Mohammed-Najib Benbourhim, Patrick Chenin, Abdelhak Hassouni & Jean-Baptiste Hiriart-Urruty, Editors
SUBDIVISION SCHEME OF QUARTIC BIVARIATE SPLINES ON A
FOUR-DIRECTIONAL MESH
El Bachir Ameur
1, Domingo Barrera Rosillo
2and Driss Sbibih
3Abstract. In this paper we give a new definition of minimally and quasi-minimally supported C2 quartic bivariate B-splines associated with the four-directional mesh of the plane, introduced in [7, 19], which is convenient to show that theses B-splines satisfy the refinement equation and we determine the associated matrix mask, we prove that the family of these B-splines is stable and the associated subdivision scheme converges. These results can be extended to various cases in the spline space of classC3k+2and degree 4k+ 4, but in these cases the supports of the masks are larger.
R´esum´e. Dans cet article nous donnons une nouvelle d´efinition des B-splines quartiques de classeC2 `a support minimal et quasi-minimal sur le r´eseau quadridirectionnel du plan, introduites dans [7, 19]. Nous utilisons cette d´efinition pour montrer que ces B-splines v´erifient une ´equation de raffinement et d´eterminer les matrices du filtre associ´e. Nous montrons que la famille de ces B-splines est stable et que le sch´ema de subdivision associ´e converge. Ces r´esultats peuvent ˆetre g´en´eralis´es aux espaces de fonctions splines de classeC3k+2et de degr´e 4k+ 4, mais dans ce cas les supports des filtres associ´es deviennent plus grands.
Introduction
Refinable function vectors and vector subdivision schemes, as two of the most important and extensively studied fundamental objects in the literature of wavelet analysis, are useful in many applications such as signal processing and computer aided geometric design ( [4,5,9,10,13,16,22]). In [9,10] the authors give the subdivision scheme associated with the function vectors of two linear and cubic B-splines. In this paper we are interested in the function vector of threeC2quartic B-splines on the four-directional mesh.
We begin with some notations and definitions used throughout this paper. Letτbe the uniform triangulation of the plane, whose set of vertices isZ2, and whose edges are parallel to the four directionse
1= (1,0),e2= (0,1),
e3= (1,1) ande4= (1,−1). This type of triangulation is called a four-directional mesh. For any integersrand
d, letPd be the space of bivariate polynomials of total degree at mostd, and
Sdr(τ) := ©
s∈ Cr(R2) : s|T ∈Pd for allT ∈τ ª
1 Universit´e Moulay Ismail, Facult´e des sciences et Techniques, D´epartement d’Informatique, 52000 Errachdia, Maroc;e-mail:[email protected]
2Departamento de Matem´atica Aplicada, Faculdad de Ciencias, Universidad de Granada, Campus Universitario de Fuentenueva s/n, 18071, Granada, Spain; e-mail:[email protected]
3 Universit´e Mohammed I, Ecole Suprieure de Technologie, Laboratoire MATSI, Oujda, Maroc; e-mail:[email protected] c
°EDP Sciences, SMAI 2007
be the space of bivariate piecewise polynomial functions of classCron the plane and whose restrictions to each
triangular cell of τ are inPd.
LetT be a triangle ofτ andλ= (λ1, λ2, λ3) be the barycentric coordinates of a pointM ofR2relative toT.
Each polynomial pofPd(T) has a unique representation in the Bernstein-B´ezier form:
p(M) = X
µ∈4d
b(µ)Bd µ(λ)
where
4d= ©
µ= (µ1, µ2, µ3)∈Z3+:|µ|=µ1+µ2+µ3=d
ª
and
Bµd(λ) =
d!
µ! λ
µ = d!
µ1!µ2!µ3! λ
µ1
1 λµ22 λµ33 .
The family of the ¡d+22 ¢ polynomials Bd
µ, µ ∈ 4d, forms a basis for the space Pd(T). The coefficients
{b(µ), µ∈ 4d}, are called the B-net of pon the triangle T.
