238 P. Sablonni`ere
and form a partition of unity blending system. The boundary B-splines are
linearly independent
as the univariate ones. But the inner B-splines are
linearly dependent
, the dependence relationship being:
X
i, j ∈ ˆ K
mn
− 1
i+ j
h
i
k
j
B
i j
= 0 It is well known that S
1
is exact on bilinear polynomials, i.e. S
1
e
rs
= e
rs
f or 0 ≤ r, s ≤ 1 In [36], we obtained the following dQI, which is exact on P
2
: S
2
f = X
i, j ∈K
mn
µ
i j
f B
i j
where the coefficient functionals are given by µ
i j
f =
b
i
+ ¯ b
j
− 1 f M
i j
+ a
i
f M
i−1, j
+ c
i
f M
i+1, j
+ ¯a
j
f M
i, j −1
+ ¯ c
j
f M
i, j +1
. As in Section 2, we introduce the fundamental functions:
˜ B
i j
= b
i
+ ¯ b
j
− 1B
i j
+ a
i+1
B
i+1, j
+ c
i−1
B
i−1, j
+ ¯a
j +1
B
i, j +1
+ ¯c
j −1
B
i, j −1
. We also proved the following theorems, by bounding above the Lebesgue function of
S
2
: 3
2
= X
i, j ∈K
mn
| ˜ B
i j
| T
HEOREM
7. The infinite norm of S
2
is uniformly bounded independently of the partition T
mn
of the domain: kS
2
k
∞
≤ 5 T
HEOREM
8. For uniform partitions, we have the following bound: kS
2
k
∞
≤ 2.4 These bounds are probably not optimal and can still be slightly reduced.
4. A biquadratic blending sum of univariate dQIs
In this section, we study a biquadratic dQI on a rectangular domain = [a
1
, b
1
] × [a
2
, b
2
] which is a blending sum of bivariate extensions of quadratic spline dQIs of Section 2. We use the same notations as in Section 2 for the domain , the partitions
Quadratic spline quasi-interpolants 239
of I = [a
1
, b
1
], J = [a
2
, b
2
] and data sites. The partition considered on is the tensor product of partitions of I and J . We use the two sets of univariate B-splines
{B
i
x , 0 ≤ i ≤ m + 1}, {B
j
y, 0 ≤ j ≤ n + 1} and the two sets of univariate fundamental functions introduced in Section 2:
{ ˜ B
i
x , 0 ≤ i ≤ m + 1}, { ˜ B
j
y, 0 ≤ j ≤ n + 1} The associated
extended bivariate
dQIs are respectively see e.g. [14] for bivariate extensions of univariate operators
P
1
f x , y :=
m+1
X
i=0
f s
i
, yB
i
x , P
2
f x , y :=
m+1
X
i=0
f s
i
, y ˜
B
i
x
Q
1
f x , y :=
n+1
X
j =0
f x , t
j
B
j
y, Q
2
f x , y :=
n+1
X
j =0
f x , t
j
˜ B
j
y The bivariate dQI considered in this section is now defined as the blending sum
R := P
1
Q
2
+ P
2
Q
1
− P
1
Q
1
and it can be written in the following form R f x , y =
X
i, j ∈K
mn
f M
i j
¯ B
i j
x , y where the biquadratic fundamental functions are defined by
B
♭ i j
x , y := B
i
x ˜ B
j
y + ˜ B
i
x B
j
y − B
i
x B
j
y In terms of tensor-product B-splines B
i j
x , y = B
i
x B
j
y, we have: R f x , y =
X
i, j ∈K
mn
µ
i j
f B
i j
x , y, where the coefficient functionals are given by
µ
i j
f :=
a
i
f M
i−1, j
+ c
i
f M
i+1, j
+ ¯ a
j
f M
i, j −1
+ ¯c
j
f M
i, j +1
+ b
i
+ ¯ b
j
− 1 f M
i j
We have proved in [36] the following T
HEOREM
9. The operator R is exact on the 8-dimensional subspace P
12
[x , y]⊕ P
21
[x , y] of biquadratic polynomials. Moreover, its infinite norm is bounded above independently of the nonuniform partition X
m
⊗ Y
n
of the domain kRk
∞
≤ 5
240 P. Sablonni`ere
5. A trivariate blending sum of univariate and bivariate quadratic dQIs
In this section, we study a trivariate dQI on a parallelepiped = [a
1
, b
1
] × [a
2
, b
2
] × [a
3
, b
3
