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Near-best operators based on a C2 quartic spline on the uniform four-directional mesh
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Mathematics and Computers in Simulation 77 (2008) 151–160
Near-best operators based on a C2 quartic spline on the uniform four-directional mesh夽 El Bachir Ameur a , Domingo Barrera b , Mar´ıa J. Ib´an˜ ez b , Driss Sbibih c,∗ a
Universit´e Moulay Ismail, Facult´e des Sciences et Techniques, D´epartement d’Informatique, 52000 Errachdia, Morocco b Departamento de Matem´ atica Aplicada, Facultad de Ciencias, Universidad de Granada, Campus Universitario de Fuentenueva s/n, 18071 Granada, Spain c Universit´ e Mohammed I, Ecole Sup´erieure de Technologie, Laboratoire MATSI, Oujda, Morocco Available online 31 August 2007
Abstract We present some results about the construction of quasi-interpolant operators based on a special C2 quartic B-spline. We show that these operators, called near-best quasi-interpolants, have the best approximation order and small infinity norms. They are obtained by solving a minimization problem that admits always a solution. We give an error bound of these quasi-interpolants and we illustrate our results by a numerical example. © 2007 IMACS. Published by Elsevier B.V. All rights reserved. PACS: 41A05; 41A15; 65D05; 65D07 Keywords: B-splines; Box-splines; Subdivision scheme; Refinable function vector; Near-best quasi-interpolants
1. Introduction Let τ be the uniform triangulation of R2 whose set of vertices is Z2 ∪ (Z + (1/2))2 , and whose edges are parallel to the four directions e1 = (1, 0), e2 = (0, 1), e3 = (1, 1) and e4 = (−1, 1). Let Pd be the space of bivariate polynomials of total degree at most d, and let Srd (τ) be the space of bivariate piecewise polynomial functions of class Cr on the plane and whose restrictions to each triangular cell of τ are in Pd . Let T be a triangle of τ and λ = (λ1 , λ2 , λ3 ) be the barycentric coordinates of a point M of R2 relative to T. Each polynomial p in the space Pd (T ) of polynomials defined on T has a unique representation in the Bernstein–B´ezier form p(M) =
b(i) Bid (λ),
i ∈ d
夽 Research supported in part by PROTARS III, D11/18, Ministerio de Educacin y Ciencia (Research project MTM2005-01403) and Junta de Andaluca (research group FQM/191). ∗ Corresponding author. E-mail addresses: [email protected] (E.B. Ameur), [email protected] (D. Barrera), [email protected] (M.J. Ib´an˜ ez), [email protected] (D. Sbibih).
0378-4754/$32.00 © 2007 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.matcom.2007.08.005
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Fig. 1. B-nets and supports of ϕi for i = 1, 2.
where d = {i = (i1 , i2 , i3 ) ∈ Z3+ : |i| = i1 + i2 + i3 = d} and Bid (λ) =
d! i d! λ = λi1 λi2 λi3 . i! i1 !i2 !i3 ! 1 2 3
The family of the (1/2)(d + 1)(d + 2) polynomials Bid , i ∈ d , forms a basis for the space Pd (T ). The coefficients {b(i), i ∈ d } are called the B-net of p on the triangle T. When r = 0 and d = 1, the space S01 (τ) of linear bivariate splines, on the four-directional mesh τ, is generated by the two minimally supported linear B-splines ϕ1 and ϕ2 , whose B-nets and supports are given in Fig. 1. In this paper we are interested in the space S24 (τ) of quartic bivariate splines which is generated by three independent locally supported B-splines, denoted ψ1 , ψ2 and ψ3 . The B-splines ψ1 and ψ2 , constructed by Sablonni`ere [12], are minimally supported. The B-spline ψ3 , constructed by Chui and He [7], is quasi-minimally supported. Using only the B-nets of these B-splines, it is not easy to show that neither these B-splines satisfy a refinement equation nor determine their associated matrix masks. That is why we introduced in [1] a new definition of these B-splines which is convenient to prove that the function vector (ψ1 , ψ2 , ψ3 )T satisfies the refinement equation and to determine explicitly the associated refinement matrix mask. We briefly recall these results in Section 2. In Section 3, we construct spline quasi-interpolants with optimal approximation orders and small uniform norms in the space generated by a linear combination of ψ1 , ψ2 and ψ3 , which we call near-best quasi-interpolants. They are obtained by solving a minimization problem that admits always a solution. Finally, in Section 4, we illustrate with two examples that the norms of these quasi-interpolants are small in comparison with those of the ones based on the quartic box-splines available in the literature (see e.g. [11]). 2. New definition, refinement equation and subdivision scheme Let := (ψ1 , ψ2 , ψ3 )T be the function vector of the above three quartic B-splines in the space S24 (τ). We recall some results of [1], say a convolution-based definition of these B-splines and the refinement equation satisfied by . Theorem 1. The function vector can be expressed in terms of ϕ1 and ϕ2 . More precisely we have T = (ψ1 , ψ2 , ψ3 ) = (ϕ1 ∗ ϕ1 , 2ϕ1 ∗ ϕ2 , ϕ2 ∗ ϕ2 ). Theorem 2. The function vector satisfies the refinement equation = Pj (2 · −j), j ∈ Z2
(1)
(2)
E.B. Ameur et al. / Mathematics and Computers in Simulation 77 (2008) 151–160
where the refinement matrix mask P := (Pj )j ∈ Z2 , which is a sequence of 3 × 3-matrices, is given by ⎡ 1 ⎡1 ⎡ 1 ⎤ ⎤ ⎤ 0 0 0 0 0 0 ⎢ 16 ⎢8 ⎢ 16 ⎥ ⎥ ⎥ ⎢ ⎢ ⎢ ⎥ ⎥ ⎥ 1 ⎢ 1 3 ⎢1 3 1⎥ ⎢ ⎥ ⎥ P(0,0) = ⎢ P(1,0) = ⎢ P(2,0) = ⎢ 0 ⎥, 0 ⎥, 0 ⎥, ⎢ 2 8 ⎢4 8 4⎥ ⎢ ⎥ ⎥ 16 ⎣ 1 1 1⎦ ⎣ ⎣ 1 1⎦ 1 ⎦ 0 0 0 16 4 2 16 4 16 ⎡1 1 ⎤ ⎡ ⎤ ⎡ 1 1 1⎤ 1 1 0 ⎢4 8 ⎥ ⎢8 8 0⎥ ⎢ ⎥ ⎢ ⎢ 16 8 4 ⎥ ⎥ ⎢1 3 1⎥ ⎥ 1 1 ⎥, P(1,1) = ⎢ P(2,2) = ⎢ P(2,1) = ⎢ ⎥, ⎢ ⎥ ⎣ 0 0 0 ⎦, ⎢8 8 2⎥ ⎣ 0 16 4 ⎦ ⎣ ⎦ 1 0 0 0 0 0 0 0 0 8 ⎡ ⎡ ⎡ ⎤ ⎤ ⎤ 0 0 0 0 0 0 0 0 0 ⎢1 ⎢1 1 ⎢1 1 ⎥ ⎥ ⎥ ⎢ ⎢ ⎢ ⎥ ⎥ ⎥ 0 0 0 0 ⎢ ⎢ ⎢ ⎥ ⎥ ⎥, P(−1,−1) = ⎢ 8 P(0,−1) = ⎢ 4 16 P(1,−1) = ⎢ 8 16 ⎥, ⎥, ⎥ ⎣1 1 1⎦ ⎣1 1 1⎦ ⎣ 1 1⎦ 0 4 4 8 8 4 4 16 8 ⎡ ⎡ ⎡ ⎤ ⎤ ⎤ 0 0 0 0 0 0 0 0 0 ⎢ 0 ⎢ 0 0 0⎥ ⎢ 0 0 0⎥ 0 0 ⎥ ⎢ ⎢ ⎥, ⎥, ⎥, = = P(0,−2) = ⎢ P P (−1,−2) (−2,−2) ⎣ 1 ⎣1 1 ⎣ 1 ⎦ ⎦ 1 1 ⎦ 0 0 0 16 16 16 8 16 16 and PG0 j , j ∈ {(−2, −1), (−2, 0), (−1, 0), (−1, 1), (0, 1), (0, 2), (1, 2)}, Pj = 0, j ∈ Z2 \ H2 ,
0 1 and H2 is the subset of Z2 which intersects the hexagon of where G0 = 1 0 {(−2, −2), (0, −2), (2, 0), (2, 2), (0, 2), (−2, 0)}.