Whenr= 0 andd= 1, the linear bivariate splines space on the four-directional meshτ,S0
1(τ), is generated
by two minimally supported linear B-splinesϕ1and ϕ2, whose B-nets and supports are given in Figure 1.
1
1 0
0 0
0
0
0 0
0 0
0 0
0
i=1 i=2
Figure 1. B-nets and supports ofϕi fori= 1,2.
A bivariate compactly supported function vector Ψ = (ψ1,· · ·, ψm)T is said to be refinable if it satisfies the
refinement equation
Ψ = X
j∈Z2
PjΨ(2· −j) (1)
where the sequenceP:= (Pj)j∈Z2 is the refinement matrix mask withPj being real (m×m)-matrices.
Let¡`(Z2)¢and¡`∞(Z2)¢be the linear space of all sequences onZ2and the subspace of all bounded sequences
respectively. We say that Ψ is stable if its integer translates satisfy the stability condition, i.e., if there exist constants 0< c1≤c2<∞such that
c1 kλk∞≤ k
X
j∈Z2
λT
j Ψ(· −j)k∞≤c2kλk∞far allλ∈
¡
`∞(Z2¢)m. (2)
wherekλk∞= maxikλik∞andkfk∞ denote the sup-norms of the vector sequenceλand the bivariate function
f respectively.
The subdivision operator associated withPis given by
SP : ¡
`(Z2)¢m−→¡`(Z2)¢m
(SP λ)i:= X
j∈Z2P T
In this paper we are interested in the quartic bivariate splines space S2
4(τ) on the four-directional mesh τ,
which is generated by three independent locally supported B-splines, two are minimally supported ψ1 andψ2,
they are constructed by Sablonni`ere in [18], whose B-nets and supports are given in the Figure 2, and one is quasi-minimally supportedψ3constructed by Chui and He in [7], whose B-net and support are given in Figure
3. 2 0 1 0 0 0 0 0 0 0 0 i=1 0 0 0 8 8 8 4
2 1 0 0
0 0 4 20 4 4 0 0 0 0 0 1 16 8 8 16 16 20 20 16 16 20 8 8 8 8 4 4 4 2 4 42 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0
0 0
0 0 0 0
0 0 0
i=2
Figure 2. B-nets and supports of 48ψi fori= 1,2.
16 16 8 4 2 0 0 0 0 16
12 8 5
2 0 0 0 8 6 5 4 3 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 12
Figure 3. B-net and support of 48ψ3.
Using only the B-nets of these B-splines, it is difficult to show that neither these B-splines satisfy a refinement equation nor determine their associated matrix masks. So that, in this section 1, we give a new definition of these B-splines which is convenient to prove that the function vector (ψ1, ψ2, ψ3)T satisfies the refinement equation
1.
Definition of quartic B-splines
ψ1
,
ψ2
,
ψ3
by convolution
Let Ψ := (ψ1, ψ2, ψ3)T be the function vector of the above three quartic B-splines in the space S42(τ). In
this section we give a new definition of these B-splines which is convenient to prove that the function vector Ψ verifies the refinement equation:
Ψ = X
j∈Z2
PjΨ(2· −j) (4)
and we can determine explicitly the associated refinement matrix mask P := (Pj)j∈Z2 where Pj is a real
3×3-matrix, for allj∈Z2.
Lemma 1.1. The B-splines ϕ1∗ϕ1,ϕ1∗ϕ2 andϕ2∗ϕ2 satisfy the following symmetry properties:
ϕ1∗ϕ1(· ) =ϕ1∗ϕ1(Gi· + 2 ηi), (5)
ϕ1∗ϕ2(·) =ϕ1∗ϕ2(Gi· + ηi) (6) and
ϕ2∗ϕ2( ·) =ϕ2∗ϕ2(Gi· ) (7)
fori= 0,1, where
G0=
µ 0 1 1 0 ¶
, G1=−G0, η0 = (0,0), η1= (1,1)
and∗ denote the convolution operator of the scaling functions.
