General Mathematics Vol. 17, No. 3 (2009), 161–172
A note on the Bernstein’s cubature
formula 1
Dan Bărbosu, Ovidiu T. Pop
Abstract
The Bernstein’s cubature formula is revisited and the evaluation
of it’s remainder term is corrected.
2000 Mathematics Subject Classification: 65D32, 41A10, 41A63
Key words and phrases: Bernstein’s operator, Bernstein’s bivariate
operator, Bernstein’s cubature formula, remainder term
1
Preliminaries
Let us to denote N = {1, 2, . . . } and N0 = N ∪ {0}. The Bernstein’s 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:
µ
¶
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 23 November, 2008
Accepted for publication (in revised form) 26 December, 2008
161
162
D. Bărbosu, O. T. Pop
where
µ ¶
m k
pm,k (x) =
x (1 − x)m−k
k
(2)
and
µ ¶
n j
y (1 − y)n−j
pn,j (y) =
j
(3)
are the fundamental Bernstein’s polynomials.
Many approximation properties of the operator (1) are well known [1].
Let f ∈ C([0, 1] × [0, 1]) be given. The following
(4)
f = Bm,n f + Rm,n f
is known as the ”Bernstein bivariate approximation formula”, Rm,n f denoting the remainder term.
In [14], pp. 325, is mentioned the following:
”If f ∈ C (2,2) ([0, 1] × [0, 1]) the remainder term of (4) can be expressed
under the form
(5)
y(1 − y) (0,2)
x(1 − x) (2,0)
f
(x, η) −
f
(ξ, y)
2m
2n
xy(1 − x)(1 − y) (2,2)
+
f
(ξ, η).”
4mn
(Rm,n f )(x, y) = −
Next, using (4) with the expression of remainder term from (5), the Bernstein’s cubature formula is constructed.
In our recent paper [4], was obtained the correct form for the remainder
term of (4) when the approximated function f belong to C([0, 1]×[0, 1]) and
an upper bound estimation for Rm,n f for the case when f is ”sufficiently”
differentiable on [0, 1] × [0, 1].
A note on the Bernstein’s cubature formula
163
Let X be a linear space, L1 , L2 : X → X be projectors, I : X → X
be the identity operator and R1 , R2 : X → X be the remainder operators
associated to L1 and respectively L2 . If L1 and L2 commute on X, the
following decomposition of the identity operator
(6)
I = L1 L2 + R1 ⊕ R2
with
(7)
R1 ⊕ R2 = R1 + R2 − R1 R2
is well known [6], [7].
x
Suppose now that X := C([0, 1] × [0, 1]), L1 := Bm
, L2 := Bny , where
x
Bm
, Bny denote the parametrical extensions [1] of the Bernstein’s univariate
operator, i.e.
(8)
x
(Bm
f ) (x, y)
=
n
m X
X
k=0 j=0
(9)
(Bny f ) (x, y)
=
µ
pm,k (x)pn,j (y)f
¶
k
,y ,
m
µ
¶
j
pm,k (x)pn,j (y)f x,
.
n
j=0
n
m X
X
k=0
It is well known [1] that (8) and (9) are not projectors. Is also well known
[1] that for f ∈ C 2,2 ([0, 1] × [0, 1]) the remainder operators associated to (8)
and (9) are defined respectively by
(10)
(11)
¡
¢
x(1 − x) (2,0)
x
f (x, y) = −
f
(x, η)
Rm,n
2m
¡
¢
y(1 − y) (0,2)
y
f (x, y) = −
Rm,n
f
(ξ, y)
2n
164
D. Bărbosu, O. T. Pop
for any (x, y) ∈ [0, 1] × [0, 1] and any m, n ∈ N, where (ξ, η) ∈]0, 1[×]0, 1[.
It is immediately that the operator (1) is the ”tensorial product” [6], [7] of
operators (10) and (11), i.e
x
Bny .
Bm,n = Bm
(12)
Computing the boolean sum of operators (10) and (11) one arrives to the
x
expression (5) which is false, because Bm
, Bny are not projectors and the
decomposition formula (6) doesn’t holds.
By the above motives, we corrected (5) as follows.
Theorem 1 [4] 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:
(13)
(Rm,n f )(x, y) = −
−
+
x(1−x)
m
y(1 − y)
n
n
m−1
XX

pm−1,k (x)pn,j (y) 
k k+1
x, m
, m
j
k
k=0 j=0
n−1
m X
X

pm,k (x)pn−1,j (y) 
j
n
y, ,
k=0 j=0
xy(1−x)(1−y)
mn
k
m
n−1
m−1
XX
k=0 j=0
j+1
n

pm−1,k (x)pn−1,j (y)

