Bernstein-type operators on a triangle with
one curved side
Petru Blaga, Teodora Cătinaş and Gheorghe Coman
Abstract. We construct Bernstein-type operators on a triangle with one
curved side. We study univariate operators, their product and Boolean
sum, as well as their interpolation properties, the order of accuracy (degree of exactness, precision set) and the remainder of the corresponding
approximation formulas. We also give some illustrative examples.
Mathematics Subject Classification (2010). Primary 41A36; Secondary
41A35, 41A25, 41A80.
Keywords. Bernstein operator, product and boolean sum operators,
modulus of continuity, degree of exactness, error evaluation.
1. Introduction
Starting with the paper [2] of R.E. Barnhill, G. Birkhoff and W.J. Gordon,
there have been constructed interpolation operators of Lagrange, Hermite and
Birkhoff type, that interpolate the values of a given function or the values of
the function and of certain of its derivatives on the boundary of a triangle
with straight sides (see, e.g., [3], [10], [11], [14], [15], [21]). In order to match
all the boundary information on a curved domain (as in Dirichlet, Neumann
or Robin boundary conditions for differential equation problems) there were
considered interpolation operators on a triangle with curved sides (see, e.g.,
[16], [17]).
Since the Bernstein-type operators interpolate a given function at the
endpoints of the interval, these operators can also be used as interpolation
operators both on triangles with straight sides (see, e.g., [8], [25], [26]) and
with curved sides. The aim of this paper is to construct Bernstein-type operators, and also their product and boolean sum, (see, e.g., [18], [19], [20]), for
a triangle with one curved side and to study such operators especially from
the theoretical point of view. We study here only the local problem and not
consider the global problem of assembling the curved elements in a triangulation of a domain with curved boundaries, as there are, for example, in [6],
2
Petru Blaga, Teodora Cătinaş and Gheorghe Coman
[7], [13], [12]. Using modulus of continuity, respectively Peano’s theorem we
also study the remainders of the corresponding approximation formulas.
As in the case of a triangle with straight sides, by affine invariance, it is
sufficient to consider the standard triangle T̃h with vertices V1 = (0, h), V2 =
(h, 0) and V3 = (0, 0), with two straight sides Γ1 , Γ2 , along the coordinate
axes, and with the third edge Γ3 (opposite to the vertex V3 ) defined by the
one-to-one functions f and g, where g is the inverse of the function f, i.e.,
y = f (x) and x = g(y), with f (0) = g(0) = h, for h > 0. Also, we have
f (x) ≤ h and g(y) ≤ h, for x, y ∈ [0, h] . The functions f and g are defined
as in [3]. In the sequel we denote by eij (x, y) = xi y j , for i, j ∈ N.
2. Univariate operators
Let F be a real-valued function defined on Teh and (0, y), (g(y), y), respectively, (x, 0), (x, f (x)) be the points in which the parallel lines to the coordinate axes, passing through the point (x, y) ∈ Teh , intersect the sides Γi ,
i = 1, 2, 3. (See Figure 1.)
V
1
(x,f(x))
Γ3
Γ2
(0,y)
V3
(x,y)
(x,0)
(g(y),y)
Γ1
V
2
Figure 1. Triangle T̃h .
x
and Bny defined by
One considers the Bernstein-type operators Bm
(
)
m
∑
i
x
F ) (x, y) =
pm,i (x, y) F
(Bm
g(y), y ,
m
i=0
with
( )(
)i (
)m−i
m
x
x
pm,i (x, y) =
1−
,
i
g(y)
g(y)
respectively,
(Bny F ) (x, y) =
n
∑
0 6 x + y 6 g(y),
(
)
qn,j (x, y) F x, nj f (x)
j=0
with
( )(
)j (
)n−j
n
y
y
qn,j (x, y) =
1−
,
j
f (x)
f (x)
0 ≤ x + y ≤ f (x),
Bernstein-type operators on a triangle with one curved side
where
{
∆xm =
}
{
}
i
g(y) i = 0, m and ∆yn = nj f (x) j = 0, n
m
are uniform partitions of the intervals [0, g(y)] and [0, f (x)].