153
vertices
Let (Z2 ) and ∞ (Z2 ) be respectively the linear space of all sequences and the subspace of all bounded sequences defined on Z2 . Recall that is stable if its integer translates satisfy the stability condition, i.e., if there exist two constants 0 < c1 ≤ c2 < ∞ such that 3 λTj (· − j)∞ ≤ c2 λ∞ , λ ∈ (∞ (Z2 )) , (3) c1 λ∞ ≤ j ∈ Z2
where λ∞ = maxi λi ∞ denotes the sup-norm of the vector sequence λ. 3 3 The subdivision operator SP : ((Z2 )) → ((Z2 )) associated with the matrix mask P is given by P T λ , i ∈ Z2 . (SP λ)i := 2 i−2j j j∈Z
(4)
Now, using the refinement Eq. (2), we define the subdivision scheme associated with the refinable function vector as follows λTi (· − j) For a given spline function Sλ = j ∈ Z2
Put λ(0)
=λ
(n) Compute λi
=
j ∈ Z2
(5) (n−1) T Pi−2j λj ,
i ∈ Z2 ,
n = 1, 2, . . . .
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3
Following [9], we say that this scheme converges for λ = (λ1 , λ2 , λ3 ) ∈ (∞ (Z2 )) if there exists a continuous function Sλ : R2 → R such that lim Sλ (·/2n )e − λ(n) ∞ = 0
n→∞
for e = (1, 1, 1)T . 3
Theorem 3. is stable, the corresponding subdivision scheme given by (5) converges for all λ ∈ (∞ (Z2 )) , and the limit function is a C2 quartic spline given by Sλ = λ1j ψ1 (· − j) + λ2j ψ2 (· − j) + λ3j ψ3 (· − j). j ∈ Z2
j ∈ Z2
j ∈ Z2
3. Near-best quasi-interpolants Let S() be the spline space generated by the integer translates of ψ1 , ψ2 and ψ3 . From [6], we have S() = S24 (τ). Then, in order to build up the initial spline function Sλ = j ∈ Z2 λTj (· − j) in the subdivision scheme, we are interested in this section in the construction of spline quasi-interpolants defined on S() and having optimal approximation orders and small norms. Let us denote by Q a quasi-interpolant defined by Qf = cα f (j + α)M(· − j), (6) j ∈ Z2 α ∈ K
where M is a non-negative compactly supported spline function, K is a finite subset of Z2 and f is a real function defined on R2 . More specifically, we have f − Qf ∞ ≤ (1 + Q∞ ) dist (f, S(M)),
(7)
on the other hand, it is simple to see that (6) is equivalent to Qf = f (j) L(. − j), j ∈ Z2
where L denotes the fundamental function defined by L= cα M(· − α). α∈K
Moreover, when the integer translates of M form a partition of unity, i.e., |cα |. Q∞ ≤ ν(c) :=
j M(· − j)
= 1, we easily verify that (8)
α∈K
In our case, we first construct a spline function M ∈ S() so that the associated quasi-interpolants satisfy the optimal approximation order in S(). Then we determine the corresponding sequence c = (cα )α ∈ Z2 which minimizes the upper bound ν(c). To do this, we need the following lemmas. Lemma 4 (see e.g. [6]). The quasi-interpolation operator Q has an approximation order m in the space S() if there exists a spline function M in S() that satisfies the following Strang-Fix conditions of order m ˆ ˆ = 0, β ∈ Z2 \ {0} and |μ| < m, M(0) = 1, Dμ M(2πβ)
ˆ where M(ω) = M(t) e−iωt dt is the Fourier transform of M. Lemma 5 (see [8]). Put = (ϕ1 , ϕ2 )T and let N be the linear spline in the space S( ) = S01 (τ) defined by 1 N = ϕ2 + (ϕ1 + ϕ1 (· + e1 ) + ϕ1 (· + e2 ) + ϕ1 (· + e3 )). 4 Then N satisfies the Strang-Fix conditions of order 2.