; f

; f
k k+1
x, m
, m
j
k
y, ,
j+1
n

; f.
Note that in (13) the brackets denote bivariate divided differences [2], [4].
In the Section 2, we use the following mean-value theorem for divided
differences (see [8]).
Theorem 2 Let m ∈ N, a ≤ x0 < x1 < · · · < xm ≤ b distinct knots and
f : [a, b] → R be a given function. If f is continuous on [a, b] and has a mth
A note on the Bernstein’s cubature formula
165
derivatives on (a, b), then there exists ξ ∈ (a, b) such that
(14)
2
[x0 , x1 , . . . , xm ; f ] =
1 (m)
f (ξ).
m!
Main results
Theorem 3 Let p, q ∈ N0 , p + q ≥ 1, x0 , x1 , . . . , xp ∈ [a, b] and
y0 , y1 , . . . , yq ∈ [c, d] be a distinct knots and f : [a, b] × [c, d] → R be a
function. If f (·, y) ∈ C([a, b]) for any y ∈ [c, d],
for any y ∈ [c, d],
∂ pf
∂xp
(·, y) exists on ]a, b[
∂ pf
∂ p+q
(x, ∗) ∈ C([c, d]) for any x ∈]a, b[ and
(x, ∗)
∂xp
∂xp ∂y q
exists on ]c, d[ for any x ∈]a, b[, then there exists (ξ, η) ∈]a, b[×]c, d[ such
that
(15)



x0 , x 1 , . . . , x p
y0 , y 1 , . . . , y q
; f =
1 ∂ p+q f
(ξ, η),
p!q! ∂xp ∂y q
where ”·” and ”∗” stand for the first and second variable.
Proof. Applying the method of parametric extension (see [3]) and the
mean-value theorem for one dimensional divided differences, there exist ξ ∈
]a, b[ and respectively η ∈]c, d[, such that


x , x , . . . , xp
 0 1
; f  = [y0 , y1 , . . . , yq ; [x0 , x1 , . . . , xp ; f ]x ]y
y0 , y 1 , . . . , y q
¸
¸
·
·
1
1 ∂ pf
∂ pf
(ξ, ∗)) =
= y0 , y 1 , . . . , y q ;
y0 , y1 , . . . , yq ; p (ξ, ∗)
p! ∂xp
p!
∂x
y
y
=
so the equality (15) holds.
1 ∂ p+q f
(ξ, η),
p!q! ∂xp ∂y q
166
D. Bărbosu, O. T. Pop
Remark 1 In the conditions of Theorem 3, if p = 0 then q ∈ N, and we
consider that f has the properties that f (x0 , ∗) ∈ C([c, d]) and
∂qf
(x0 , ∗)
∂y q
exists on ]c, d[. If q = 0, then we consider similarly above conditions about
function f .
Theorem 4 Let p, q ∈ N0 , p + q ≥ 1, x0 , x1 , . . . , xp ∈ [a, b] and
y0 , y1 , . . . , yq ∈ [c, d] be a distinct knots. If f : [a, b]×[c, d] → R is a function
with the property that f ∈ C (p,q) ([a, b]×[c, d]), then exists (ξ, η) ∈]a, b[×]c, d[
such that


(16)

