Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 609431, 13 pages
doi:10.1155/2011/609431
Research Article
A de Casteljau Algorithm for
q-Bernstein-Stancu Polynomials
Grzegorz Nowak
The Great Poland University of Social and Economics in Środa Wielkopolska, Paderewskiego 27,
63-000 Środa Wielkopolska, Poland
Correspondence should be addressed to Grzegorz Nowak, grzegnow2@gmail.com
Received 17 September 2010; Accepted 7 January 2011
Academic Editor: Wolfgang Ruess
Copyright q 2011 Grzegorz Nowak. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
This paper is concerned with a generalization of the q-Bernstein polynomials and Stancu operators,
where the function is evaluated at intervals which are in geometric progression. It is shown that
these polynomials can be generated by a de Casteljau algorithm, which is a generalization of that
relating to the classical case and q-Bernstein case.
1. Introduction
Let q > 0. For any fixed real number q > 0 and for n ∈ Z {0, ±1, ±2, . . .}, the q-integers of the
number n are defined by
1 − qn
,
n
1−q
for q / 1, n n, for q 1.
1.1
The q-factorial n!, for n ∈ N0 {0, 1, 2, . . .}, is defined by
n! 12 · · · n
n 1, 2, . . ., 0! 1.
1.2
2
Abstract and Applied Analysis
For the integers n, k, n ≥ k ≥ 0, the q-binomial or the Gaussian coefficients are defined by
see 1, page 12
n
k
n!
.
k!n − k!
1.3
For f ∈ C0; 1, q > 0, α ≥ 0 and each positive integer n, we introduce see 2 the following
generalized q-Bernstein operators:
n
k
q,α
,
pn,k xf
f; x n
k0
q,α Bn
1.4
where
q,α
pn,k x
k−1
n−1−k n
1 − qs x αs
s0
i0 x αi
.
n−1
k
i0 1 αi
1.5
q,α
Note, that an empty product in 1.5 denotes 1. In the case where α 0, Bn f; x reduces to
the well-known q-Bernstein polynomials introduced by Phillips 3, 4 in 1997
Bn,q f; x n n
k0
k
xk
k 1 − qi x f
.
n
n−k−1
i0
1.6
q,α
In the case where q 1, Bn f; x reduces to Bernstein-Stancu polynomials, introduced by
Stancu 5 in 1968
k−1
n−k−1
n
n
k
s0 1 − x sα
i0 x αi
.
Sn f; x f
n−1
n
k0 k
i0 1 iα
1.7
When q 1 and α 0, we obtain the classical Bernstein polynomial defined by
Bn f; x n
n
k0
k
xk 1 − xn−k f
k
.
n
1.8
Basic facts on Bernstein polynomials, their generalizations, and applications can be found
for example in 6–8. In recent years, the q-Bernstein polynomials have attracted much
interest, and a great number of interesting results related to the Bn,q f polynomials have
been obtained see 3, 4, 9–12. Some approximation properties of the Stancu operators are
presented in 5, 13–15.
Let Δ0q fj fj , for j 0, 1, . . . , n, and recursively,
k
k k
Δk1
q fj Δq fj1 − q Δq fj ,
1.9
Abstract and Applied Analysis
3
for k 0, 1, . . . , n − j − 1 and fj fj/n. It is easily established by induction that qdifferences satisfy the relation
Δkq fj
k
k
k ii−1/2
fjk−i .
−1 q
i
i0
1.10
q,α
In 2, we prove that the operators Bn f; x defined by 1.4 can be expressed in terms of
q-differences
q,α
Bn fx
k−1
n n
x αi
Δkq f0
,
1 αi
i0
k0 k
1.11
which generalized the well-known result 3, 4 for the q-Bernstein polynomial. In this paper,
we show that polynomials defined by 1.4 can be generated by a de Castljau algorithm,
which is a generalization of that relating to the classical case 16 and q-Bernstein case 4, 11.
2. Auxiliary Results
q,α
We note that Bn f; x defined by 1.4, is a monotone linear operator for any 0 < q ≤ 1 and
α ≥ 0. These operators reproduces linear functions 2, that is,
q,α
Bn ax b; x ax b,
2.1
a, b ∈ R.
q,α
q,α
They also satisfy the end point interpolation conditions Bn f; 0 f0 and Bn f; 1 f1.
These properties are significant in designing curves and surfaces.
Moreover, the following holds.
Lemma 2.1. Let 0 < q ≤ 1, α ≥ 0. Then,
s−1
m−r−1
m
m s ss−1/2m−sr
q − q x αu − r 1α j ,
−1 q
x αi
s i0
s0
js−r
m−1
u0
r
u
2.2
for all m ∈ N, r ∈ N0 N ∪ {0} and x ∈ 0; 1.
