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Convergence of irregular Hermite subdivision schemes
Computer Aided Geometric Design 27 (2010) 372–381
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Computer Aided Geometric Design www.elsevier.com/locate/cagd
Convergence of irregular Hermite subdivision schemes ✩ Yao Zhao, Di-Rong Chen ∗ Department of Mathematics, LMIB, Beijing University of Aeronautics and Astronautics, Beijing 100083, PR China
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 13 December 2008 Received in revised form 13 December 2009 Accepted 12 February 2010 Available online 19 February 2010
We consider in this paper the convergence of Hermite subdivision schemes related to the irregular grids X. It is reduced to the convergence of a scalar subdivision scheme, so the results concerning perturbation of convergent scalar subdivision schemes are applied. For interpolatory subdivision schemes HX , with the irregular grid X which is equivalent to the regular grid X∗ in a suitable sense, we proved that the convergence of HX is implied by that of HX∗ . Our arguments work in the settings of L p -convergence and uniform convergence. © 2010 Elsevier B.V. All rights reserved.
Keywords: Hermite subdivision scheme Associated subdivision scheme Difference subdivision scheme Irregular grid Interpolatory subdivision scheme
1. Introduction A Hermite subdivision scheme recursively computes a function φ and its derivative φ from the initial state of the scheme, sequence of vectors in R2 . In this paper, we consider the convergence of Hermite subdivision schemes with respect to irregular grids. In order to introduce the Hermite subdivision schemes, we need two components. One is a sequence of matrices, which shows how the data on new level is computed; the other is the grid, which specifies where to put the data. Let F = {( f (i ))i ∈Z | f (i ) ∈ R2 } be the set of all sequences of vectors in R2 . Suppose that, for n = 1, 2, . . . , S n = ( S n (i , j ))i , j ∈Z is a bi-infinite matrix with its entry S n (i , j ) being 2 × 2 matrix. With an initial sequence f 0 = ( f 0 (k)) ∈ F , we can construct recursively f n = S n f n−1 , for n = 1, 2, . . . , by { S n }, which
S n f (i ) =
S n (i , j ) f ( j ),
i ∈ Z.
(1)
j ∈Z
Definition 1. A grid is a set X = {xn,k | n 0, k ∈ Z} of real numbers satisfying the following conditions. (1) At the coarsest level, we have limk→−∞ x0,k = −∞, limk→∞ x0,k = ∞. (2) At any level n 0, the sequence xn,k is strictly increasing. (3) The set of points of a given level is contained in the set of points of the next level n + 1, in the sense that xn+1,2k = xn,k , for any n 0, k ∈ Z. (4) limn→∞ supk dn,k = 0, where dn,k = xn,k+1 − xn,k . ✩ Research supported in part by NSF of China under grants 10871015 and 10972018, and National Basic Research Program of China under grants 9732010CB731900 and 973-2006CB303102. Corresponding author. Tel.: +86 010 82317934; fax: +86 010 82317933. E-mail addresses: [email protected] (Y. Zhao), [email protected] (D.-R. Chen).
*
0167-8396/$ – see front matter doi:10.1016/j.cagd.2010.02.004
© 2010
Elsevier B.V. All rights reserved.
Y. Zhao, D.-R. Chen / Computer Aided Geometric Design 27 (2010) 372–381
373
A grid X is quasi-regular if there exist positive numbers a, b such that a2−n dn,k b2−n for any n 0, k ∈ Z. In particular, the grid X∗ = {k/2n | n 0, k ∈ Z} is said to be regular. A Hermite subdivision scheme H is consisted of { S n } and X , which generates a sequence { F n } of vector valued functions (0) (1) F n = ( F n , F n ) T : R −→ R2 , satisfying ( 0)
(1 )
F n (xn, j ) = f n ( j ), F n , F n
are affine on [xn, j , xn, j +1 ],
n 0, j ∈ Z.
(2)
We say that a Hermite subdivision scheme is interpolatory if in each subdivision step the values at the even gridpoints are kept, i.e.,
f n+1 (2k) = f n (k),
n 0, k ∈ Z.
(3)
Definition 2. We define that the Hermite subdivision scheme H = {{ S n }, X} converges if, for every sequence ( f 0 (i ))i ∈Z , the sequence { F n } of vector valued functions, given as above, converges uniformly in any finite interval to a vector valued function F = ( F (0) , F (1) ) : R −→ R2 , with property
dF (0) (x) dx
= F (1) (x), i.e. F (1) (x) is just the derivative of F (0) (x).