Proof. Using the fact that the linear B-splinesϕ1andϕ2 satisfy the symmetry properties
ϕ1(·) =ϕ1(·Gi + ηi) andϕ2(·) =ϕ2(·Gi),
we deduce that, fori= 0,1,
ϕ1∗ϕ1(x, y) =
Z
R2
ϕ1(u, v)ϕ1(x−u, y−v)dudv
= Z
suppϕ1
ϕ1((u, v)Gi + ηi) ϕ1((x−u, y−v)Gi + ηi) dudv.
If we put (s, t) = (u, v)Gi + ηi, then, by using a variable change, we obtain
ϕ1∗ϕ1(x, y) =|detGi|2 Z
suppϕ1 Gi+ηi
ϕ1(s, t) ϕ1((x, y) Gi + 2ηi−(s, t)) dsdt,
since ( suppϕ1)Gi+ηi = suppϕ1 and|detGi| = 1, we obtain
ϕ1∗ϕ1(x, y) =
Z
suppϕ1
ϕ1(s, t) ϕ1((x, y) Gi + 2ηi−(s, t)) dsdt,
consequently, we have
ϕ1∗ϕ1(x, y) =ϕ1∗ϕ1((x, y) Gi + 2ηi).
A similar technique can be applied to obtain the equations (6) and (7). ¤
Theorem 1.2. The minimally and quasi-minimally supported quartic B-splines ψ1,ψ2 andψ3 are defined by:
Ψ :=
ψψ12
ψ3
=
2ϕϕ11∗∗ϕϕ12
ϕ2∗ϕ2
Proof. Firstly, it is easy to verify that the supports of the B-splinesϕ1∗ϕ1, ϕ1∗ϕ2 andϕ2∗ϕ2 are the same
of those ofψ1,ψ2andψ3 respectively illustrated in figures 2 and 3.
Hence, to prove this theorem, it suffices to compute the B-nets of these B-splines on their supports. From Lemma 1.1, the symmetry properties of the B-splines ϕ1∗ϕ1, ϕ1∗ϕ2 and ϕ2∗ϕ2 allow us to compute their
B-nets only on a part of their supports. For this, since the linear box spline B1,1,1,0 lies inS10(τ) (see [6]), we
can easily verify that
B1,1,1,0 = 1
2 ϕ1+ 1
2 ϕ1(· −e3) +ϕ2(· −e3), consequently
ϕ1∗B1,1,1,0 =
1
2 ϕ1∗ϕ1+ 1
2 ϕ1∗ϕ1(· −e3) +ϕ1∗ϕ2(· −e3), (9) and
ϕ2∗B1,1,1,0 = 1
2 ϕ1∗ϕ2+ 1
2 ϕ1∗ϕ2(· −e3) +ϕ2∗ϕ2(· −e3). (10) However, we can compute easily the B-nets of the B-splinesϕ1∗B1,1,1,0andϕ2∗B1,1,1,0by using the classical
algorithm for generating the B-nets of bivariate B-splines defined on the four-direction mesh developed in [8].