E.B. Ameur et al. / Mathematics and Computers in Simulation 77 (2008) 151–160
155
Now, if we put M = N ∗ B1110 (· + e3 ),
(9) S24 (τ)
then it is clear that M is a spline function in the space and it is a linear combination of the integer translates of the B-splines ψ1 , ψ2 and ψ3 . Moreover, we have the following result. Theorem 6. M can be written in the form bjT (· − j), M= j ∈ H1
where H1 = {0, −e1 , −e2 , −e3 , −e1 − e3 , −e2 − e3 , −2e3 } and b0 = [1/8, 3/8, 1]T , b−e1 = b−e2 = [1/8, 1/8, 0]T , b−e3 = [1/4, 3/8, 0]T , b−e1 −e3 = b−e2 −e3 = b−2e3 = [1/8, 0, 0]T . Furthermore, it satisfies the Strang-Fix conditions of order 4. Proof. Using the expression of N, given in Lemma 5 and Eq. (9), we have 1 M = ϕ2 ∗ B1110 (· + e3 ) + (ϕ1 ∗ B1110 (· + e3 ) + ϕ1 (· + e1 ) ∗ B1110 (· + e3 ) + ϕ1 (· + e2 ) ∗ B1110 (· + e3 ) 4 + ϕ1 (· + e3 ) ∗ B1110 (· + e3 )). As 1 B1110 (· + e3 ) = ϕ2 + (ϕ1 + ϕ1 (· + e3 )), 2 we obtain 1 M = (ϕ1 ∗ ϕ1 + ϕ1 ∗ ϕ1 (· + e1 ) + ϕ1 ∗ ϕ1 (· + e2 ) + 2ϕ1 ∗ ϕ1 (· + e3 ) + ϕ1 ∗ ϕ1 (· + e1 + e3 ) 8 1 + ϕ1 ∗ ϕ1 (· + e2 + e3 ) + ϕ1 ∗ ϕ1 (· + 2e3 )) + (3ϕ1 ∗ ϕ2 + ϕ1 ∗ ϕ2 (· + e1 ) + ϕ1 ∗ ϕ2 (· + e2 ) 4 + 3ϕ1 ∗ ϕ2 (· + e3 )) + ϕ2 ∗ ϕ2 . Hence, since ϕ1 ∗ ϕ1 = ψ1 , 2ϕ1 ∗ ϕ2 = ψ2 and ϕ2 ∗ ϕ2 = ψ3 we get the expression of M. On the other hand, it is well known that the box-spline B1110 satisfies the Strang-Fix conditions of order 2, then from Lemma 5 we get ˆ ˆ B ˆ 1110 (0) = 1. M(0) = N(0) Moreover, for all β ∈ Z2 \ {0} and |μ| < m, it holds μ! (μ−ν) ˆ ˆ ˆ Dμ M(2πβ) = (μ − ν)! Dν N(2πβ)D B1110 (2πβ) = 0, ν! ν≤μ and the claim follows.
Remark 7. Using the expression of the C2 quartic B-spline M described above, we can easily verify that M(x1 , x2 ) = M(x2 , x1 )
and M(x1 , x2 ) = M(−x1 − x2 ). (8)
Fig. 2 shows the graph of M obtained by the computation of the iterate sequences λj with the initial sequence (0) λj
= bj , for j ∈ H1 and 0 for j ∈ Z2 \ H1 . Now, in order to construct quasi-interpolants based on the B-spline M, we determine its associated Marsden identity.
Lemma 8. For all μ ∈ N2 and |μ| < 4, we have xμ mμ (x) = gμ (j)M(x − j), = μ! 2 j∈Z
(10)
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E.B. Ameur et al. / Mathematics and Computers in Simulation 77 (2008) 151–160
Fig. 2. Graph of the B-spline M.
where the sequence of polynomials (gμ (j))|μ|<4 is given by g(0,0) (j) = 1,
g(1,0) (j) = j1 ,
g(0,1) (j) = j2 ,
g(1,1) (j) = j1 j2 −
5 , 48
1 2 1 1 1 1 1 j1 − , g(0,2) (j) = j22 − , g(3,0) (j) = j13 − j1 , 2 12 2 12 6 12 1 2 5 1 1 2 1 5 g(2,1) (j) = j1 j2 − j1 − j2 , g(1,2) (j) = j1 j2 − j1 − j2 . 2 48 12 2 12 48
g(2,0) (j) =
Let Ks = [−s, s]2
g(0,3) (j) =
1 3 1 j 2 − j2 , 6 12
Z2 and c a sequence such that
c(α1 ,α2 ) = c(α2 ,α1 )
and c(α1 ,α2 ) = c(−α1 ,−α2 ) .
Then c is fully determined by the list c˜ = [c(0,0) , c(1,0) , c(1,1) , c(1,−1) , c(2,0) , c(2,1) , c(2,2) , c(2,−1) , c(2,−2) , . . . , c(s,0) , c(s,1) , . . . , c(s,s) , c(s,−1) , . . . , c(s,−s) ]T = [cα1 , cα2 , . . . , cαn ]T , where n = (s + 1)2 , and the associated quasi-interpolant Qs given in (6) becomes (λs,j (f ))T (· − j), Qs =
(11)
j ∈ Z2
with λs,j (f ) =
cα f (j + α − i) bi .
α ∈ Ks i ∈ H1
It is easy to prove that Qs is exact on P3 if and only if c satisfies the three following conditions α ∈ Ks
cα = 1,
α ∈ Ks
α 1 α 2 cα = −
5 , 48
and
α ∈ Ks
1 α21 cα = − . 6
(12)
Our objective is to construct a quasi-interpolant Qs with small norm. To do this, we choose a priori a sequence c with a larger support than that of the associated B-spline M and afterwards minimize the l1 -norm ν(c) of c under the linear constraints (12) for reproducing all monomials in P3 .
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157
It is clear that the exactness of Qs on P3 implies that there exist a 3 × n matrix A of rank 3 < n and a vector b in R3 such that A c˜ = b. Set V = {x ∈ R3 : A x = b}, then the construction of Qs with small norm is equivalent to solving the following minimization problem Solve min{x1 , x ∈ V }.
(13)
Definition 9. If c is a solution of Problem (13), then the associated quasi-interpolant Qs defined by (11) is called a near-best quasi-interpolant. Proposition 10. For s ≥ 1, Problem (13) has at least one solution. Proof. Since the rank of A is 3, the system Ax = b can be solved and each solution xαj , j = 1, . . . , n, is an affine function of n − 3 parameters of x. However, by substituting the affine functions xαj in the expression of x1 , we obtain ˜ 1 . Consequently, solving Problem (13) is equivalent ˜ − b ˜ and a vector b˜ such that x1 = Ax a n × (n − 3) matrix A ˜ and the existence of at least one solution is guaranteed. ˜ of b, to determine the best l1 -approximation Ax Proposition
11.
13 ∗ c(s,s) = − 192s 2,
∗ c(0,0) =1+
1 , 6s2 2 T (s+1) ] ∈R
Let
and
1 ∗ c(s,−s) = − 64s 2.
Then
c˜ ∗ =
∗ ∗ , 0, . . . , 0, c∗ , 0, . . . , 0, c(s,s) is the solution of Problem (13). If Q∗s is its associated quasi[c(0,0) (s,−s) interpolant, then Q∗s ∞ ≤ 1 + 3s12 . Moreover the sequence (Q∗s )s≥1 converges in the infinity norm to the Schoenberg’s operator.
Proof. For all s ≥ 1 the expression of c1 is given by c1 = |c(0,0) | + 4
s
|c(i,0) | + 4
i=1
s i−1
(|c(i,j) | + |c(i,−j) |) + 2
i=2 j=1
s
(|c(i,i) | + |c(i,−i) |),
i=1
and the associated linear constraints in Problem (13) are c(0,0) + 4
s
c(i,0) + 4
i=1
4
s i−1
s i−1 i=2 j=1
ij(c(i,j) − c(i,−j) ) + 2
i=2 j=1
4
s i=1
i c(i,0) + 4 2
(c(i,j) + c(i,−j) ) + 2 s
i2 (c(i,i) − c(i,−i) ) = −
i=2 j=1
i (c(i,j) + c(i,−j) ) + 2 2
(c(i,i) + c(i,−i) ) = 1,
i=1
i=1 s i−1
s
s i=1
5 , 48
1 i2 (c(i,i) + c(i,−i) ) = − . 6
(14)
If we put c1 = w(c(0,0) , c(1,0) , c(1,1) , c(1,−1) , . . . , c(s,0) , c(s,1) , . . . , c(s,s) , c(s,−1) , . . . , c(s,−s) ), then, by using Eq. (14) we can express c(0,0) , c(s,s) and c(s,−s) in terms of the other coefficients on the square sequence 2 c. Therefore, minimizing c1 under the linear constraints given in (14) becomes equivalent to minimizing in R(s+1) the polyhedral convex function w of variables c(1,0) , c(1,1) , c(1,−1) , . . . , c(s,0) , c(s,1) , . . . , c(s,s−1) , c(s,−1) , . . . , c(s,−s+1) .
