . , p + k} \ {p} and f (p) (x) = 1. Hence
¶
µ
p+k
P
k
(j−p)
k
Rn,p
(x) =
Mn,j (x), n ≥ 1, k ≥ 0, p ≥ 0.
j=p p + k − j
In order to come up with a description of the asymptotic behavior of the ratio
k
in question, we investigate the quantities Rn,p
(x) further in case that Ln = Bn .
We have the following
Lemma 2. For the Bernstein operators Bn we have
n
k
Rn,4
≤ A,
k
Rn,2
n ≥ 1,
for some positive constant A.
Proof. For Bn we have
Mn,2j (x) =
Mn,2j+1 (x) =
X
j−1
P
n2j−1
i=0
aji (n)X i ,
P
XX 0 j−1
bji (n)X i ,
2j
n i=0
where aji (n) and bji (n) are polynomials in n, of degree i.
By the previous lemma,
µ
¶
k+4
P
k
1
(j−4)
k
Rn,4 =
Mn,j = 2 (X 2 uk+1 (n) + Xvk (n) + wk−1 (n)),
k
+
4
−
j
n
j=4
where uk+1 , vk and wk−1 are polynomials of degrees indicated by the corresponding
indices. Analogously,
1
k
= (Xqk+1 (n) + rk (n)).
Rn,2
n
Now the claim of the lemma is a consequence of the latter two representations.
Continuation of the proof of Theorem 4. All we have to observe is that
s
s
r
k (x)
4! · Rn,4
Qkn ((e1 − x)4 ; x
1
=
≤ 6A · ,
k (x)
Qkn ((e1 − x)2 ; x)
n
2! · Rn,2
where A is the uniform bound from Lemma 2. The final statement follows from
the inequality
ω̃(f ; c²) ≤ (c + 1) · ω̃(f ; ²), c, ² ≥ 0.
60
H. Gonska, I. Raşa
Remark 1. We noted earlier that Akn = Bnk = Cnk = 0 for k = 0. In this case
the inequality of Theorem 4 can be replaced by
¶
µ
1
1
|n[Bn f (x) − f (x)] − x(1 − x) · f 00 (x)| ≤ O(1) · ω̃ f 00 ; √
.
2
n
In fact, looking at the proof again shows that we even get the right hand side
s
Ã
!
¶
µ
n
Bn ((e1 − x)4 ; x)
x(1 − x)
1
00
2
00 1
√
· Bn ((e1 − x) ; x) · ω̃ f ;
≤
·
ω̃
f
;
.
2
3 Bn ((e1 − x)2 ; x)
2
3 n
This is the quantitative version of the classical Voronovskaya theorem given first
in [4].
REFERENCES
[1] S. N. Bernstein, Complément à l’article de E. Voronovskaya “Détermination de la forme
asymptotique de l’approximation des fonctions par les polynômes de M. Bernstein”, C. R.
(Dokl.) Acad. Sci. URSS A (1932), 4, 86–92.
[2] M. S. Floater, On the convergence of derivatives of Bernstein approximation, J. Approx.
Theory, 134 (2005), 130–135.
[3] H. Gonska, On the degree of approximation in Voronovskaja’s theorem, Studia Univ. BabeşBolyai, Mathematica 52, 3 (2007), 103–116.
[4] H. Gonska, P. Piţul and I. Raşa, On Peano’s form of the Taylor remainder, Voronovskaja’s
theorem and the commutator of positive linear operators, In: “Numerical Analysis and Approximation Theory” (Proc. Int. Conf. Cluj-Napoca 2006; ed. by O. Agratini & P. Blaga),
55–80. Cluj-Napoca: Casa Carţii de Ştiinţǎ 2006.
[5] D. P. Kacsó, Certain Bernstein-Durrmeyer-type Operators Preserving Linear Functions,
Schriftenreihe des Fachbereichs Mathematik, Universität Duisburg-Essen, SM-DU-675 (2008).
[6] H. B. Knoop and P. Pottinger, Ein Satz vom Korovkin-Typ für C k -Räume, Math. Z. 148
(1976), 23–32.
[7] R. G. Mamedov, On the asymptotic value of the approximation of repeatedly differentiable
functions by positive linear operators (Russian), Dokl. Akad. Nauk, 146 (1962), 1013–1016.
Translated in Soviet Math. Dokl., 3 (1962), 1435–1439.
[8] Bl. Sendov and V. A. Popov, Convergence of the derivatives of linear positive operators
(Bulgarian), Bulgar. Akad. Nauk Izv. Mat. Inst. 11 (1970), 107–115.
[9] V. S. Videnskiı̆, Linear Positive Operators of Finite Rank (in Russian), Leningrad: “A.I.
Gerzen” State Pedagocical Institute 1985.
[10] E. V. Voronovskaya, Détermination de la forme asymptotique d’approximation par les
polynômes de M. Bernstein (in Russian), C. R. (Dokl.) Acad. Sci. URSS A, 4, 79–85 (1932).
(received 31.7.2008, in revised form 7.2.2009)
Heiner Gonska, University of Duisburg-Essen, Department of Mathematics, D-47048 Duisburg,
Germany
E-mail: heiner.gonska@uni-due.de
Ioan Raşa, Technical University, Department of Mathematics, RO-400020 Cluj-Napoca, Romania
E-mail: Ioan.Rasa@math.utcluj.ro