i=1
12
22
12
0 0 0
4
1 2
0 0
0 4
4 4 8 4 8 8 2 2 1 0 0 0 0 12 2 1 0 0 0 0 0 0 0 0 0 0 0
4 4 3 8 8 8 8 6 16 16 16 16 16 16 16 16 22 22 22 22 22 20 20 20 20 16 12 0
0 0 0 0 0 0 0 0 0 0 0 0 0 T T T T T T 0 1 2 7 5 4 3 6 T T i=2 36 20 20 6 2 2 2 1 1/2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000 0 0 0 0 0 0 0 0 0 0 00 8 8 8 4 4 4 2 2 1 4 4 8 14 14 16 10 12 0 0 0 0 0 0 32 24 36 36 26 12 32 26 20
Figure 4. B-nets and supports of 96ϕi∗B1,1,1,0, i= 1,2, and the trianglesTi fori= 0,· · ·,7
Now, by using the equations (9) and (10) we obtain
(ϕ1∗ϕ1)|T0 = 2 (ϕ1∗B1,1,1,0)|T0, (11)
(ϕ1∗ϕ1)|T1 = 2 (ϕ1∗B1,1,1,0)|T1−4 (ϕ2∗B1,1,1,0(· −e3))|T1, (12)
(ϕ1∗ϕ2(· −e3))|T1ST2ST3 = 2 (ϕ1∗B1,1,1,0)|T1ST2ST3, (13)
(ϕ1∗ϕ2(· −e3))|T4ST5 = (ϕ1∗B1,1,1,0)|T4ST5 − 1
2 (ϕ1∗ϕ1)|T4 S
T5, (14)
(ϕ1∗ϕ2(· −e3))|T6 = (ϕ1∗B1,1,1,0)|T6 − 1
2 (ϕ1∗ϕ1)|T6 − 1
2 (ϕ1∗ϕ1(· −e3))|T6, (15) (ϕ2∗ϕ2(· −2e3))|T4ST5ST6 = (ϕ2∗B1,1,1,0(· −e3))|T4ST5ST6 − 1
(ϕ2∗ϕ2(· −2e3))|T7 = (ϕ2∗B1,1,1,0(· −e3))|T7 − 1
2 (ϕ1∗ϕ2(· −e3))|T7 − 1
2 (ϕ1∗ϕ2(· −2e3))|T7. (17) where the trianglesTi, i= 0,· · · ,7 are illustrated in Figure 4.
In the first step, we compute the B-net of the B-spline ϕ1∗ϕ1 by using the equations (11) and (12), and the
symmetry properties of this B-spline (5). In the second step, we use the equations (6), (13), (14) and (15), and the B-net ofϕ1∗ϕ1 for computing that ofϕ1∗ϕ2. Finally, by using the the equations (7), (16) and (17), and
the B-net ofϕ1∗ϕ2, we compute that ofϕ2∗ϕ2. After the computation, we obtain the equation (8). ¤
2.
Refinement equation and subdivision scheme
In this section, we determine the refinement equation associated with the function vector Ψ of the three quartic B-splinesψ1,ψ2andψ3by using the new definition described above and the following lemmas (see [9]).
Furthermore, we prove that this refinable function vector Ψ is stable and the associated subdivision scheme converges.
Lemma 2.1. Let Λ = (λ1,· · · , λn)T and Θ = (θ1,· · · , θm)T be two refinable function vectors of compactly supported bivariate functions, satisfying the refinement equations
Λ = X
j∈Z2
Aj Λ(2· −j) (18)
and
Θ = X
j∈Z2
Bj Θ(2· −j) (19)
with corresponding matrix masks A := (Aj)j∈Z2 and B := (Bj)j∈Z2, which are matrix sequences of (n×n)
-matrices and(m×m)-matrices respectively. Then the Kronecker convolved function vector
Λ∗Θ :=
λ1∗Θ
.. .
λn∗Θ :=
λ1∗θ1
.. .
λ1∗θm .. .
λn∗θ1
.. .
λn∗θm
verifies the following refinement equation
Λ∗Θ = X
j∈Z2
CjΛ∗Θ(2· −j) (20)
where the refinement mask C:= (Cj)j∈Z2, which is a matrix sequence of (nm×nm)-matrices, is computed as
follows:
Cj = 1
4 X
i∈Z2
Ai⊗Bj−i, j ∈Z2 (21)
with ⊗denotes the Kronecker product of matrices.
Lemma 2.2. The function vectorΦ := (ϕ1, ϕ2)T of linear bivariate B-splines on a four-direction mesh satisfies
the matrix refinement equation
Φ = X
j∈Z2
where the refinement matrix maskH:= (Hj)j∈Z2 ,which is a matrix sequence of(2×2)-matrices, is given by
j1= 0
H= .. . ... ... ... ...