(15)
¯ (i,j) ) the restriction of w obtained by replacing its variables by zero Let c(i,j) be any variable in (15). Denote by w(c except c(i,j) . We will prove that this univariate function admits a minimum at 0. Indeed, assume, for example, that c(i,j) = c(1,0) , then by annulling the other variables in (14), we get the expressions of c(0,0) , c(s,s) and c(s,−s) in terms of c(1,0) . More precisely, we obtain 1 1 1 ∗ ∗ ∗ + 4 2 − 1 c(1,0) , c(s,s) = c(s,s) − 2 c(1,0) , c(s,−s) = c(s,−s) − 2 c(1,0) . c(0,0) = c(0,0) s s s
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¯ (i,j) ) takes the following expression Thus, w(c ¯ (1,0) ) = |c(0,0) | + 4|c(1,0) | + 2|c(s,s) | + 2|c(s,−s) | w(c ∗ ∗ ∗ 1 1 1 = c(0,0) + 4 2 − 1 c(1,0) + 4|c(1,0) | + 2 c(s,s) − 2 c(1,0) + 2 c(s,−s) − 2 c(1,0) . s s s ¯ (1,0) ) attains a unique minimum at c(1,0) = 0 and we have w(c ¯ (1,0) ) ≥ w(0) ¯ It is easy to verify that w(c = 1 + (1/3s2 ). A similar technique can be applied for each of the other variables in (15). On the other hand, using (11) and the above expression of c˜ ∗ , we can easily show that for all f ∈ C(R2 ) such that f ∞ ≤ 1, we have |Q∗ f | ≤ 1 +
1 3s2
and |Q∗ f − Sf | ≤
1 , 3s2
where Sf = i ∈ Z2 f (i)M(· − i) is the classical Schoenberg’s operator. Hence the claim follows. 4. Examples of near-best quasi-interpolants based on M ∗ ∗ Case 1 (s = 1). According to Proposition 11, a sequence c ∈ K1 can be determined only in terms of c(0,0) , c(1,1) and ∗ ∗ c(1,−1) , and there are three exactness conditions. The associated near-best quasi-interpolant Q1 is given by
Q∗1 f =
cα f (j + α)M(. − j),
j ∈ Z2 α ∈ K1
where
∗ ∗ ∗ ∗ c˜ ∗ = [c(0,0) , c(1,0) , c(1,1) , c(1,−1) ]T =
13 1 7 , 0, − ,− 6 192 64
T .
Thus, Q∗1 ∞ ≤
4 = 1.3333 . . . . 3
Case 2 (s = 2). Once again, according to Proposition 11, a sequence c ∈ K2 can be determined only in terms of ∗ ∗ ∗ c(0,0) , c(2,2) and c(2,−2) . Hence the near-best quasi-interpolant Q∗2 is defined by Q∗2 f =
cα f (j + α)M(. − j),
j ∈ Z2 α ∈ K2
where
c˜ ∗ =
25 13 1 , 0, 0, 0, 0, 0, − , 0, − 24 768 256
T .
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159
Fig. 3. Graphs of f in the left f, Q∗1 f in the medium and Q∗2 f in the right.
Thus, Q∗2 ∞ ≤
13 = 1.08333 . . . . 12
In order to test our subdivision scheme we use the following test function f (x, y) = 3(1 − x)2 e(−x
2 −(y+1)2 )
− 10
1 2 2 2 2 − x3 − y5 e(−x −y ) − e(−(x+1) −y ) . 5 3
x
We give in Fig. 3 the graphs of the function f and of Q∗s f for s = 1, 2, respectively. Q∗1 f and Q∗2 f are (6) (0) obtained by the computation of the iterate sequences λs,j (f ) with the initial sequences λs,j (f ) given in Eq. (11). Remark 12. The near-best quasi-interpolants based on the two quartic C2 classical box-splines, generated by the sets X1 = {2e1 , 2e2 , e3 , e4 } and X2 = {e1 , e2 , 2e3 , 2e4 }, have been studied in [11]. If we denote their corresponding quasi-interpolants by Q1s and Q2s , s ≥ 1, then using the Bernstein–B´ezier forms of these box-splines, it is easy to compute their infinity norms for the first values of s. More specifically, for s = 1, 2 we have 763 1718 1.44781 and Q21 ∞ = 1.53256 527 1121 475 2009 1.23698 and Q22 ∞ = 1.30794 = 384 1536
Q11 ∞ = Q12 ∞
In a similar way, one can compute the exact values of the infinity norms of Q∗s . For instance, if s = 1, 2 we get Q∗1 ∞ =
6911 1.12484 6144
and Q∗2 ∞ =
4975 1.07964 4608
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E.B. Ameur et al. / Mathematics and Computers in Simulation 77 (2008) 151–160
Therefore, we remark that the norms of Q∗s are small in comparison with those of Q1s and Q2s . Moreover, according to the above examples, the bounds of Q∗s ∞ , s = 1, 2, are very close to the exact values of Q∗s ∞ and better than those of Q1s ∞ and Q2s ∞ , s = 1, 2. Hence, the quasi-interpolants developed in this paper seem interesting. References [1] [6] [7] [8] [9]
B. Ameur, D. Barrera, D. Sbibih, Subdivision scheme of quartic bivariate splines on a four-directional mesh, submitted for publication. C.K. Chui, Multivariate splines, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1988. C.K. Chui, T.X. He, On minimal and quasi-minimal supported bivariate splines, J. Approximaion Theory 52 (2) (1988) 217–238. C. Conti, K. Jetter, A New subdivision method for bvariate splines on the four-directional mesh, J. Comput. Appl. Math. 119 (1/2) (2000) 81–96. C. Conti, K. Jetter, A. Coehn, C. Rabut, L.L. Schumaker, A Note on Convolving Refinable Function Vectors, Curves and Surfaces Fitting, Nashboro Press, 2000, pp. 135–142. [11] M.J. Ib´an˜ ez, Quasi-interpolantes discretos de norma casi m´ınima: teor´ıa y aplicaciones, Tesis Doctoral, Universidad de Granada, Spain, 2003. [12] P. Sablonni`ere, Bases de Bernstein et approximants splines, Th`ese d’Etat, Universit´e des Sciences et Techniques de Lille, France, 1982.
