General Mathematics Vol. 16, No. 4 (2008), 165–170
A lower bound for the second moment of
Schoenberg operator
Gancho T. Tachev
Abstract
In this paper we represent a new lower bound for the second moment for Schoenberg variation-diminishing spline operator. We apply
this estimate for f ∈ C 2 [0, 1] and generalize the results obtained earlier by Gonska, Pitul and Rasa.
2000 Mathematics Subject Classification: 41A10, 41A15, 41A17,
41A25, 41A36
1
Main result
We start with the definition of variation-diminishing operator, introduced
by I.Schoenberg. For the case of equidistant knots we denote it by Sn,k .
Consider the knot sequence Δn = {xi }n+k
−k , n ≥ 1, k ≥ 1 with equidistant
”interior knots”, namely
Δn : x−k = · · · = x0 = 0 < x1 < x2 < · · · < xn = · · · = xn+k = 1
165
A lower bound for the second moment of Schoenberg operator
166
i
for 0 ≤ i ≤ n. For a bounded real-valued function f defined
n
over the interval [0, 1] the variation-diminishing spline operator of degree k
and xi =
w.r.t. Δn is given by
n−1
Sn,k (f, x) =
(1)
f (ξj,k )Ṅj,k (x)
j=−k
for 0 ≤ x < 1 and
Sn,k (f, 1) =
lim Sn,k (f, y)
y→1, y<1
with the nodes (Greville abscissas)
ξj,k :=
(2)
xj+1 + · · · + xj+k
, −k ≤ j ≤ n − 1,
k
and the normalized B− splines as fundamental functions
Nj,k (x) := (xj+k+1 − xj )˙[xj , xj+1 , . . . , xj+k+1](· − x)k+ .
The first quantitative variant of Voronovskaja’s Theorem for a broad
class of linear positive operators L was obtained very recently by H.Gonska,
P.Pitul and I.Rasa in [3](see the proof of Theorem 6.2). We cite this result
in the following
Theorem A. Let L : C[0, 1] → C[0, 1] be a positive, linear operator reproducing linear functions. If f ∈ C 2 [0, 1] and x ∈ [0, 1] then
1
L(f ; x) − f (x) − · f (x) · L((e1 − x)2 ; x)
2
(3)
-
≤
1
1
· L((e1 − x)2 ; x) · ω̃ f , ·
2
3
$
L((e1 −
L((e1 − x)2 ; x)
x)4 ; x)
.
.
A lower bound for the second moment of Schoenberg operator
167
Here en : x ∈ [0, 1] → xn , n = 0, 1, . . . are the monomial functions and
ω̃(f, ·) denotes the least concave majorant of ω(f, ·) given by
ω̃(f, ε) =
(ε − x)ω(f, y) + (y − ε)ω(f, x)
,
y−x
0≤x≤ε≤y≤1, x=y
sup
for 0 ≤ ε ≤ 1.
Here we point out that to estimate the argument of ω̃(f , ·) we need a
”good” upper bound for L((e1 − x)4 ; x) and a ”good” lower bound for the
second moment L((e1 − x)2 ; x). If f is convex function it is known that
Sn,k (f, x) ≤ Bk (f, x),
where Bk (f, x) is the Bernstein operator of degree k. Therefore from the
well-known representation of the fourth moment of Bk (f, x) we have
x(1−x)
2
1
4
4
· 3(1− )x(1−x)+ ·
(4) Sn,k ((e1 − x) , x) ≤ Bk ((e1 −x) ,x) =
k2
k
k
Next we are going to prove a lower bound for the second moment of Sn,k .