Theorem 2.1. If F is a real-valued function defined on Teh then:
x
(i) Bm
F = F on Γ2 ∪ Γ3 ,
(ii) Bny F = F on Γ1 ∪ Γ3 ,
and
x
(iii) (Bm
eij ) (x, y) = xi y j , [ i = 0, 1; j ∈ N,]
x
(Bm
e2j ) (x, y) = x2 + x(g(y)−x)
y j , j ∈ N,
m
(iv) (Bny eij ) (x, y) = xi y j , i [∈ N, j = 0, 1, ]
(Bny ei2 ) (x, y) = xi y 2 + y(f (x)−y)
, i ∈ N.
n
Proof. The interpolation properties (i) and (ii) follow from the relations:
{
1, for i = 0,
pm,i (0, y) =
0, for i > 0,
and
{
pm,i (g(y), y) =
respectively by
{
qn,j (x, 0) =
and
1, for j = 0,
0, for j > 0,
{
qn,j (x, f (x)) =
0, for i < m,
1, for i = m,
0, for j < n,
1, for j = n.
Regarding the properties (iii), we have
x
x
ei0 )(x, y),
eij ) (x, y) = y j (Bm
(Bm
and
(
x
(Bm
e00 ) (x, y)
x
Bm
e10 (x, y)
=
x
x
+1−
g(y)
g(y)
j∈N
)m
= 1,
)i (
)m−i
m ( )(
∑
x
i
m
x
=
1−
g(y)
g(y)
g(y)
m
i
i=0
)m−1−i
m−1
∑ (m − 1) ( x )i (
x
=x
1−
i
g(y)
g(y)
i=0
(
)m−1
x
x
=x
+1−
= x,
g(y)
g(x)
3
4
Petru Blaga, Teodora Cătinaş and Gheorghe Coman
x
Bm
e20 (x, y)
)i (
)m−i (
)2
m ( )(
∑
m
x
x
g(y)
2
=
1−
i
i
g(y)
g(y)
m
i=0
(
)2 ∑
(
)
(
)
(
)m−i
m
i
g(y)
m
x
x
g(y)
=
i(i − 1)
1−
+x
m
i
g(y)
g(y)
m
i=0
m−1 2
g(y)
x[g(y) − x]
x +x
= x2 +
.
m
m
m
Properties (iv) are proved in the same way.
=
Now, let us consider the approximation formula
x
x
F = Bm
F + Rm
F.
Theorem 2.2. If F (·, y) ∈ C[0, g(y)] then
)
x
(
h
(Rm F )(x, y) ≤ 1 + √
ω(F (·, y); δ),
2δ m
y ∈ [0, h],
where ω(F (·, y); δ) is the modulus of continuity of the function F with regard
to the variable x.
√
Moreover, if δ = 1/ m then
x
(
)
(Rm F ) (x, y) ≤ 1 + h ω(F (·, y); √1 ), y ∈ [0, h].
(2.1)
2
m
x
Proof. From the property (Bm
e00 )(x, y) = 1, it follows that
m
∑
x
i
(Rm F ) (x, y) ≤
pm,i (x, y)F (x, y) − F ( g(y), y).
m
i=0
Using the inequality
( )
F (x, y) − F ( i g(y), y) ≤ 1 x − i g(y) + 1 ω(F (·, y); δ)
δ m
m
one obtains
x
|(Rm
F )(x, y)|
( )
i
1 ≤
pm,i (x, y) δ x − g(y) + 1 ω(F (·, y); δ)
m
i=0
]
[
m
)2 )1/2
(
1( ∑
i
ω(F (·, y); δ)
≤ 1+
pm,i (x, y) x − g(y)
δ i=0
m
]
[
√
1 x(g(y)−x)
ω(F (·, y); δ).
= 1+
m
δ
Since,
max
0≤x≤g(y)
it follows that
hence
m
∑
[
]
x(g(y) − x) =
g 2 (y)
4
and max g 2 (y) = h2 ,
0≤y≤h
[
] h2
max x(g(y) − x) =
,
4
Teh
x
(Rm F )(x, y) ≤
(
1+
h
√
2δ m
)
ω(F (·, y); δ).
Bernstein-type operators on a triangle with one curved side
√
Now, for δ = 1/ m, one obtains (2.1).