· · · 0 0 0 0 0 · · ·
· · · 0 0
µ1
2 0
0 1 2
¶ µ1
2 1
0 0 ¶
0 · · ·
· · · 0 µ
0 0
1
2 12
¶ µ1
2 0
1
2 1
¶ µ1
2 0
0 1 2
¶
0 · · ·
· · · 0 µ 0 0 1 2 0 ¶ µ 0 0 1
2 12
¶
0 0 · · ·
· · · 0 0 0 0 0 · · ·
.. . ... ... ... ...
j2= 0
Theorem 2.3. The function vectorΨ := (ψ1, ψ2, ψ3)T of aC2quartic minimally and quasi-minimally supported
bivariate B-spline on a four-direction mesh satisfies the following refinement equation:
Ψ = X
j∈Z2
Pj Ψ(2· −j) (23)
where the refinement matrix maskP:= (Pj)j∈Z2 ,which is a matrix sequence of(3×3)-matrices, is given by
P(0,0)=
1
16 0 0
1
2 38 0
1
16 14 12
, P(1,0)=
1
8 0 0
1
4 38 14
0 1
16 14
, P(2,0)=
1
16 0 0
0 1
16 0
0 0 1 16
,
P(1,1)=
1
4 18 0
1
8 38 12
0 0 1 8
, P(2,1)=
1
8 18 0
0 1
16 14
0 0 0
, P(2,2)=
1
16 18 14
0 0 0 0 0 0
,
P(−1,−1)=
01 0 0
8 0 0
1
4 14 18
, P(0,−1)=
01 0 0
4 161 0
1
8 14 14
, P(1,−1)=
01 0 0
8 161 0
0 1
16 18
,
P(0,−2)=
00 00 00
1
16 161 161
, P(−1,−2)=
00 00 00
1
8 161 0
, P(−2,−2)=
00 0 00 0
1
16 0 0
,
Pj=PG0j for all j∈ {(−2,−1),(−2,0),(−1,0),(−1,1),(0,1),(0,2),(1,2)},
and
Pj= 0 for allj∈Z2\ H2
whereH2is the subset ofZ2which intersect the hexagon of vertices{(−2,−2),(0,−2),(2,0),(2,2),(0,2),(−2,0)}.
Proof. From lemmas 2.1 and 2.2, it follows that the Kronecker convolved function vector Φ∗Φ satisfies the
following refinement equation
Φ∗Φ = X
j∈Z2 e
where the matrix sequence of (4×4)-matrices³Hej ´
j∈Z2 is given by
e
Hj = 1
4 X
i∈Z2
Hi⊗Hj−i, j∈Z2.
On the other hand
Ψ = R Φ∗Φ, (24)
with
R =
1 0 0 00 1 1 0 0 0 0 1
,
and
Φ∗Φ = T Ψ, (25)
with
T =
1 0 0 0 1
2 0
0 1
2 0
0 0 1 .
Hence, by using the equations (24) and (30) we obtain
Ψ = RΦ∗Φ = X
j∈Z2
RHejΦ∗Φ(2· −j) = X
j∈Z2
R Hej T Ψ(2· −j),
consequently we have
Pj = R Hej T = 1
4 X
i∈Z2
R (Hi⊗Hj−i) T, j∈Z2.
Then, by using the matrix maskHj described in Lemma 2.2, we determine explicitly the matrix maskPj. ¤
Now, we use the refinement equation (23) to define the subdivision scheme associated with the refinable function vector Ψ as follows.
For a given fuction splineSλ= X
j∈Z2
λT
j Ψ(· −j).
Putλ(0)=λand
Computeλ(in)=Pj∈Z2PiT−2j λ(jn−1), i∈Z2, n= 1,2,· · ·
(26)
Following [10], we say that the subdivision scheme converges forλ= (λ1, λ2, λ3)T ∈ ¡`∞(Z2)¢3 if there exists
a continuous functionSλ:R2−→Rsuch that
lim
n→∞kSλ ³ ·
2n ´
e − λ(n)k
∞ fore= (1,1,1)
T
,
The symbolSλ ¡ ·
2n
¢
stands for the scalar-valued sequence¡Sλ ¡j
2n
¢¢ j∈Z2.