Mathematics and Computers in Simulation 77 (2008) 151–160
Near-best operators based on a C2 quartic spline on the uniform four-directional mesh夽 El Bachir Ameur a , Domingo Barrera b , Mar´ıa J. Ib´an˜ ez b , Driss Sbibih c,∗ a
Universit´e Moulay Ismail, Facult´e des Sciences et Techniques, D´epartement d’Informatique, 52000 Errachdia, Morocco b Departamento de Matem´ atica Aplicada, Facultad de Ciencias, Universidad de Granada, Campus Universitario de Fuentenueva s/n, 18071 Granada, Spain c Universit´ e Mohammed I, Ecole Sup´erieure de Technologie, Laboratoire MATSI, Oujda, Morocco Available online 31 August 2007
Abstract We present some results about the construction of quasi-interpolant operators based on a special C2 quartic B-spline. We show that these operators, called near-best quasi-interpolants, have the best approximation order and small infinity norms. They are obtained by solving a minimization problem that admits always a solution. We give an error bound of these quasi-interpolants and we illustrate our results by a numerical example. © 2007 IMACS. Published by Elsevier B.V. All rights reserved. PACS: 41A05; 41A15; 65D05; 65D07 Keywords: B-splines; Box-splines; Subdivision scheme; Refinable function vector; Near-best quasi-interpolants
1. Introduction Let τ be the uniform triangulation of R2 whose set of vertices is Z2 ∪ (Z + (1/2))2 , and whose edges are parallel to the four directions e1 = (1, 0), e2 = (0, 1), e3 = (1, 1) and e4 = (−1, 1). Let Pd be the space of bivariate polynomials of total degree at most d, and let Srd (τ) be the space of bivariate piecewise polynomial functions of class Cr on the plane and whose restrictions to each triangular cell of τ are in Pd . Let T be a triangle of τ and λ = (λ1 , λ2 , λ3 ) be the barycentric coordinates of a point M of R2 relative to T. Each polynomial p in the space Pd (T ) of polynomials defined on T has a unique representation in the Bernstein–B´ezier form p(M) =
b(i) Bid (λ),
i ∈ d
夽 Research supported in part by PROTARS III, D11/18, Ministerio de Educacin y Ciencia (Research project MTM2005-01403) and Junta de Andaluca (research group FQM/191). ∗ Corresponding author. E-mail addresses: [email protected] (E.B. Ameur), [email protected] (D. Barrera), [email protected] (M.J. Ib´an˜ ez), [email protected] (D. Sbibih).
0378-4754/$32.00 © 2007 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.matcom.2007.08.005
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Fig. 1. B-nets and supports of ϕi for i = 1, 2.
where d = {i = (i1 , i2 , i3 ) ∈ Z3+ : |i| = i1 + i2 + i3 = d} and Bid (λ) =
d! i d! λ = λi1 λi2 λi3 . i! i1 !i2 !i3 ! 1 2 3
The family of the (1/2)(d + 1)(d + 2) polynomials Bid , i ∈ d , forms a basis for the space Pd (T ). The coefficients {b(i), i ∈ d } are called the B-net of p on the triangle T. When r = 0 and d = 1, the space S01 (τ) of linear bivariate splines, on the four-directional mesh τ, is generated by the two minimally supported linear B-splines ϕ1 and ϕ2 , whose B-nets and supports are given in Fig. 1. In this paper we are interested in the space S24 (τ) of quartic bivariate splines which is generated by three independent locally supported B-splines, denoted ψ1 , ψ2 and ψ3 . The B-splines ψ1 and ψ2 , constructed by Sablonni`ere [12], are minimally supported. The B-spline ψ3 , constructed by Chui and He [7], is quasi-minimally supported. Using only the B-nets of these B-splines, it is not easy to show that neither these B-splines satisfy a refinement equation nor determine their associated matrix masks. That is why we introduced in [1] a new definition of these B-splines which is convenient to prove that the function vector (ψ1 , ψ2 , ψ3 )T satisfies the refinement equation and to determine explicitly the associated refinement matrix mask. We briefly recall these results in Section 2. In Section 3, we construct spline quasi-interpolants with optimal approximation orders and small uniform norms in the space generated by a linear combination of ψ1 , ψ2 and ψ3 , which we call near-best quasi-interpolants. They are obtained by solving a minimization problem that admits always a solution. Finally, in Section 4, we illustrate with two examples that the norms of these quasi-interpolants are small in comparison with those of the ones based on the quartic box-splines available in the literature (see e.g. [11]). 2. New definition, refinement equation and subdivision scheme Let := (ψ1 , ψ2 , ψ3 )T be the function vector of the above three quartic B-splines in the space S24 (τ). We recall some results of [1], say a convolution-based definition of these B-splines and the refinement equation satisfied by . Theorem 1. The function vector can be expressed in terms of ϕ1 and ϕ2 . More precisely we have T = (ψ1 , ψ2 , ψ3 ) = (ϕ1 ∗ ϕ1 , 2ϕ1 ∗ ϕ2 , ϕ2 ∗ ϕ2 ). Theorem 2. The function vector satisfies the refinement equation = Pj (2 · −j), j ∈ Z2
(1)
(2)
E.B. Ameur et al. / Mathematics and Computers in Simulation 77 (2008) 151–160
where the refinement matrix mask P := (Pj )j ∈ Z2 , which is a sequence of 3 × 3-matrices, is given by ⎡ 1 ⎡1 ⎡ 1 ⎤ ⎤ ⎤ 0 0 0 0 0 0 ⎢ 16 ⎢8 ⎢ 16 ⎥ ⎥ ⎥ ⎢ ⎢ ⎢ ⎥ ⎥ ⎥ 1 ⎢ 1 3 ⎢1 3 1⎥ ⎢ ⎥ ⎥ P(0,0) = ⎢ P(1,0) = ⎢ P(2,0) = ⎢ 0 ⎥, 0 ⎥, 0 ⎥, ⎢ 2 8 ⎢4 8 4⎥ ⎢ ⎥ ⎥ 16 ⎣ 1 1 1⎦ ⎣ ⎣ 1 1⎦ 1 ⎦ 0 0 0 16 4 2 16 4 16 ⎡1 1 ⎤ ⎡ ⎤ ⎡ 1 1 1⎤ 1 1 0 ⎢4 8 ⎥ ⎢8 8 0⎥ ⎢ ⎥ ⎢ ⎢ 16 8 4 ⎥ ⎥ ⎢1 3 1⎥ ⎥ 1 1 ⎥, P(1,1) = ⎢ P(2,2) = ⎢ P(2,1) = ⎢ ⎥, ⎢ ⎥ ⎣ 0 0 0 ⎦, ⎢8 8 2⎥ ⎣ 0 16 4 ⎦ ⎣ ⎦ 1 0 0 0 0 0 0 0 0 8 ⎡ ⎡ ⎡ ⎤ ⎤ ⎤ 0 0 0 0 0 0 0 0 0 ⎢1 ⎢1 1 ⎢1 1 ⎥ ⎥ ⎥ ⎢ ⎢ ⎢ ⎥ ⎥ ⎥ 0 0 0 0 ⎢ ⎢ ⎢ ⎥ ⎥ ⎥, P(−1,−1) = ⎢ 8 P(0,−1) = ⎢ 4 16 P(1,−1) = ⎢ 8 16 ⎥, ⎥, ⎥ ⎣1 1 1⎦ ⎣1 1 1⎦ ⎣ 1 1⎦ 0 4 4 8 8 4 4 16 8 ⎡ ⎡ ⎡ ⎤ ⎤ ⎤ 0 0 0 0 0 0 0 0 0 ⎢ 0 ⎢ 0 0 0⎥ ⎢ 0 0 0⎥ 0 0 ⎥ ⎢ ⎢ ⎥, ⎥, ⎥, = = P(0,−2) = ⎢ P P (−1,−2) (−2,−2) ⎣ 1 ⎣1 1 ⎣ 1 ⎦ ⎦ 1 1 ⎦ 0 0 0 16 16 16 8 16 16 and PG0 j , j ∈ {(−2, −1), (−2, 0), (−1, 0), (−1, 1), (0, 1), (0, 2), (1, 2)}, Pj = 0, j ∈ Z2 \ H2 ,
0 1 and H2 is the subset of Z2 which intersects the hexagon of where G0 = 1 0 {(−2, −2), (0, −2), (2, 0), (2, 2), (0, 2), (−2, 0)}.