4
Abstract and Applied Analysis
Proof. We use induction on m. First, we see from equality −r −q−r r, r ∈ N, that 2.2
is evident for m 1. Let us assume that 2.2 holds for a given m ∈ N. Then, using 2.2, we
obtain
m
qr − qu x αu − r
u0
m
m
s ss−1/2m−sr
q − q x αm − r
−1 q
s
s0
·
r
m
s−1
m−r−1
1α j
x αi
i0
js−r
m
m
s ss−1/2m−sr r
m
q αm − αr αq s
−1 q
s
s0
·
s−1
m−r−1
1α j
x αi
i0
js−r
m1
−1s qs−1s−2/2m−s1rm 1 αs − r − 1
s1
·
m
2.3
s−1
s−1
m−r−1
1α j
x αi
i0
js−r
m−r−1
qmr qr αm − αr
1α j
j−r
−1m1 qmm−1/2m 1 αm − r
m
x αi
i0
m−r−1
s−1
m
1α j ,
−1s qss−1/2m1−sr Us
x αi
i0
s1
js−r
where
Us m s
r
m
q αm − αr αq s q
−r
q
m−s1
m
s−1
1 αs − r − 1.
2.4
Abstract and Applied Analysis
5
Using the obvious equalities
qr αm − αr q−r 1 αm − r,
m
m
s m − s 1,
s
s−1
2.5
we have
Us m
s
1 αm − r
m
s−1
qm−s1
1 α m − s 1qs−r−1 s − r − 1 .
2.6
It is easy to see that
m − s 1qs−r−1 s − r − 1 m − r,
m
m
m1
m−s1
q
.
s
s−1
s
2.7
Therefore,
Us 1 αm − r
m1
.
s
2.8
From last equality and 2.3, we obtain
m
qr − qu x αu − r
u0
m−r−1
qmr qr αm − αr
1α j
j−r
−1m1 qmm−1/2m 1 αm − r
m
s ss−1/2m1−sr
−1 q
m1
s
s1
m1
−1s qss−1/2m1−sr
s0
m
1 αm − r
This completes the proof of the lemma.
2.9
s−1
m−r−1
1α j
x αi
i0
s−1
m1 s
x αi
i0
i0
x αi
m−r
1α j .
js−r
js−r
6
Abstract and Applied Analysis
input: q; f0/n, f1/n, . . . , fn/n
for r 0 to n
r
0
fr : f
n
next r
for m 1 to n
for r 0 to n − m
m−1
m−1
x αrfr1 }
{qr − qm−1 x αm − 1 − rfr
m
fr :
1 αm − 1
next r
next m
Algorithm 1: De Casteljau type algorithm.
3. Main Result
The generalized q-Bernstein polynomials, defined by 1.4, may be evaluated by Algorithm 1.
In the case, where α 0, this is the de Casteljau algorithm for evaluating the qBernstein polynomial 3, 4. Note that with q 1 and α 0, we recover the original
classical de Casteljau algorithm see Hoschek and Lasser 16. The algorithm is justifed by
the following theorem.
m
Theorem 3.1. Each intermediate point fr
m
fr
of the algorithm can be expressed as
−1
m−1
1 αi
i0
·
m
t0
frt
t−1
m t
x αr s
s0
m−t−1
3.1
qr − qu x αu − r ,
u0
and, in particular
n
f0
q,α Bn
f; x .
3.2
0
Proof. We use induction on m. From the initial conditions in the algorithm, fr fr/n fr , 0 ≤ r ≤ n, it is clear that 3.1 holds for m 0 and 0 ≤ r ≤ n. Let us assume that 3.1 holds
for some m such that 0 ≤ m < n, and for all r such that 0 ≤ r ≤ n−m. Then, for 0 ≤ r ≤ n−m−1,
it follows from the algorithm that
m1
fr
:
m
m
qr − qm x αm − r fr x αrfr1
1
,
1 αm
3.3
Abstract and Applied Analysis
7
and using 3.1, we obtain
m1
fr
m
1 αi : qr − qm x αm − r
i0
·
m
frt
t−1
m t
t0
m
u0
frt1
t−1
m t
t0
m−t−1
qr − qu x αu − r
s0
x αr ·
·
m−t−1
x αr s ·
x αr s 1
s0
qr1 − qu x αu − r 1
u0
m−1
r
qr − qm x αm − r fr
q − qu x αu − r
u0
m
m
q − q x αm − r · frt
t
t1
·
r
t−1
m
x αr s ·
m−t−1
s0
u0
x αr ·
m
frt
t1
·
qr − qu x αu − r
t−2
m
t−1
x αr s 1
3.4
s0
qr1 − qu x αu − r 1
m−t u0
x αrfrm1
m−1
x αr s 1
s0
fr
m
qr − qu x αu − r
u0
m
m
q − q x αm − r
t
t1
·
r
t−1
m
x αr s
s0
m−t−1
u0
qr − qu x αu − r
t−2
m x αr
x αr s 1
t − 1 s0
m−t r1
u
q − q x αu − r 1 frt
u0
frm1
m
s0
x αr s.