Lemma 2.4. Let Θ = (θ1,· · · , θm)T be a refinable function vector of continuous compactly supported bivariate functions, satisfying the refinement equation
Θ = X
j∈Z2
If Θis stable and the associated matrix maskB:= (Bj)j∈Z2 satisfies
vX
j∈Z2
Bi−2j = v, i∈Z2, forv= (1,1,· · · ,1), (27)
then the associated subdivision scheme is convergent for allλ∈¡`∞(Z2¢m, and the limit function is given by
Sλ= X
j∈Z2
λT
j Θ(· −j).
Lemma 2.5. The C2 quartic spline function vectorΨis stable.
Proof. From [16], Ψ is stable if and only if the sequences
³ b
ψr(ω+ 2πj) ´
j∈Z2 r= 1,2,3 (28)
are linearly independent for everyω∈R2. Suppose that Ψ is not stable, then there existω
0∈R2,j0∈Z2 and
r0∈ {1,2,3} such that
S(ω0) = 3
X
r=1
X
j∈Z2
cr
jψbr(ω0+ 2πj) = 0 with cr0j0 6= 0.
By using the new definition of Ψ described in Theorem 1.2, we obtain
S(ω0) =
X
j∈Z2 ©
c1
jϕb1(ω0+ 2πj)ϕb1(ω0+ 2πj) + 2c2jϕb1(ω0+ 2πj)ϕb2(ω0+ 2πj)
+ c3
jϕb2(ω0+ 2πj)ϕb2(ω0+ 2πj)
ª
= X
j∈Z2 ©
c1
jϕb1(ω0+ 2πj) + c2jϕb2(ω0+ 2πj)
ª b
ϕ1(ω0+ 2πj)
+ ©c2
jϕb1(ω0+ 2πj) + c3jϕb2(ω0+ 2πj)
ª b
ϕ2(ω0+ 2πj).
If we put
d1
j =c1jϕb1(ω0+ 2πj) + c2jϕb2(ω0+ 2πj)
and
d2
j =c2jϕb1(ω0+ 2πj) + c3jϕb2(ω0+ 2πj),
then we have
S(ω0) =
X
j∈Z2 ©
d1
j ϕb1(ω0+ 2πj) +d2j ϕb2(ω0+ 2πj)
ª
= 0. (29)
Now we show that at leastd1
j06= 0 ord2j06= 0, for this consider the caser0= 1, i.e.,c1j0 6= 0.
If we suppose thatd1
j0 =c1j0ϕb1(ω0+ 2πj0) + c2j0ϕb2(ω0+ 2πj0) = 0, then
e
S(ω0) =
X
j∈Z2 n
e
dj1 ϕb1(ω0+ 2πj) +de2j ϕb2(ω0+ 2πj)
o = 0,
wherede1
j =de2j= 0 for allj∈Z2\{j0}and dej01 =c1j0 6= 0,de2j0 =c2j0.
Consequently there existω0∈R2 andj0∈Z2 such that Se(ω0) = 0 withde1j0 6= 0, which is absurd because the
linear piecewise function vector Φ is stable, i.e., the sequences
are linearly independent for everyω ∈R2. Therefored1
j0 6= 0. A similar technique can be applied in the cases
ofr0= 2 or 3. So that, there existj0such that at least d1j0 6= 0 ord2j0 6= 0.
Consequently there existω0∈R2,j0∈Z2andt0∈ {1,2} such that the equation (29) hold withdtj00 6= 0, which
is in contradiction with the fact that Φ is stable. ¤
Theorem 2.6. The subdivision scheme associated to the quartic function vector Ψ converges for all λ ∈ ¡
`∞(Z2)¢3, and the limit function isC2 quartic spline in the four-directional mesh given by
Sλ= X
j∈Z2
λ1
j ψ1(· −j) +
X
j∈Z2
λ2
jψ2(· −j) +
X
j∈Z2
λ3
jψ3(· −j).