153
vertices
Let (Z2 ) and ∞ (Z2 ) be respectively the linear space of all sequences and the subspace of all bounded sequences defined on Z2 . Recall that is stable if its integer translates satisfy the stability condition, i.e., if there exist two constants 0 < c1 ≤ c2 < ∞ such that 3 λTj (· − j)∞ ≤ c2 λ∞ , λ ∈ (∞ (Z2 )) , (3) c1 λ∞ ≤ j ∈ Z2
where λ∞ = maxi λi ∞ denotes the sup-norm of the vector sequence λ. 3 3 The subdivision operator SP : ((Z2 )) → ((Z2 )) associated with the matrix mask P is given by P T λ , i ∈ Z2 . (SP λ)i := 2 i−2j j j∈Z
(4)
Now, using the refinement Eq. (2), we define the subdivision scheme associated with the refinable function vector as follows λTi (· − j) For a given spline function Sλ = j ∈ Z2
Put λ(0)
=λ
(n) Compute λi
=
j ∈ Z2
(5) (n−1) T Pi−2j λj ,
i ∈ Z2 ,
n = 1, 2, . . . .
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3
Following [9], we say that this scheme converges for λ = (λ1 , λ2 , λ3 ) ∈ (∞ (Z2 )) if there exists a continuous function Sλ : R2 → R such that lim Sλ (·/2n )e − λ(n) ∞ = 0
n→∞
for e = (1, 1, 1)T . 3
Theorem 3. is stable, the corresponding subdivision scheme given by (5) converges for all λ ∈ (∞ (Z2 )) , and the limit function is a C2 quartic spline given by Sλ = λ1j ψ1 (· − j) + λ2j ψ2 (· − j) + λ3j ψ3 (· − j). j ∈ Z2
j ∈ Z2
j ∈ Z2
3. Near-best quasi-interpolants Let S() be the spline space generated by the integer translates of ψ1 , ψ2 and ψ3 . From [6], we have S() = S24 (τ). Then, in order to build up the initial spline function Sλ = j ∈ Z2 λTj (· − j) in the subdivision scheme, we are interested in this section in the construction of spline quasi-interpolants defined on S() and having optimal approximation orders and small norms. Let us denote by Q a quasi-interpolant defined by Qf = cα f (j + α)M(· − j), (6) j ∈ Z2 α ∈ K
where M is a non-negative compactly supported spline function, K is a finite subset of Z2 and f is a real function defined on R2 . More specifically, we have f − Qf ∞ ≤ (1 + Q∞ ) dist (f, S(M)),
(7)
on the other hand, it is simple to see that (6) is equivalent to Qf = f (j) L(. − j), j ∈ Z2
where L denotes the fundamental function defined by L= cα M(· − α). α∈K
Moreover, when the integer translates of M form a partition of unity, i.e., |cα |. Q∞ ≤ ν(c) :=
j M(· − j)
= 1, we easily verify that (8)
α∈K
In our case, we first construct a spline function M ∈ S() so that the associated quasi-interpolants satisfy the optimal approximation order in S(). Then we determine the corresponding sequence c = (cα )α ∈ Z2 which minimizes the upper bound ν(c). To do this, we need the following lemmas. Lemma 4 (see e.g. [6]). The quasi-interpolation operator Q has an approximation order m in the space S() if there exists a spline function M in S() that satisfies the following Strang-Fix conditions of order m ˆ ˆ = 0, β ∈ Z2 \ {0} and |μ| < m, M(0) = 1, Dμ M(2πβ)
ˆ where M(ω) = M(t) e−iωt dt is the Fourier transform of M. Lemma 5 (see [8]). Put = (ϕ1 , ϕ2 )T and let N be the linear spline in the space S( ) = S01 (τ) defined by 1 N = ϕ2 + (ϕ1 + ϕ1 (· + e1 ) + ϕ1 (· + e2 ) + ϕ1 (· + e3 )). 4 Then N satisfies the Strang-Fix conditions of order 2.
E.B. Ameur et al. / Mathematics and Computers in Simulation 77 (2008) 151–160
155
Now, if we put M = N ∗ B1110 (· + e3 ),
(9) S24 (τ)
then it is clear that M is a spline function in the space and it is a linear combination of the integer translates of the B-splines ψ1 , ψ2 and ψ3 . Moreover, we have the following result. Theorem 6. M can be written in the form bjT (· − j), M= j ∈ H1
where H1 = {0, −e1 , −e2 , −e3 , −e1 − e3 , −e2 − e3 , −2e3 } and b0 = [1/8, 3/8, 1]T , b−e1 = b−e2 = [1/8, 1/8, 0]T , b−e3 = [1/4, 3/8, 0]T , b−e1 −e3 = b−e2 −e3 = b−2e3 = [1/8, 0, 0]T . Furthermore, it satisfies the Strang-Fix conditions of order 4. Proof. Using the expression of N, given in Lemma 5 and Eq. (9), we have 1 M = ϕ2 ∗ B1110 (· + e3 ) + (ϕ1 ∗ B1110 (· + e3 ) + ϕ1 (· + e1 ) ∗ B1110 (· + e3 ) + ϕ1 (· + e2 ) ∗ B1110 (· + e3 ) 4 + ϕ1 (· + e3 ) ∗ B1110 (· + e3 )). As 1 B1110 (· + e3 ) = ϕ2 + (ϕ1 + ϕ1 (· + e3 )), 2 we obtain 1 M = (ϕ1 ∗ ϕ1 + ϕ1 ∗ ϕ1 (· + e1 ) + ϕ1 ∗ ϕ1 (· + e2 ) + 2ϕ1 ∗ ϕ1 (· + e3 ) + ϕ1 ∗ ϕ1 (· + e1 + e3 ) 8 1 + ϕ1 ∗ ϕ1 (· + e2 + e3 ) + ϕ1 ∗ ϕ1 (· + 2e3 )) + (3ϕ1 ∗ ϕ2 + ϕ1 ∗ ϕ2 (· + e1 ) + ϕ1 ∗ ϕ2 (· + e2 ) 4 + 3ϕ1 ∗ ϕ2 (· + e3 )) + ϕ2 ∗ ϕ2 . Hence, since ϕ1 ∗ ϕ1 = ψ1 , 2ϕ1 ∗ ϕ2 = ψ2 and ϕ2 ∗ ϕ2 = ψ3 we get the expression of M. On the other hand, it is well known that the box-spline B1110 satisfies the Strang-Fix conditions of order 2, then from Lemma 5 we get ˆ ˆ B ˆ 1110 (0) = 1. M(0) = N(0) Moreover, for all β ∈ Z2 \ {0} and |μ| < m, it holds μ! (μ−ν) ˆ ˆ ˆ Dμ M(2πβ) = (μ − ν)! Dν N(2πβ)D B1110 (2πβ) = 0, ν! ν≤μ and the claim follows.