x0 , x 1 , . . . , x p
y0 , y 1 , . . . , y q
; f =
1 ∂ p+q f
(ξ, η).
p!q! ∂xp ∂y q
Proof. It results from Theorem 3.
Theorem 5 Let f : [0, 1] × [0, 1] → R be a function.
1
If f (·, y) ∈ C ([0, 1] for any y ∈ [0, 1], exists
y ∈ [0, 1],
∂ 2f
∂x2
(·, y) on ]0, 1[ for any
∂ 2f
(x, ∗) ∈ C 1 ([0, 1]) for any x ∈]0, 1[, exists
2
∂ 4f
(x, ∗) on
∂x
∂x2 ∂y 2
]0, 1[ for any x ∈]0, 1[, then for any (x, y) ∈ [0, 1] × [0, 1], any m, n ∈ N,
there exist (ξi (k, j), ηi (k, j)) ∈ [0, 1] × [0, 1], i ∈ {1, 2, 3}, such that
(17) (Rm,n f )(x, y) = −
m−1 n
x(1 − x) X X ∂ 2 f
(ξ1 (k, j), η1 (k, j))
2
2m
∂x
k=0 j=0
m n−1
y(1 − y) X X ∂ 2 f
(ξ2 (k, j), η2 (k, j))
−
2n
∂y 2
k=0 j=0
m−1 n−1
xy(1 − x)(1 − y) X X ∂ 4 f
+
(ξ3 (k, j), η3 (k, j)).
2 ∂y 2
4mn
∂x
k=0 j=0
A note on the Bernstein’s cubature formula
If
∂ 2f
∂ 2f
167
∂ 4f
,
and
are bounded on ]0, 1[×]0, 1[, the following ine∂x2 ∂y 2
∂x2 ∂y 2
qualities
(18)
x(1−x)
y(1−y)
xy(1−x)(1−y)
M1 (f ) +
M2 (f ) +
M3 (f )
2m
2n
4mn
1
1
1
≤
M1 (f ) +
M2 (f ) +
M3 (f )
8m
8n
64mn
|(Rm,n f )(x, y)| ≤
and
(19)
µ
|(Rm,n f )(x, y)| ≤
1
1
1
+
+
8m 8n 64mn
¶
M (f )
hold, for any (x, y) ∈ [0, 1] × [0, 1] and any m, n ∈ N, where
¯ 2
¯
¯∂ f
¯
¯
¯
(20)
M1 (f ) =
sup
¯ ∂x2 (x, y)¯ ,
(x,y)∈]0,1[×]0,1[
¯
¯ 2
¯
¯∂ f
¯
¯
M2 (f ) =
sup
¯ ∂y 2 (x, y)¯ ,
(x,y)∈]0,1[×]0,1[
(21)
(22)
M3 (f ) =
sup
(x,y)∈]0,1[×]0,1[
¯
¯ 4
¯
¯ ∂ f
¯
¯
(x,
y)
¯
¯ ∂x2 ∂y 2
and
(23)
M (f ) = max{M1 (f ), M2 (f ), M3 (f )}.
Proof. In the relation (13) we apply Theorem 3 and the relation (17)
results. Because x(1 − x) ≤
m−1
n
XX
1
4
, y(1 − y) ≤
pm−1,k (x)pn,j (y) =
k=0 j=0
1
4
m X
n−1
X
,
pm,k (x)pn−1,j (y)
k=0 j=0
=
m−1
n−1
XX
k=0 j=0
pm−1,k (x)pn−1,j (y) = 1
168
D. Bărbosu, O. T. Pop
and transforming into modulus in the relation above and taking into account
that the partial derivatives of f are bounded on ]0, 1[×]0, 1[, the inequalities
from (18) are obtained.
Integrating the Bernstein’s bivariate approximation formula (4) one arrives to the following Bernstein’s cubature formula
Z
1
Z
1
(24)
f (x, y)dx dy =
0
0
m X
n
X
i=0 j=0
µ
Ai,j f
i j
,
m n
¶
+ Rm,n [f ].
Theorem 6 [14] The coefficients of the cubature formula (24) are given by
the equalities:
(25)
Aij =
1
,
(m + 1)(n + 1)
i = 0, m, j = 0, n.
Regarding the remainder term of (23), we have the following:
Theorem 7 In the conditions of Theorem 5, the following upper-bound estimation for the remainder term of Bernstein’s cubature formula (24) is
(26)
|Rm,n [f ]| ≤
1
1
1
M1 (f ) +
M2 (f ) +
M3 (f ),
12m
12n
144mn
where M1 (f ), M2 (f ) and M3 (f ) were defined at (20), (21) and (22).
Proof. The inequality (26) follows by integrating the Bernstein’s bivariate
approximation formula (4) and taking the first inequality (18) into account.
Theorem 8 Let f : [0, 1] × [0, 1] → R be a function. If f ∈ C (2,2) ([0, 1] ×
[0, 1]), the relations (17) and (26) hold, where
¯ 2
¯
¯∂ f
¯
¯
¯,
M1 (f ) =
sup
(x,
y)
¯ 2
¯
(x,y)∈[0,1]×[0,1] ∂x
A note on the Bernstein’s cubature formula
169
¯ 2
¯
¯∂ f
¯
¯
¯
M2 (f ) =
sup
¯ ∂y 2 (x, y)¯ , and
(x,y)∈[0,1]×[0,1]
¯ 4
¯
¯ ∂ f
¯
¯
¯.
M3 (f ) =
sup
(x,
y)
¯ ∂x2 ∂y 2
¯
(x,y)∈[0,1]×[0,1]
Proof. It results from Theorem 7
Remark 2 In Theorem 7 we give a new proof for the known inequality