5
Theorem 2.3. If F (·, y) ∈ C 2 [0, h] then
x
(Rm
F )(x, y) =
and
x[x − g(y)] (2,0)
F
(ξ, y), for ξ ∈ [0, g(y)]
2m
2
x
(Rm F )(x, y) ≤ h M20 F, (x, y) ∈ Teh ,
8m
Mij F = max F (i,j) (x, y) .
where
Teh
x
Proof. Taking into account that the degree of exactness of the operator Bm
x
is 1, i.e., dex(Bm ) = 1, by Peano’s theorem, it follows
∫ g(y)
x
(Rm
F )(x, y) =
K20 (x, y; s)F (2,0) (s, y)ds,
0
where
K20 (x, y; s) = (x − s)+ −
m
∑
pm,i (x, y)
i=0
)
(i
g(y) − s + .
m
ν
For a given ν ∈ {1, ..., m} one denotes
by K20
(x, y;]·) the restriction of the
[
g(y)
kernel K20 (x, y; ·) to the interval (ν − 1) m , ν g(y)
, i.e.,
m
ν
K20
(x, y; ν) = (x − s)+ −
m
∑
(
pm,i (x, y)
i=ν
whence,
ν
K20
(x, y; s)
)
i
g(y) − s ,
m

m
)
(
∑


pm,i (x, y) mi g(y) − s ,
 x−s−
=
s<x
i=ν
m
(
)
∑


pm,i (x, y) mi g(y) − s ,
 −
s ≥ x.
i=ν
ν
(x, y; s) ≤ 0, for s ≥ x. For s < x we have
It follows that K20
(
) ν−1
(
)
m
∑
∑
i
i
ν
K20
(x, y; s) = x−s−
pm,i (x, y)
g(y) − s +
pm,i (x, y)
g(y) − s .
m
m
i=0
i=0
As,
m
∑
(
pm,i (x, y)
i=0
)
i
g(y) − s = x − s,
m
it follows that
ν
K20
(x, y; s)
=
ν−1
∑
i=0
(
pm,i (x, y)
)
i
g(y) − s ≤ 0.
m
ν
So, K20
(x, y; ·) ≤ 0 for any ν ∈ {1, ..., m}, i.e., K20 (x, y; s) ≤ 0, for s ∈
[0, g(y)].
6
Petru Blaga, Teodora Cătinaş and Gheorghe Coman
By mean value theorem, one obtains
∫ g(y)
x
(2,0)
(Rm F )(x, y) = F
(ξ, y)
K20 (x, y; s)ds, 0 ≤ ξ ≤ g(y).
0
Since,
∫
g(y)
K20 (x, y; s)ds =
0
x(x − g(y))
2m
and
|x(x − g(y))|
g 2 (y)
h2
=
≤
, y ∈ [0, h]
2m
8m
8m
0≤x≤g(y)
max
the conclusion follows.
Remark 2.4. Analogous results are obtained for the remainder of the formula
F = Bny F + Rny F,
i.e.,
y
(
(Rn F )(x, y) ≤ 1 +
and
h
√
2δ n
)
ω(F (x, ·); δ),
F (x, ·) ∈ C[0, f (x)]
)
) (
(
(Rny F )(x, y) ≤ 1 + h2 ω F (x, ·); √1n
respectively,
(Rny F )(x, y) =
and
y[y − f (x)] (0,2)
F
(x, η),
2n
2
y
(Rn F )(x, y) ≤ h M02 F,
8n
η ∈ [0, f (x)]
(x, y) ∈ Teh .
3. Product operator
x
x
Let Pmn = Bm
Bny , respectively, Qnm = Bny Bm
be the products of the opery
x
ators Bm and Bn .
We have
m ∑
n
∑
(
) (
(
))
(Pmn F ) (x, y) =
pm,i (x, y) qn,j mi g(y), y F mi g(y), nj f mi g(y)
i=0 j=0
and
(Qnm F ) (x, y) =
m ∑
n
∑
i=0 j=0
( (
)
)
)
(
pm,i x, nj f (x) qn,j (x, y) F mi g nj f (x) , nj f (x) .
Bernstein-type operators on a triangle with one curved side
7
Remark 3.1. The nodes of the operator Pmn , respectively, Qnm are given in
Figure 2.
Figure 2. The nodes for Pmn and Qnm , for m = n = 4.
Theorem 3.2. If F is a real-valued function defined on Teh then:
(i) (Pmn F )(V3 ) = F (V3 ),
Pmn F = F, on Γ3
and
(ii) (Qnm F )(V3 ) = F (V3 ),
Qnm F = F, on Γ3 .
Proof. The proof follows from the properties
x
(Pmn F )(x, 0) = (Bm
F )(x, 0),
(Pmn F )(0, y) = (Bny F )(0, y),
(Pmn F )(x, f (x)) = F (x, f (x)),
x, y ∈ [0, h]
and
x
(Qnm F )(x, 0) = (Bm
F )(x, 0),
(Qnm F )(0, y) = (Bny F )(0, y),
(Qnm F )(g(y), y) = F (g(y), y),
x, y ∈ [0, h],
which can be verified by a straightforward computation.
y
For example, the property (Pmn F )(x, 0) = (Bm
F )(x, 0) implies (Pmn F )(0, 0) =
F (0, 0).