] which is a blending sum of trivariate extensions of univariate and bivariate dQIs seen in Sections 2 and 3. We consider the three partitions
X
m
:= {x
i
, 0 ≤ i ≤ m}, Y
n
= {y
j
, 0 ≤ j ≤ n}, Z
p
:= {z
k
, 0 ≤ k ≤ p}
respectively of the segments I = [a
1
, b
1
] = [x ,
x
m
], J = [a
2
, b
2
] = [y ,
y
n
] and K = [a
3
, b
3
] = [z ,
z
p
]. For the projection
′
= [a
1
, b
1
] × [a
2
, b
2
] of on the x y − plane, the notations are those of Section 3. For the projection
′′
= [a
3
, b
3
] of
on the z − axi s, we use the following notations, for 1 ≤ k ≤ p: l
k
= z
k
− z
k−1
, K
k
= [z
k−1
, z
k
], u
k
= 1
2 z
k−1
+ z
k
, with u
= z and u
p+1
= z
p
. For mesh ratios of subintervals, we set respectively ω
k
= l
k
l
k−1
+ l
k
, ω
′ k
= l
k−1
l
k−1
+ l
k
= 1 − ω
k
for 1 ≤ k ≤ p, with l = l
p+1
= 0 all these ratios lie between 0 and 1, and ˆa
k
= − ω
2 k
ω
′ k+1
ω
k
+ ω
′ k+1
, ˆ b
k
= 1 + ω
k
ω
′ k+1
, ˆ c
k
= − ω
k
ω
′ k+1
2
ω
k
+ ω
′ k+1
. Let K = K
mnp
= {i, j, k, 0 ≤ i ≤ m + 1, 0 ≤ j ≤ n + 1, 0 ≤ k ≤ p + 1}, then the set of data sites is
D = D
mnp
= {N
i j k
= x
i
, y
j
, z
k
, i, j, k ∈ K
mnp
}, The partition of considered here is the tensor product of partitions on
′
and
′′
, i.e. a partition into
vertical prisms with triangular horizontal sections
. Setting K
′ mn
= {i, j , 0 ≤ i ≤ m + 1, 0 ≤ j ≤ n + 1}, we consider the bivariate B-splines and
fundamental splines on
′
= [a
1
, b
1
] × [a
2
, b
2
] defined in Section 3 above: {B
i j
x , y, i, j ∈ K
′ mn
}, and { ˜ B
i j
x , y, i, j ∈ K
′ mn
} and the univariate B-splines and fundamental splines on [a
3
, b
3
] defined in Section 2: {B
k
z, 0 ≤ k ≤ p + 1} and { ˜ B
k
z, 0 ≤ k ≤ p + 1}. The extended trivariate dQIs that we need for the construction are the following
P
1
f x , y, z := X
i, j ∈K
′mn
f s
i
, t
j
, zB
i j
x , y, P
2
f x , y, z := X
i, j ∈K
′ mn
f s
i
, t
j
, z ˜
B
i j
x , y,
Quadratic spline quasi-interpolants 241
Q
1
f x , y, z :=
p+1
X
k=0
f x , y, u
k
B
k
z, Q
2
f x , y, z :=
p+1
X
k=0
f x , y, u
k
˜ B
k
z. For the sake of clarity, we give the expressions of P
2
and Q
2
in terms of B-splines: P
2
f x , y, z = X
i, j ∈K
′mn
µ
i j
f B
i j
x , y µ
i j
f = a
i
f s
i−1
, t
j
, z + c
i
f s
i+1
, t
j
, z + ¯a
j
f s
i
, t
j −1
, z + ¯c
j
f s
i
, t
j +1
, z
+b
i
+ ¯ b
j
− 1 f s
i
, t
j
, z
Q
2
f x , y, z :=
p+1
X
k=0
{ ˆa
k
f x , y, u
k−1
+ ˆ b
k
f x , y, u
k
+ ˆ c
k
f x , y, u
k+1
} B
k
z We now define the trivariate blending sum
R = P
1
Q
2
+ P
2
Q
1
− P
1
Q
1
Setting B
♭ i j k
x , y, z = B
i j
x , y ˜ B
k
z + ˜ B
i j
x , yB
k
z − B
i j
x , yB
k
z we obtain
R f = X
i, j,k∈K
mnp
f N
i j k
B
♭ i j k
In terms of tensor product B-splines B
i j k
= B
i j
B
k
, one has R f =
X
i, j,k∈K
mnp
ν
i j k
f B
i j k
where ν
i j k
f is based on the 7 neighbours of N
i j k
in R
3
: ν
i j k
f =
ˆa
k
f N
i, j,k−1
+ ˆ c
k
f N
i, j,k+1
+ a
i
f N
i−1, j,k
+ c
i
f N
i+1, j,k
+ ¯a
j
f N
i, j −1,k
+ ¯ c
j
f N
i, j +1,k
+ b
i
+ ¯ b
j
+ ˆc
k
− 1 f N
i j k
. In [36], we proved the following
T
HEOREM
10. The operator R is exact on the 15-dimensional subspace P
1
[x , y] ⊗ P
2
[z] ⊕ P
2
[x , y] ⊗ P
1
[z] of the 18-dimensional space P
2
[x , y] ⊗ P
2
[z]. Moreover, its infinite norm is bounded above independently of the nonuniform partition
of the domain kRk
∞
≤ 8.
242 P. Sablonni`ere
6. Some applications
