General Mathematics Vol. 18, No. 3 (2010), 73–81
On the composite Bernstein type cubature
formula 1
Dan Bărbosu, Dan Miclăuş
Abstract
Considering a given function f ∈ C([0, 1] × [0, 1]), the bivariate interval [0, 1] × [0, 1] is divided in mn equally spaced bivariate subintervals
k−1 k j−1 j m , m ×
n , n , k = 1, m, j = 1, n. On each such type of subintervals the Bernstein bivariate approximation formula is applied and a corresponding Bernstein type cubature formula is obtained. Making the sum
of mentioned cubature formulas, the composite Bernstein type cubature
formula is obtained. The coefficients of above formula are determined
and an upper-bound for its remainder term is given.
2010 Mathematics Subject Classification: 65D32, 41A10
Key words and phrases: Bernstein bivariate operator, Bernstein bivariate
approximation formula, Bernstein quadrature formula, Bernstein cubature
formula, Bivariate divided difference, Remainder term
1
Preliminaries
Let us to denote N = {1, 2, ...} and N0 = N ∪ {0}.
The Bernstein bivariate operator Bm,n : C([0, 1] × [0, 1]) → C([0, 1] × [0, 1]) is
defined for any f ∈ C([0, 1]×[0, 1]), any (x, y) ∈ [0, 1]×[0, 1] and any m, n ∈ N,
by
m X
n
X
k j
(1)
(Bm,n f )(x, y) =
,
,
pm,k (x)pn,j (y)f
m n
k=0 j=0
1
Received 24 April, 2009
Accepted for publication (in revised form) 18 May, 2010
73
74
D. Bărbosu, D. Miclăuş
where
(2)
and
(3)
m k
pm,k (x) =
x (1 − x)m−k
k
n j
y (1 − y)n−j
pn,j (y) =
j
are the fundamental Bernstein’s polynomials. The polynomial (1) is called the
Bernstein bivariate polynomial.
Let f ∈ C([0, 1] × [0, 1]) be given. The following equality
(4)
f (x, y) = (Bm,n f ) (x, y) + (Rm,n f ) (x, y)
is known as the Bernstein bivariate approximation formula, where Rm,n is
the remainder operator associated to the Bernstein bivariate operator B m,n ,
i.e. Rm,n f is the remainder term of the bivariate approximation formula (4).
Regarding the remainder term of (4), D. Bărbosu and O. T. Pop [7] established
the following:
Theorem 1 For any f ∈ C([0, 1] × [0, 1]) and any (x, y) ∈ [0, 1] × [0, 1], the
remainder term of (4) can be expressed under the form
(5)
#
"
m−1 n
k k+1
, m
x, m
x(1 − x) X X
(Rm,n f )(x, y) =−
;f
pm−1,k (x)pn,j (y)
j
m
k=0 j=0
m
#
"
m n−1
k
y(1 − y) X X
m
−
;f
pm,k (x)pn−1,j (y)
j j+1
n
,
y,
k=0 j=0
n
n
#
"
m−1
n−1
k k+1
x, m
, m
xy(1 − x)(1 − y) X X
+
;f .
pm−1,k (x)pn−1,j (y)
mn
y, j , j+1
k=0 j=0
k
n
In (5) the brackets denote the bivariate divided differences.
Theorem 2 ([7]) Let be p, q ∈ N0 , a ≤ x0 < x1 < ... < xp ≤ b, c ≤ y0 <
y1 < ... < yq ≤ d and the function f : [a, b] × [c, d] → R be given. If f ∈
∂ p+q f
C (p−1,q−1) ([a, b]×[c, d]) and there exists ∂x
p ∂y q on ]a, b[×]c, d[ then, there exists
(ξ, η) ∈]a, b[×]c, d[ such that
"
#
1 ∂ p+q f
x0 , x1 , . . . , xp
(6)
;f =
(ξ, η).