We can not use the estimate established in Theorem 12 in [2] because it is
valid only when 2 ≤ k ≤ n−1. Here we point out that the new lower bound
for the second moment of Sn,k is valid for all k ≥ 2, n ≥ 2. In Theorem 3 in
[1] it was proved that
(5)
Sn,k ((e1 − x)2 , x) = Sn,k (g2 , x),
where the function g2 is given by
A lower bound for the second moment of Schoenberg operator
168
⎧
⎪
1
1
y 8k
k+1 n−1
⎪
2
⎪
· (−y +
· y + 2 ), 0 ≤ y ≤ min
,
,
⎪
⎪
k−1
3 n
n
2n
2k
⎪
⎪
⎪
2
⎪
n−1
1
⎨ 1 · (y − y 2 − n − 1 ),
≤y≤ ,
k−1
6nk
2k
2
g2 (y) =
⎪
(k
+
1)(k
−
1)
1
1
k
+
1
⎪
⎪
·
≤y≤ ,
,
⎪
⎪
⎪
k−1
12n2
2n
2
⎪
⎪
1
⎪
⎩ g2 (1 − y),
≤ y ≤ 1.
2
The function g2 (y) is not concave and our goal is to bound it from below
by an appropriate concave function h2 (y). If this is possible it is easy to
calculate
Sn,k (g2 , x) ≥ Sn,k (h2 , x) ≥ Bk (h2 , x)
and consequently
1
1
1
=
≤
.
2
Sn,k ((e1 − x) , x)
Sn,k (g2 , x)
Bk (h2 , x)
(6)
To define the function h2 we observe that
g2 (0) =
1
3n(k − 1)
for all n, k ≥ 2. If
(7)
h2 (y) =
1
y(1 − y), y ∈ [0, 1]
3n(k − 1)
we verify that
g2 (y) ≥ h2 (y).
Further we compute
(8)
1
x(1 − x)
2
Bk (h2 , x) =
· x − (x +
)
3n(k − 1)
k
x(1 − x)(1 − k1 )
=
.
3n(k − 1)
A lower bound for the second moment of Schoenberg operator
169
The last estimate is our lower bound for the second moment valid for all
n ≥ 2, k ≥ 2. Thus we obtain
2
3n(k − 1)
Sn,k ((e1 − x)4 , x)
x(1 − x)
1
· 3(1 − )x(1 − x) +
·
≤
=
Sn,k ((e1 − x)2 , x)
k2
k
k
x(1 − x)(1 − k1 )
2
1
n
3 · 3(1 − )x(1 − x) +
:= Δn,k (x).
k
k
k
(9)
k
→ ∞ (the polynomial case ) then lim Δn,k (x) = 0. We apply
k
n
→∞
n
Theorem A to arrive at
When
Theorem 1 For f ∈ C 2 [0, 1] we have
1
|Sn,k (f, x) − f (x) − Sn,k ((e1 − x)2 , x)f (x)|
2
(10)
8
1
2
1
≤ Sn,k ((e1 − x) , x) · ω̃ f , · Δn,k (x) ,
2
3
where Δn,k (x) is defined in (8).
Corollary 1 If we set n = 1 in (9) and (10) we get exactly the result of
Gonska in [4].
Acknowledgment. This work was done during my stay in January 2008
as DAAD Fellow by Prof. H.Gonska at the University of Duisburg-Essen.
A lower bound for the second moment of Schoenberg operator
170
References
[1] L.Beutel, H.H.Gonska, D.Kacso, G.Tachev, On variation-diminishing
Schoenberg operators: new quantitative statements, Multivariate Approximation and Interpoltaion with Applications (ed. by M.Gasca),
Monogr. Academia Ciencas de Zaragoza 20 (2002), 9-58.
[2] L.Beutel, H.H.Gonska, D.Kacso, G.Tachev, On the Second Moments of
Variation-Diminishing Splines, J. of Concr. and Appl. Math., vol.2,1
(2004), 91-117.
[3] H.Gonska,P.Pitul and I.Rasa, On Peano’s form of the Taylor remainder,
Voronovskaja’s theorem and the commutator of positive linear operators,
”Numerical Analysis and Approximation Theory” -Proceedings of the
Int. Conf. 2006 in Cluj-Napoca, (2006), 55-80.
[4] H.Gonska, On the degree of Approximation in Voronovskaja’s theorem,
Studia Univ. ”Babes-Bolyai”, Mathematica, vol.LII,3(2007), 103-115.
Gancho T. Tachev
University of Architecture, Civil Engineering and Geodesy
Department of Mathematics
BG-1046, Sofia, Bulgaria
e-mail: gtt− fte@uacg.acad.bg