Remark 3.3. The product operators Pmn and Qnm interpolate the function F
at the vertex (0, 0) and on the side y = f (x) (or x = g(y)).
Let us consider now the approximation formula
P
F = Pmn F + Rmn
F,
P
is the corresponding remainder operator.
where Rmn
Theorem 3.4. If F ∈ C(Teh ) then
)
(
( P
)
Rmn F (x, y) ≤ (1 + h)ω F ; √1 , √1 ,
m
n
(x, y) ∈ Teh
8
Petru Blaga, Teodora Cătinaş and Gheorghe Coman
Proof. We have
[
P
(Rmn F )(x, y) ≤
m n
(
)
1 ∑∑
pm,i (x, y)qn,j mi g(y), y x −
δ1 i=0 j=0
i
m g(y)
m n
(
)
(
)
1 ∑∑
pm,i (x, y)qn,j mi g(y), y y − nj f mi g(y) δ2 i=0 j=0
]
m ∑
n
∑
(i
)
pm,i (x, y)qn,j m g(y), y ω(F ; δ1 , δ2 ).
+
+
i=0 j=0
Since,
m ∑
n
∑
(
)
x −
√
x(g(y) − x)
,
m
i=0 j=0
√
m ∑
n
∑
)
(i
)
(i
y(f (x) − y)
j
,
pm,i (x, y)qn,j m g(y), y y − n f m g(y) ≤
n
i=0 j=0
m ∑
n
∑
pm,i (x, y)qn,j
pm,i (x, y)qn,j
(
i
m g(y), y
i
m g(y), y
)
i
m g(y)
≤
= 1,
i=0 j=0
it follows that
P
(Rmn F )(x, y) ≤
But
(
1
1+
δ1
√
x(g(y) − x)
1
+
m
δ2
√
y(f (x) − y)
n
)
ω(F ; δ1 , δ2 ).
(
) h2
(
) h2
x g(y) − x ≤
and y f (x) − y ≤
,
4
4
whence,
P
(Rmn F )(x, y) ≤
and
(
)
1 h
1 h
√ +
√
1+
ω(F ; δ1 , δ2 )
δ1 2 m δ2 2 n
(
)
P
(Rmn F )(x, y) ≤ (1 + h) ω F ; √1 , √1 .
m
n
4. Boolean sum operators
x
Finally, we consider the Boolean sums of the operators Bm
and Bny , i.e.,
x
x
x
Smn := Bm
⊕ Bny = Bm
+ Bny − Bm
Bny ,
respectively,
x
x
x
Tnm := Bny ⊕ Bm
= Bny + Bm
− Bny Bm
.
Bernstein-type operators on a triangle with one curved side
9
Theorem 4.1. If F is a real-valued function defined on Teh then
Smn F ∂ Te = F ∂ Te
and
Tnm F ∂ Te = F ∂ Te .
Proof. As,
x
(Pmn F ) (x, 0) = (Bm
F ) (x, 0) ,
(Pmn F ) (0, y) = (Bny F ) (0, y) ,
x
(Bm
F ) (x, h − x) = (Bny F ) (x, h − x) = (Pmn F )(x, h − x) = F (x, h − x)
the conclusion follows.
For the remainder of the Boolean sum approximation formula,
S
F = Smn F + Rmn
F,
we have the following result.
Theorem 4.2. If F ∈ C(Teh ) then
S
(Rmn F )(x, y) ≤(1 + h )ω(F (·, y); √1 ) + (1 + h )ω(F (x, ·); √1 )
2
2
m
n
(4.1)
+ (1 + h)ω(F ; √1m , √1n ), (x, y) ∈ Teh .
Proof. The identity
x
F + F − Bny F − (F − Pmn F )
F − Smn F = F − Bm
implies that
S
x
P
(Rmn F )(x, y) ≤ (Rm
F ) (x, y) + (Rny F ( x, y) + (Rmn
F )(x, y)
and the conclusion follows.
5. Numerical examples
Example 5.1. We consider the following test functions, generally used in the
literature, (see, e.g., [22]):
2
2
exp[− 81
16 ((x − 0.5) + (y − 0.5) )],
F1 (x, y) =
1
3
F2 (x, y) =
1.25 + cos 5.4y
.