p!q! ∂xp ∂y q
y0 , y1 , . . . , yq
On the composite Bernstein type cubature formula
75
Theorem 3 ([7]) Let f : [0, 1] × [0, 1] → R be a function with the properties
4f
∂2f
∂2f
that f ∈ C([0, 1] × [0, 1]), there exists ∂x∂2 ∂y
2 on ]0, 1[×]0, 1[ and ∂x2 , ∂y 2 ,
∂4f
∂x2 ∂y 2
are bounded on ]0, 1[×]0, 1[. Then, the inequalities
(7)
y(1 − y)
xy(1 − x)(1 − y)
x(1 − x)
M1 [f ] +
M2 [f ] +
M3 [f ]
2m
2n
4mn
1
1
1
≤
M1 [f ] +
M2 [f ] +
M3 [f ]
8m
8n
64mn
hold, for any (x, y) ∈ [0, 1] × [0, 1] and any m, n ∈ N, where
2
∂ f
M1 [f ] =
sup
∂x2 (x, y) ,
(x,y)∈]0,1[×]0,1[
2
∂ f
,
(8)
(x,
y)
M2 [f ] =
sup
2
(x,y)∈]0,1[×]0,1[ ∂y
4
∂ f
M3 [f ] =
sup
∂x2 ∂y 2 (x, y) .
(x,y)∈]0,1[×]0,1[
|(Rm,n f )(x, y)| ≤
Integrating the Bernstein bivariate approximation formula (4) one arrives to
the following Bernstein’s cubature formula [6]:
(9)
Z1 Z1
0
f (x, y)dxdy =
m X
n
X
i=0 j=0
0
Ai,j f
i j
,
m n
+ Rm,n [f ],
where
(10)
Ai,j =
1
, i = 0, m, j = 0, n
(m + 1)(n + 1)
and
1
1
1
M1 [f ] +
M2 [f ] +
M3 [f ],
12m
12n
144mn
where M1 [f ], M2 [f ] and M3 [f ] were defined at (8).
The focus of the present paper is to construct the composite Bernstein type
cubature formula. For this aim, the bivariate interval [0, 1] ×h[0, 1] will
i be
k−1 k j−1 j
divided in mn equally spaced bivariate subintervals m , m × n , n , k =
h j−1 j i
k
1, m, j = 1, n. On each such type of subintervals k−1
,
m m ×
n , n , the
Bernstein’s cubature formula (9) will be applied. Next, adding the mentioned
cubature formulas, the composite Bernstein cubature formula on [0, 1] × [0, 1]
will be obtained.
(11)
|Rm,n [f ]| ≤
76
2
D. Bărbosu, D. Miclăuş
Main results
We start by the simple results contained in the following two lemmas.
Lemma 1 Suppose that a, b, c, d ∈ R, a < b and c < d, f ∈ C([a, b] × [c, d]).
Then, the bivariate Bernstein polynomial associated to the function f is defined
by
n m X
X
1
m n
(12) (Bm,n f )(x, y) =
(x − a)k (b − x)m−k
k
(b − a)m (d − c)n
j
k=0 j=0
d−c
b−a
j
n−j
,c + j
.
· (y − c) (d − y) f a + k
m
n
Proof. The parametric extensions [5] of the Bernstein’s univariate operator
[8] are defined by
n m X
X
m n
1
x
(13) (Bm f )(x, y) =
(x − a)k (b − x)m−k
m
n
j
k
(b − a) (d − c)
k=0 j=0
b−a
j
n−j
· (y − c) (d − y) f a + k
,y ,
m
respectively
(14)
(Bny f )(x, y)
m X
n X
1
m n
=
(x − a)k (b − x)m−k
m
n
(b − a) (d − c)
k
j
k=0 j=0
d−c
j
n−j
.
· (y − c) (d − y) f x, c + j
n
Their tensorial product [9] is the bivariate Bernstein operator (12).