8
Abstract and Applied Analysis
We see that
m−t qr1 − qu x αu − r 1
u0
m−t−1
qr1 − x − αr 1
qr1 − qu1 x αu 1 − r 1
u0
q
r1
− x − αr 1
m−t−1
q
r1
−q
u1
x αqu − r
3.5
u0
m−t−1
qr1 − x − αr 1 qm−t
qr − qu x αu − r ,
u0
and hence,
m1
fr
m
1 αi
i0
: fr
m
qr − qu x αu − r
u0
m
m t
t1
· frt
t−1
qr − qm x αm − r x αr s
s0
m−t−1
m
t−1
3.6
qr1 − x − αr 1 qm−t
m
qr − qu x αu − r frm1
x αr s.
u0
s0
It is easy to verify that
m
t
q
m
t−1
m−t1
q
m1
,
t−1
t
m
m
t
t
m1
t
3.7
.
Abstract and Applied Analysis
9
Therefore,
m r
m
q − q x αm − r t
m
r
m
t−1
qr1 − x − αr 1 qm−t
− xq
m
m−t
t
m
q
t
t−1
m
m
m
m
α
r
m−t1
m−t
t
q
q
−q
q
1−q
t
t−1
t−1
t
m 1 r
q − xqm−t αm − t − r .
t
q
t
q
m−t1
m
t−1
3.8
Consequently,
m1
fr
m
m
r
q − qu x αu − r
1 αi : fr
u0
i0
m m1
t
t1
· frt
t−1
qr − xqm−t αm − t − r
x αr s
s0
frm1
m−t−1
qr − qu x αu − r
u0
m
x αr s
s0
m1
m1
t
t0
· frt
t−1
x αr s
s0
m−t
qr − qu x αu − r .
u0
3.9
Thus, one has the desired result.
Theorem 3.2. For 0 ≤ m ≤ n and 0 ≤ r ≤ n − m, we have
m
fr
sr−1
m
m s
x αi
m−sr
Δq fr ir
q
,
s−1 s
s0
j0 1 α j
3.10
for all x ∈ 0; 1.
Proof. Using 2.2 and 3.1, we have
m
fr
m m
frt St m,
1 αi t0 t
i0
m−1
3.11
10
Abstract and Applied Analysis
where
St m m−t
u uu−1/2m−t−ur
−1 q
m−t
u
u0
×
tr−1
u−1
sr
i0
x αs
m−t−r−1
x αi
1α j
3.12
0 ≤ t ≤ m.
ju−r
First, we prove that
St m m−t
u uu−1/2m−t−ur
−1 q
m−t
u
u0
·
tur−1
x αi
ir
m−1
1α j
jut
3.13
for all m ∈ N0 {0, 1, 2, . . .}, t ∈ N0 , and x ∈ 0; 1. Note that an empty sum denotes 0.
We use the induction on m. First, we see that 3.13 holds for m 0 and all t ∈ N0 . Let
us assume that 3.13 holds for a given m, and for all t ∈ N0 . Then, from 3.12 and 3.13, we
obtain
St m 1 x αt r − 1
m1−t
u uu−1/2m−t1−ur
−1 q
u
u0
·
tu−2r
x αi
ir
m−1
1α j
jut−1
m1−t
u uu−1/2m−t1−ur
−1 q
m−t1
u
u0
·
tur−1
x αi
ir
3.14
m−1
1α j
jut−1
m−t1
u uu−1/2m−t1−ur
−1 q
α
m−t1
m−t1
u
u1
· t r 1 − t u r 1
tu−2r
x αi
ir
m−1
1α j .
jut−1
We see that
m−t1
u
t r − 1 − t u r − 1 −q
tr−1
m−t1
,
m − t − u 2
u−1
3.15
Abstract and Applied Analysis
11
and hence,
St m 1 m−t1
u uu−1/2m−t1−ur
−1 q
m−t1
u
u0
·
tur−1
x αi
ir
m−1
1α j
jut−1
m−t
u uu−1/2m−t1−ur ut−1
−1 q
α
q
u
u0
· m − t − u 1
tur−1
x αi
ir
m−t1
m−1
1α j
jut
−1m−t1 qm−t1m−t/2
mr
3.16
x αi
ir
m−t
−1u quu−1/2m−t1−ur
u0
· 1 αu t − 1 αq
·
tur−1
x αi
ir
ut−1
m − t − u 1
m−t1
u
m−1
1α j .