(26)(see [14], pp.325). The inequality from (26) is demonstrate in [14] in
the conditions of Theorem 8.
Theorem 9 In the conditions of Theorem 7 or Theorem 8, it follows that
(27)
lim
m,n→∞
m X
n
X
i=0 j=0
1
f
(m + 1)(n + 1)
µ
i j
,
m n
¶
Z1 Z1
=
f (x, y)dx dy
0
0
and the convergence from (27) is uniform.
Proof. It results from inequality (26).
Remark 3 Because the Bernstein’s bivariate operator Bm,n conserve only
the lineares functions in x and respectively y, it follows that the degree of
exactness for the cubature formula (24) is (1, 1). In the case when the
approximated function f satisfies the hypotheses of Theorem 6, the above
affirmation follows directly from the mentioned theorem.
Acknowledgement. This paper is devoted to the memory of Luciana
Lupaş and Alexandru Lupaş, remarkable representatives of the Romanian
school of Approximation Theory and Numerical Analysis.
170
D. Bărbosu, O. T. Pop
References
[1] Bărbosu, D., Aproximarea funcţiilor de mai multe variabile prin sume
booleene de operatori liniari de tip interpolator, Ed. Risoprint, ClujNapoca (2002) (Romanian)
[2] Bărbosu,
D.,
On the Schurer-Stancu approximation formula,
Carpathian J. Math. 21 (2005), 7-12
[3] Bărbosu, D., Two dimensional divided differences revisited, Creative
Math.& Inf., 17 (2008), 1-7
[4] Bărbosu, D. and Pop, O. T., On the Bernstein bivariate approximation
formula (submitted)
[5] Bărbosu, D. and Pop, O. T., A note on the GBS Bernstein’s approximation formula, Annals Univ. of Craiova, Math. Comp. Sci. Ser. 35(2008),
1-6
[6] Delvos, F. J. and Schempp, W., Boolean methods in interpolation and
approximation, Pitman Research Notes in Math., Sevies 230 New York,
1989
[7] Gordon, W. J., Distributive lattices and the approximation of multivariate functions, in Proc. Symp. Approximation with Emphasis on Spline
Functions ed. by I. J. Schoenberg, Acad. Press, New York (1969), 223277
[8] Ivan, M., Elements of Interpolation Theory, Mediamira Science Publisher, Cluj-Napoca (2004), 61-68
A note on the Bernstein’s cubature formula
[9] Popoviciu,
T.,
171
Sur le reste dans certains formules lineares
d’approximation de l’analyse, Mathematica I (24) (1959), 95-142
[10] Stancu, D. D., On the remainder term in approximation formulae by
Bernstein polynomials, Notices, Amer. Math. Soc. 9, 26 (1962)
[11] Stancu, D. D., Evaluation of the remainder term in approximation formulas by Bernstein polynomials, Math. Comput. 17 (1963), 270-278
[12] Stancu, D. D., The remainder term of certain approximation formulas
in two variables, J. SIAM Numer. Anal., 1 (1964), 137-163
[13] Stancu, D. D., On the use of divided differences in the investigation
of interpolating positive operators, Studia Scient. Math. Hungarica,
XXXV (1996), 65-80
[14] Stancu, D. D., Coman, Gh., Agratini, O., Trı̂mbiţaş, R., Analiză numerică şi teoria aproximării, II, Presa Univ. Clujeană, Cluj-Napoca
(2002) (Romanian)
Dan Bărbosu
North University of Baia Mare
Department of Mathematics and Computer Science
Victoriei 76, 430122 Baia Mare Romania,
e-mail: barbosudan@yahoo.com
172
Ovidiu T. Pop
National College ”Mihai Eminescu”
5 Mihai Eminescu Street
440014 Satu Mare Romania,
e-mail: ovidiutiberiu@yahoo.com
D. Bărbosu, O. T. Pop