Lemma 2 Suppose that a, b, c, d ∈ R, a < b, c < d and let f : [a, b]×[c, d] → R
4f
be a function with the properties that f ∈ C 2,2 ([a, b]×[c, d]), there exists ∂x∂2 ∂y
2
2
2
4
f
on ]a, b[×]c, d[ and ∂∂xf2 , ∂∂yf2 , ∂x∂2 ∂y
2 are bounded on ]a, b[×]c, d[. Then, the
remainder term of the Bernstein bivariate approximation formula on [a, b] ×
[c, d] verifies the following inequality
(15)
|(Rm,n f )(x, y)| ≤
(y − c)(d − y) 0
(x − a)(b − x) 0
M1 [f ] +
M2 [f ]
2m(b − a)2
2n(d − c)2
(x − a)(b − x)(y − c)(d − y) 0
M3 [f ],
+
4mn(b − a)2 (d − c)2
On the composite Bernstein type cubature formula
77
for any (x, y) ∈]a, b[×]c, d[ and any m, n ∈ N, where
2
∂ f
=
sup
∂x2 (x, y) ,
(x,y)∈]a,b[×]c,d[
2
∂ f
0
M2 [f ] =
sup
∂y 2 (x, y) ,
(x,y)∈]a,b[×]c,d[
4
∂ f
0
.
(x,
y)
M3 [f ] =
sup
2 2
(x,y)∈]a,b[×]c,d[ ∂x ∂y
M10 [f ]
(16)
Proof. One applies (7) and the method of parametric extensions [9].
In what follows, let us to consider the bivariate interval [0,
1]×[0, 1] divided
k−1 k h j−1 j i
in the mn equally spaced bivariate subintervals m , m × n , n , k = 1, m,
h j−1 j i
k
,
j = 1, n. In each interval k−1
m m ×
n , n one considers the distinct knots
jq−q+l
xh = kp−p+h
mp , h = 0, p, respectively yl =
nq , l = 0, q.
Applying Lemma 1 yields the following Bernstein type bivariate polynomial
(17)
p X
q X
p
q
p−h
k−1 h k
(Bp,k,q,j f )(x, y) = m n
−x
x−
m
m
h
l
h=0 l=0
q−l j−1 l j
kp − p + h jq − q + l
.
−y
,
· y−
f
n
n
mp
nq
p q
The following Bernstein type bivariate approximation formula
(18)
f (x, y) = (Bp,k,q,j f )(x, y) + (Rp,k,q,j f )(x, y)
holds, on any interval
k−1
m
C 2,2 ([0, 1]
h j−1 j i
k
,m
× n , n , k = 1, m, j = 1, n.
For any f ∈
× [0, 1]) the following upper-bound estimation for the
bivariate remainder term
(19)
|(Rp,k,q,j f )(x, y)| ≤
m x−
k−1
m
2
mn x −
+
k
m
k−1
m
−x
k
m
M100 [f ] +
−x y−
4
n y−
j−1
n
2 j
j−1
n
n −y
j
n
−y
M300 [f ]
M200 [f ]
78
D. Bărbosu, D. Miclăuş
holds, where
2
∂ f
=
sup
(x, y) ,
2
∂x
, k × j−1 , j
(x,y)∈] k−1
m m[ ] n n[
2
∂ f
00
(x, y) ,
sup
M2 [f ] =
2
∂y
, k × j−1 , j
(x,y)∈] k−1
m m[ ] n n[
4
∂ f
00
M3 [f ] =
sup
(x, y) .