Remark 7. Using the expression of the C2 quartic B-spline M described above, we can easily verify that M(x1 , x2 ) = M(x2 , x1 )
and M(x1 , x2 ) = M(−x1 − x2 ). (8)
Fig. 2 shows the graph of M obtained by the computation of the iterate sequences λj with the initial sequence (0) λj
= bj , for j ∈ H1 and 0 for j ∈ Z2 \ H1 . Now, in order to construct quasi-interpolants based on the B-spline M, we determine its associated Marsden identity.
Lemma 8. For all μ ∈ N2 and |μ| < 4, we have xμ mμ (x) = gμ (j)M(x − j), = μ! 2 j∈Z
(10)
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Fig. 2. Graph of the B-spline M.
where the sequence of polynomials (gμ (j))|μ|<4 is given by g(0,0) (j) = 1,
g(1,0) (j) = j1 ,
g(0,1) (j) = j2 ,
g(1,1) (j) = j1 j2 −
5 , 48
1 2 1 1 1 1 1 j1 − , g(0,2) (j) = j22 − , g(3,0) (j) = j13 − j1 , 2 12 2 12 6 12 1 2 5 1 1 2 1 5 g(2,1) (j) = j1 j2 − j1 − j2 , g(1,2) (j) = j1 j2 − j1 − j2 . 2 48 12 2 12 48
g(2,0) (j) =
Let Ks = [−s, s]2
g(0,3) (j) =
1 3 1 j 2 − j2 , 6 12
Z2 and c a sequence such that
c(α1 ,α2 ) = c(α2 ,α1 )
and c(α1 ,α2 ) = c(−α1 ,−α2 ) .
Then c is fully determined by the list c˜ = [c(0,0) , c(1,0) , c(1,1) , c(1,−1) , c(2,0) , c(2,1) , c(2,2) , c(2,−1) , c(2,−2) , . . . , c(s,0) , c(s,1) , . . . , c(s,s) , c(s,−1) , . . . , c(s,−s) ]T = [cα1 , cα2 , . . . , cαn ]T , where n = (s + 1)2 , and the associated quasi-interpolant Qs given in (6) becomes (λs,j (f ))T (· − j), Qs =
(11)
j ∈ Z2
with λs,j (f ) =
cα f (j + α − i) bi .
α ∈ Ks i ∈ H1
It is easy to prove that Qs is exact on P3 if and only if c satisfies the three following conditions α ∈ Ks
cα = 1,
α ∈ Ks
α 1 α 2 cα = −
5 , 48
and
α ∈ Ks
1 α21 cα = − . 6
(12)
Our objective is to construct a quasi-interpolant Qs with small norm. To do this, we choose a priori a sequence c with a larger support than that of the associated B-spline M and afterwards minimize the l1 -norm ν(c) of c under the linear constraints (12) for reproducing all monomials in P3 .
E.B. Ameur et al. / Mathematics and Computers in Simulation 77 (2008) 151–160
157
It is clear that the exactness of Qs on P3 implies that there exist a 3 × n matrix A of rank 3 < n and a vector b in R3 such that A c˜ = b. Set V = {x ∈ R3 : A x = b}, then the construction of Qs with small norm is equivalent to solving the following minimization problem Solve min{x1 , x ∈ V }.
(13)
Definition 9. If c is a solution of Problem (13), then the associated quasi-interpolant Qs defined by (11) is called a near-best quasi-interpolant. Proposition 10. For s ≥ 1, Problem (13) has at least one solution. Proof. Since the rank of A is 3, the system Ax = b can be solved and each solution xαj , j = 1, . . . , n, is an affine function of n − 3 parameters of x. However, by substituting the affine functions xαj in the expression of x1 , we obtain ˜ 1 . Consequently, solving Problem (13) is equivalent ˜ − b ˜ and a vector b˜ such that x1 = Ax a n × (n − 3) matrix A ˜ and the existence of at least one solution is guaranteed. ˜ of b, to determine the best l1 -approximation Ax Proposition
11.
13 ∗ c(s,s) = − 192s 2,
∗ c(0,0) =1+
1 , 6s2 2 T (s+1) ] ∈R
Let
and
1 ∗ c(s,−s) = − 64s 2.
Then
c˜ ∗ =
∗ ∗ , 0, . . . , 0, c∗ , 0, . . . , 0, c(s,s) is the solution of Problem (13). If Q∗s is its associated quasi[c(0,0) (s,−s) interpolant, then Q∗s ∞ ≤ 1 + 3s12 . Moreover the sequence (Q∗s )s≥1 converges in the infinity norm to the Schoenberg’s operator.
Proof. For all s ≥ 1 the expression of c1 is given by c1 = |c(0,0) | + 4
s
|c(i,0) | + 4
i=1
s i−1
(|c(i,j) | + |c(i,−j) |) + 2
i=2 j=1
s
(|c(i,i) | + |c(i,−i) |),
i=1
and the associated linear constraints in Problem (13) are c(0,0) + 4
s
c(i,0) + 4
i=1
4
s i−1
s i−1 i=2 j=1
ij(c(i,j) − c(i,−j) ) + 2
i=2 j=1
4
s i=1
i c(i,0) + 4 2
(c(i,j) + c(i,−j) ) + 2 s
i2 (c(i,i) − c(i,−i) ) = −
i=2 j=1
i (c(i,j) + c(i,−j) ) + 2 2
(c(i,i) + c(i,−i) ) = 1,
i=1
i=1 s i−1
s
s i=1
5 , 48
1 i2 (c(i,i) + c(i,−i) ) = − . 6
(14)
If we put c1 = w(c(0,0) , c(1,0) , c(1,1) , c(1,−1) , . . . , c(s,0) , c(s,1) , . . . , c(s,s) , c(s,−1) , . . . , c(s,−s) ), then, by using Eq. (14) we can express c(0,0) , c(s,s) and c(s,−s) in terms of the other coefficients on the square sequence 2 c. Therefore, minimizing c1 under the linear constraints given in (14) becomes equivalent to minimizing in R(s+1) the polyhedral convex function w of variables c(1,0) , c(1,1) , c(1,−1) , . . . , c(s,0) , c(s,1) , . . . , c(s,s−1) , c(s,−1) , . . . , c(s,−s+1) .