jut
Next, in view of the equality
1 αu t − 1 αqut−1 m − t − u 1 1 αm,
3.17
we obtain 3.13. Consequently, in view of 3.11 and 3.13, we get
m
fr
m−1
1 αi m m
t0
i0
t
frt
m−t
−1u quu−1/2m−t−ur
u0
m−1
m − t tur−1
·
1α j
x αi
u
ir
jut
m m m
frt −1u−t qu−tu−t−1/2m−ur
t
m−1
m − t ur−1
·
1α j .
x αi
u − t ir
ju
t0 ut
3.18
12
Abstract and Applied Analysis
Next, in view of the equality
m m−t
t
m u
,
u−t
u t
3.19
we obtain
m
fr
m u m
frt −1u−t qu−tu−t−1/2m−ur
1 αi u0 t0 u
i0
ur−1
m−1
u ·
1α j
x αi
t ir
ju
m−1
m−1
ur−1
m m
qm−ur
1α j
x αi
u0 u
ir
ju
3.20
u u
·
−1u−t qu−tu−t−1/2 frt .
t0 t
The condition 1.10 completes the proof.
Theorems 3.1 and 3.2 are generalizations of Theorems 2.1 and 2.3 in 11.
Note that when m n and r 0, 3.10 does indeed reduce to 1.11
References
1 V. Kac and P. Cheung, Quantum Calculus, Springer, New York, NY, USA, 2002.
2 G. Nowak, “Approximation properties for generalized q-Bernstein polynomials,” Journal of Mathematical Analysis and Applications, vol. 350, no. 1, pp. 50–55, 2009.
3 G. M. Phillips, “Bernstein polynomials based on the q-integers,” Annals of Numerical Mathematics, vol.
4, no. 1–4, pp. 511–518, 1997.
4 G. M. Phillips, Interpolation and Approximation by Polynomials, Springer, New York, NY, USA, 2003.
5 D. D. Stancu, “Approximation of functions by a new class of linear polynomial operators,” Académie
de la République Populaire Roumaine. Revue Roumaine de Mathématiques Pures et Appliquées, vol. 13, pp.
1173–1194, 1968.
6 G. G. Lorentz, Bernstein Polynomials, vol. 8, University of Toronto Press, Toronto, Canada, 1953.
7 P. Pych-Taberska, “Some approximation properties of Bernšteı̆n and Kantorovič polynomials,”
Functiones et Approximatio Commentarii Mathematici, vol. 6, pp. 57–67, 1978.
8 P. Pych-Taberska, “On the rate of pointwise convergence of Bernstein and Kantorovič polynomials,”
Functiones et Approximatio Commentarii Mathematici, vol. 16, pp. 63–76, 1988.
9 T. N. T. Goodman, H. Oruç, and G. M. Phillips, “Convexity and generalized Bernstein polynomials,”
Proceedings of the Edinburgh Mathematical Society. Series 2, vol. 42, no. 1, pp. 179–190, 1999.
10 A. II’inskii and S. Ostrovska, “Convergence of generalized Bernstein polynomials,” Journal of
Approximation Theory, vol. 116, no. 1, pp. 100–112, 2002.
11 G. M. Phillips, “A de Casteljau algorithm for generalized Bernstein polynomials,” BIT. Numerical
Mathematics, vol. 37, no. 1, pp. 232–236, 1997.
12 G. M. Phillips, “A survey of results on the q-Bernstein polynomials,” IMA Journal of Numerical
Analysis, vol. 30, no. 1, pp. 277–288, 2010.
13 Z. Finta, “Direct and converse results for Stancu operator,” Periodica Mathematica Hungarica, vol. 44,
no. 1, pp. 1–6, 2002.
Abstract and Applied Analysis
13
14 Z. Finta, “On approximation properties of Stancu’s operators,” Mathematica, vol. 47, no. 4, pp. 47–55,
2002.
15 H. H. Gonska and J. Meier, “Quantitative theorems on approximation by Bernstein-Stancu operators,”
Calcolo, vol. 21, no. 4, pp. 317–335, 1984.
16 J. Hoschek and D. Lasser, Fundamentals of Computer Aided Geometric Design, A. K. Peters, Wellesley,
Mass, USA, 1993.
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