6 + 6(3x − 1)2
(5.1)
In Figure 3 and Figure 4 we plot the graphs of the maximum errors for
x
approximating by Bm
Fi , Bny Fi , Pmn Fi , Smn Fi , i =
√ 1, 2, on T̃h , considering
h = 1, m = 5, n = 6 and f : [0, 1] → [0, 1], f (x) = 1 − x2 .
10
Petru Blaga, Teodora Cătinaş and Gheorghe Coman
0.05
0.06
0.04
0.04
0.03
0.02
0.02
0.01
0
1
0
1
0.9
0.8
0.7
0.8
1
0.6
0.4
1
0.6
0.8
0.5
0.8
0.6
0.3
0.4
0.6
0.4
0.2
0.4
0.2
0.2
0.1
0
0.2
0
0
0
The approximation error for
Bny F1 .
The approximation error for
x
Bm
F1 .
0.01
0.1
0.08
0.06
0.005
0.04
0.02
0
1
0
1
0.8
0.9
0.8
0.7
0.6
1
0.6
0.8
0.5
0.4
1
0.8
0.4
0.6
0.6
0.3
0.4
0.2
0.4
0.2
0.2
0.2
0.1
0
0
0
The approximation error for
Pmn F1 .
0
The approximation error for
Smn F1 .
Figure 3. The maximum approximation errors for F1 .
0.09
0.09
0.08
0.08
0.07
0.07
0.06
0.06
0.05
0.05
0.04
0.04
0.03
0.03
1
0.02
0.4
0.6
0
0
0.4
0.2
0.8
0.01
0.6
0
0
1
0.02
0.8
0.01
0.8
1
0.4
0.2
0.4
0.2
0.6
0.2
0.6
0.8
0
1
The approximation error for
x
Bm
F2 .
0
The approximation error for
Bny F2 .
0.1
0.015
0.08
0.01
0.06
0.04
0.005
0.02
0
1
0
1
0.8
0.6
1
0.8
0.4
0.6
0.4
0.2
0.2
0
0
The approximation error for
Pmn F2 .
0.8
1
0.6
0.8
0.6
0.4
0.4
0.2
0.2
0
0
The approximation error
Smn F2 .
Figure 4. The maximum approximation errors for F2 .
Bernstein-type operators on a triangle with one curved side
11
Table 1 contains the maximum approximation errors for the functions
given in (5.1) for the Bernstein type operators and for some operators obtained in [16], namely Lagrange-type operators
(L1 F )(x, y) =
(P13 F )(x, y) =
g(y) − x
x
F (0, y) +
F (g(y), y),
g(y)
g(y)
g(y) − x
x
F (0, y)+
[g(y)F (y+g(y), 0)+yF (0, y+g(y))],
g(y)
g(y)[y + g(y)]
(S12 F ) (x, y) =
g(y) − x
f (x) − y
y
F (0, y) +
F (x, 0) +
F (x, f (x))
g(y)
f (x)
f (x)
[
]
g(y) − x h − y
y
−
F (0, 0) + F (0, h) ,
g(y)
h
h
Hermite-type operator
(H1 F )(x, y) =
[x − g(y)]2
x[2g(y) − x]
x[x − g(y)] (1,0)
F (0, y)+
F (g(y), y)+
F
(g(y), y),
g 2 (y)
g 2 (y)
g(y)
and Birkhoff-type operator
(B1 F ) (x, y) = F (0, y) + xF (1,0) (g (y) , y) .
F1
F2
x
Bm
F
0.0525 0.0821
Bny F
0.0452 0.0692
Pmn F
0.0858 0.0943
Qnm F
0.0857 0.0944
Smn F
0.0095 0.0144
Tnm F
0.0095 0.0112
L1 F
0.2097 0.2259
P13 F
0.2943 0.2261
S12 F
0.1718 0.1809
H1 F
0.0758 0.2210
B1 F
0.6235 0.5302
Table 1. The approximation errors.
Max error
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Petru Blaga
Babeş-Bolyai University,
Faculty of Mathematics and Computer Science,
Str. M. Kogălniceanu Nr. 1, RO-400084
Cluj-Napoca, Romania
e-mail: [email protected]
Teodora Cătinaş
Babeş-Bolyai University,
Faculty of Mathematics and Computer Science,
Str. M. Kogălniceanu Nr. 1, RO-400084
Cluj-Napoca, Romania
e-mail: [email protected]
Gheorghe Coman
Babeş-Bolyai University,
Faculty of Mathematics and Computer Science,
Str. M. Kogălniceanu Nr. 1, RO-400084
Cluj-Napoca, Romania
e-mail: [email protected]