2
2
∂x ∂y
(x,y)∈] k−1
, k × j−1 , j
m m[ ] n n[
M100 [f ]
(20)
Theorem 4 If f ∈ C([0, 1] × [0, 1]), the coefficients of the Bernstein type
cubature formula
(21)
j
k
Zm Zn
q
p X
X
kp − p + h jq − q + l
Ah,k,l,j f
f (x, y)dxdy =
+ Rk,j [f ]
,
mp
nq
h=0 l=0
k−1 j−1
m
n
can be expressed under the form
1
, h = 0, p, l = 0, q.
mn(p + 1)(q + 1)
h j−1 j i
k
Proof. Integrating (18) on k−1
m ,m ×
n , n , k = 1, m, j = 1, n and taking
(17) into account, yields
(22)
Ah,k,l,j =
k
Ah,k,l,j
j
Zm Zn p−h
k−1 h k
q
p q p
=m n
x−
−x
m
m
h
l
k−1 j−1
m
n
q−l
j−1 l j
dxdy
−y
· y−
n
n
k
Zm p−h
k−1 h k
q
p q p
x−
=m n
−x
dx
m
m
h
l
k−1
m
j
·
Zn j−1
n
j−1
y−
n
l j
−y
n
q−l
dy
On the composite Bernstein type cubature formula
79
Z1
Z1
1 h
q
p
1 l
p−h
t (1 − t) dt
u (1 − u)q−l du
=m n
l
h
mp+1
nq+1
p q
0
0
Z1
Z1
q
1 p
h
p−h
t (1 − t) dt ul (1 − u)q−l du.
=
l
mn h
0
0
The integral
R1
xi (1−x)n−i dx is the Euler function of first kind B(i+1, n−i+1).
0
Taking the well known properties of Euler function of first kind into account,
it follows that
1 p
q
Ah,k,l,j =
B(h + 1, p − h + 1)B(l + 1, q − l + 1)
l
mn h
1
p!
h!(p − h)!
q!
l!(q − l)!
=
mn h!(p − h)! (p + 1)! l!(q − l)! (q + 1)!
1
.
=
mn(p + 1)(q + 1)
Theorem 5 Let f ∈ C 2,2 ([0, 1] × [0, 1]) be given so that there exists
∂2f
∂x2 ∂y 2
on
4
∂2f ∂2f
,
, ∂ f
∂x2 ∂y 2 ∂x2 ∂y 2
]0, 1[×]0, 1[ and
are bounded on ]0, 1[×]0, 1[. Then the following upper-bound estimation for the bivariate remainder term of the Bernstein
type cubature formula (21) is
(23)
|Rk,j [f ]| ≤
M200 [f ]
M300 [f ]
M100 [f ]
+
+
,
12m2 n 12mn2 144m2 n2
where M100 [f ], M200 [f ], M300 [f ] are given at (20).
Proof. The inequality (23) follows by integrating the Bernstein type bivariate
approximation formula (18) and taking the inequality (19) into account.
Theorem 6 For any f ∈ C 2,2 ([0, 1]×[0, 1]), the following composite Bernstein
type cubature formula
(24)
Z1Z1
f (x, y)dxdy =
0 0
m
n
p
q
XXXX
1
f
mn(p + 1)(q + 1)
k=1 j=1 h=0 l=0
+ Rm,n [f ].
holds, where |Rm,n [f ]| were defined at (11).
kp − p + h jq − q + l
,
mp
nq
80
D. Bărbosu, D. Miclăuş
Proof. Adding the relation (21) for any k = 1, m, j = 1, n, we get the
following composite Bernstein type cubature formula (24).
Remark 1 It is easily to see that we get the same result for the bivariate
remainder term of the composite Bernstein type cubature formula, as the result
obtained by D. Bărbosu and O. T. Pop in [6].
Corollary 1 The following equality
n
m
(25)
p
q
XXXX
1
lim
f
m,n→∞ mn(p + 1)(q + 1)
k=1 j=1 h=0 l=0
=
Z1 Z1
0
kp − p + h jq − q + l
,
mp
nq
f (x, y)dxdy
0
holds.
Proof. Yields immediately from Theorem 6, because
lim |Rm,n [f ]| = 0.
m,n→∞
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On the composite Bernstein type cubature formula
81
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Dan Bărbosu, Dan Miclăuş
North University of Baia Mare
Department of Mathematics and Computer Science
Victoriei 76, 430122 Baia Mare, Romania
e-mail: barbosudan@yahoo.com, danmiclausrz@yahoo.com