Proof. Let Qi =
P
j∈Z2Pi−2j, for all i ∈Z2 there exist µ ∈Z2 and ν ∈ {(0,0),(1,0),(0,1),(1,1)} such that
i= 2µ+ν, so that we have Qi =Qν. Then, it suffices to prove that
vQν = v, forv= (1,1,1), andν∈ {(0,0),(1,0),(0,1),(1,1)}.
By using the expression of Pi−2j given in Theorem 2.3 we have
Q(0,0)=
1
4 18 14
1
2 12 0
1
4 38 34
, Q(1,0)=Q(0,1)=Q(1,1)=
1
4 18 0
1
2 12 12
1
4 38 12
.
and we can easily verify that the matrices Qν, forν ∈ {(0,0),(1,0),(0,1),(1,1)}, have v as a left eigenvector
associated with the eigenvalue 1, and the conditions (27) are satisfied. Since Ψ is stable, we obtain, from the
lemma, 2.4 the enounced result. ¤
We apply the subdivision scheme (26) for the initial sequences λ(0)j = (δj,0,0)T, λ(0)j = (0, δj,0)T and
λ(0)j = (0,0, δj)T to compute the B-splinesψ1,ψ2and ψ3 respectively. In figures 5, 6 and 7, we give the graph
of these B-splines approximated by the iterates sequences λ(8)j in the three cases.
Figure 5. Graph of the B-spline ψ1.
Remark 2.7. In order to construct the initial function splineSλ= X
j∈Z2
λT
j Ψ(·−j) in the subdivision scheme, we
Figure 6. Graph of the B-spline ψ2.
Figure 7. Graph of the B-spline ψ3.
small norms in the space S(Ψ). These operators, so-called near-best quasi-interpolants, are useful in several applications like approximation and estimation, numerical quadrature and numerical solutions of integral or partial differential equations.
Remark 2.8. The results described above can be extended to various cases in the spline spaceS43kk+4+2(τ) of class C3k+2 and degree 4k+ 4, k ∈IN. In this space there are three independent locally supported B-splines,
two are minimally supported,ψk
1 andψ2k, and one is quasi-minimally supportedψk3 defined by:
ψk i :=
½
ψi, fork= 0,
ψi∗Bk,k,k,k, fork= 1,2, ... (30)
for i = 1,2,3, where Bk,l,m,n := BXk,l,m,n, k, l, m, n∈ N, is the bivariate box-spline on the four-directional meshτ, associated with the directional set
Xk,l,m,n={ e1, . . . , e1
| {z }
k
, e2, . . . , e2
| {z }
l
, e3, . . . , e3
| {z }
m
, e4, . . . , e4
| {z }}
n
defined as the inverse Fourier transform of
b
Bk,l,m,n(w) =BbXk,l,m,n(w) := Z
R2
BXk,l,m,n(x)e
−i w·x dx
= µ
1−e−iw1
iw1
¶k µ
1−e−iw2
iw2
¶lµ
1−e−i(w1+w2)
i(w1+w2)
¶m µ
1−e−i(w1−w2)
i(w1−w2)
¶n
withw= (w1, w2).
If we put Ψk =¡ψk
1, ψ2k, ψ3k
¢T
, then Ψk = Ψ∗B
k,k,k,k. So that, by using Theorem 2.3, Lemma 2.1 and the
fact that the box-spline Bk,k,k,k is a refinable function, we can prove that the spline function vector Ψk is a
refinable function and we can determine its refinement matrix mask by using the equation (21). But in this case the support of the mask is larger.
Remark 2.9. If we define
Sk :=Sk(Ψ) :=©f(2k·) : f ∈ S(Ψ)ª
then
Sk⊂ Sk+1,
and since Ψ is compactly supported function vector, we deduce from [17] that [
k∈Z
Sk = L
2(R2) and
\
k∈Z
Sk = {0}.
Furthermore, since the refinable function vector Ψ is stable, the sequences of subspaces¡Sk¢
k form a
multires-olution ofL2(R2). The construction of the associated compactly supported multiwavelets and prewavelets will
be studied in a future work.
”Research supported in part by PROTARS III, D11/18”
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