(15)
¯ (i,j) ) the restriction of w obtained by replacing its variables by zero Let c(i,j) be any variable in (15). Denote by w(c except c(i,j) . We will prove that this univariate function admits a minimum at 0. Indeed, assume, for example, that c(i,j) = c(1,0) , then by annulling the other variables in (14), we get the expressions of c(0,0) , c(s,s) and c(s,−s) in terms of c(1,0) . More precisely, we obtain 1 1 1 ∗ ∗ ∗ + 4 2 − 1 c(1,0) , c(s,s) = c(s,s) − 2 c(1,0) , c(s,−s) = c(s,−s) − 2 c(1,0) . c(0,0) = c(0,0) s s s
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¯ (i,j) ) takes the following expression Thus, w(c ¯ (1,0) ) = |c(0,0) | + 4|c(1,0) | + 2|c(s,s) | + 2|c(s,−s) | w(c ∗ ∗ ∗ 1 1 1 = c(0,0) + 4 2 − 1 c(1,0) + 4|c(1,0) | + 2 c(s,s) − 2 c(1,0) + 2 c(s,−s) − 2 c(1,0) . s s s ¯ (1,0) ) attains a unique minimum at c(1,0) = 0 and we have w(c ¯ (1,0) ) ≥ w(0) ¯ It is easy to verify that w(c = 1 + (1/3s2 ). A similar technique can be applied for each of the other variables in (15). On the other hand, using (11) and the above expression of c˜ ∗ , we can easily show that for all f ∈ C(R2 ) such that f ∞ ≤ 1, we have |Q∗ f | ≤ 1 +
1 3s2
and |Q∗ f − Sf | ≤
1 , 3s2
where Sf = i ∈ Z2 f (i)M(· − i) is the classical Schoenberg’s operator. Hence the claim follows. 4. Examples of near-best quasi-interpolants based on M ∗ ∗ Case 1 (s = 1). According to Proposition 11, a sequence c ∈ K1 can be determined only in terms of c(0,0) , c(1,1) and ∗ ∗ c(1,−1) , and there are three exactness conditions. The associated near-best quasi-interpolant Q1 is given by
Q∗1 f =
cα f (j + α)M(. − j),
j ∈ Z2 α ∈ K1
where
∗ ∗ ∗ ∗ c˜ ∗ = [c(0,0) , c(1,0) , c(1,1) , c(1,−1) ]T =
13 1 7 , 0, − ,− 6 192 64
T .
Thus, Q∗1 ∞ ≤
4 = 1.3333 . . . . 3
Case 2 (s = 2). Once again, according to Proposition 11, a sequence c ∈ K2 can be determined only in terms of ∗ ∗ ∗ c(0,0) , c(2,2) and c(2,−2) . Hence the near-best quasi-interpolant Q∗2 is defined by Q∗2 f =
cα f (j + α)M(. − j),
j ∈ Z2 α ∈ K2
where
c˜ ∗ =
25 13 1 , 0, 0, 0, 0, 0, − , 0, − 24 768 256
T .
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159
Fig. 3. Graphs of f in the left f, Q∗1 f in the medium and Q∗2 f in the right.
Thus, Q∗2 ∞ ≤
13 = 1.08333 . . . . 12
In order to test our subdivision scheme we use the following test function f (x, y) = 3(1 − x)2 e(−x
2 −(y+1)2 )
− 10
1 2 2 2 2 − x3 − y5 e(−x −y ) − e(−(x+1) −y ) . 5 3
x
We give in Fig. 3 the graphs of the function f and of Q∗s f for s = 1, 2, respectively. Q∗1 f and Q∗2 f are (6) (0) obtained by the computation of the iterate sequences λs,j (f ) with the initial sequences λs,j (f ) given in Eq. (11). Remark 12. The near-best quasi-interpolants based on the two quartic C2 classical box-splines, generated by the sets X1 = {2e1 , 2e2 , e3 , e4 } and X2 = {e1 , e2 , 2e3 , 2e4 }, have been studied in [11]. If we denote their corresponding quasi-interpolants by Q1s and Q2s , s ≥ 1, then using the Bernstein–B´ezier forms of these box-splines, it is easy to compute their infinity norms for the first values of s. More specifically, for s = 1, 2 we have 763 1718 1.44781 and Q21 ∞ = 1.53256 527 1121 475 2009 1.23698 and Q22 ∞ = 1.30794 = 384 1536
Q11 ∞ = Q12 ∞
In a similar way, one can compute the exact values of the infinity norms of Q∗s . For instance, if s = 1, 2 we get Q∗1 ∞ =
6911 1.12484 6144
and Q∗2 ∞ =
4975 1.07964 4608
160
E.B. Ameur et al. / Mathematics and Computers in Simulation 77 (2008) 151–160
Therefore, we remark that the norms of Q∗s are small in comparison with those of Q1s and Q2s . Moreover, according to the above examples, the bounds of Q∗s ∞ , s = 1, 2, are very close to the exact values of Q∗s ∞ and better than those of Q1s ∞ and Q2s ∞ , s = 1, 2. Hence, the quasi-interpolants developed in this paper seem interesting. References [1] [6] [7] [8] [9]
B. Ameur, D. Barrera, D. Sbibih, Subdivision scheme of quartic bivariate splines on a four-directional mesh, submitted for publication. C.K. Chui, Multivariate splines, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1988. C.K. Chui, T.X. He, On minimal and quasi-minimal supported bivariate splines, J. Approximaion Theory 52 (2) (1988) 217–238. C. Conti, K. Jetter, A New subdivision method for bvariate splines on the four-directional mesh, J. Comput. Appl. Math. 119 (1/2) (2000) 81–96. C. Conti, K. Jetter, A. Coehn, C. Rabut, L.L. Schumaker, A Note on Convolving Refinable Function Vectors, Curves and Surfaces Fitting, Nashboro Press, 2000, pp. 135–142. [11] M.J. Ib´an˜ ez, Quasi-interpolantes discretos de norma casi m´ınima: teor´ıa y aplicaciones, Tesis Doctoral, Universidad de Granada, Spain, 2003. [12] P. Sablonni`ere, Bases de Bernstein et approximants splines, Th`ese d’Etat, Universit´e des Sciences et Techniques de Lille, France, 1982.