In the scalar case, i.e., S n (i , j ) are real numbers instead of matrices, the subdivision scheme generates a sequence of polygonal lines F n : R −→ R, n 0. Correspondingly, in this case, the convergence of subdivision schemes is simplified to the uniform convergence of the sequence { F n } in any finite interval. Subdivision schemes are tools for constructing curves and surfaces in computer aided geometric design. There are extensive study on the subdivision schemes related to the regular grid. We have only mentioned Cavaretta et al. (1991), Dyn and Levin (1995), Dyn et al. (1987), Dyn et al. (1991) and Jia (1995). However, the limitation to regular grid is too restrictive. Subdivision schemes related to irregular grids are necessary for construction of spline curves with non-regularly spaced knots. Another application that calls for the irregular setting is wavelets and multiresolution analysis for irregular samples. In Daubechies et al. (1999), Daubechies et al. (2001) and Maxim and Mazure (2004), the authors provided a sufficient condition for the convergence of the scalar subdivision schemes with irregular grids. Moreover, by comparing subdivision schemes, the convergence of a given subdivision scheme may be derived from a convergent one (Maxim and Mazure, 2004). In particular, the authors proved that if the Lagrange interpolatory subdivision schemes with respect to regular grid converges meanwhile an irregular grid is very close to the regular one, then Lagrange interpolatory subdivision scheme with respect to this irregular grid also converges. Hermite subdivision schemes, which combine the values of the function and its derivatives, are a natural extension of scalar subdivision schemes. This has drawn many researchers’ attention (Dubuc, 2006; Dubuc and Merrien, 2006, 2001; Han et al., 2005; Merrien, 1992; Zhou, 2000). In Dubuc and Merrien (2006), Merrien (1992), convergent interpolatory Hermite subdivision schemes are constructed. In practice, we need to consider the Hermite interpolatory subdivision scheme HX with the irregular grids. In this paper, we studied the convergence of Hermite subdivision schemes related to irregular grids. One of the purposes is to extend the results of Maxim and Mazure (2004) for scalar subdivision schemes to the setting of Hermite subdivision schemes mentioned above. For our purpose, the associated (scalar) subdivision schemes are introduced with respect to irregular grids. Together with a weak condition, the convergence of this scalar subdivision scheme implies the convergence of the Hermite subdivision scheme. The implicational relation enables us to work in scalar case, not necessarily in vector case, where the Hermite subdivision schemes are considered originally. Such results for subdivision schemes with regular grids satisfying
S n (i , j ) = diag 1, 2n+1 A (i − 2 j ) diag 1, 2−n ,
i , j ∈ Z,
(4)
have been established in Dubuc (2006) and Dubuc and Merrien (2006), where { A (i )} is a sequence of 2 × 2 matrices. Our method is somehow different from theirs, but the above implicational relation still holds in a more general setting, the L p -convergence setting. The convergence of subdivision schemes in L p (Jia, 1995), particularly in L 2 , had been one of the focuses in wavelet analysis. Hermite subdivision schemes in L p setting are considered in Han et al. (2004, 2005), Zhou (2000), etc. For interpolatory subdivision schemes, HX is determined by X. If an irregular grid X is equivalent to the regular grid X∗ in a suitable sense, then the slightly perturbing HX∗ results in HX . With the results for perturbation on convergent subdivision schemes, we proved that the convergence of HX is implied by the convergence of HX∗ . Our paper is organized as follows. In Section 2, the notion of associated (scalar) subdivision scheme is introduced. We then turn the problem of convergence of a Hermite subdivision scheme into convergence of its associated subdivision scheme. In Section 3, we proved that if an interpolatory Hermite subdivision scheme with regular grid converges, then interpolatory Hermite subdivision schemes with irregular girds which are very close to regular grid also converge. In Section 4, we mentioned briefly the generalizations in L p setting. Some refined results in stationary case are listed. We end this section by introducing some terminologies.
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Definition 3. (1) A subdivision scheme is local if there exists an integer n S such that for any n 0, i , j 0, S n (i , j ) = 0 ⇒ |i − 2 j | N S . (2) A subdivision scheme is bounded if there exists a number M S < +∞ such that, for all n 0, S n := sup{| S n (i , j )|, i , j ∈ Z} M S . Definition 4. (1) A Hermite subdivision scheme with { S n }n0 reproduces constants if
1 1 = , S n (i , j ) 0
j ∈Z
n 0, i ∈ Z.
0
(2) A Hermite subdivision scheme with a sequence of matrices { S n }n0 and grid X = {xn,k | n 0, k ∈ Z} reproduces linear functions if it satisfies above and if there exists a number c such that
S n (i , j )
xn, j
1
j ∈Z
=
xn+1,i + c
,
1
n 0, i ∈ Z.
2. Associated scalar subdivision schemes Given a Hermite subdivision scheme, we can construct a scalar subdivision. Then the convergence of a Hermite subdivision scheme may be deduced to the convergence of its associated (scalar) subdivision scheme. The difference operator f as usual: f (i ) = f (i + 1) − f (i ) for any sequence f = f (i ), with one index i, of vectors or matrices. Moreover, for convenience, if f = ( f (i , j ))i , j ∈Z is a sequence with two indices i and j, we use the same notation f to denote the partial difference operator with respect to the first index, i.e., f = ( f (i + 1, j ) − f (i , j ))i , j∈Z . (k,l)
Theorem 1. Denote S n (i , j ) = ( S n (i , j ))0k,l1 and S n = ( S n (i , j ))i , j ∈Z . Suppose that the local Hermite subdivision scheme H with { S n } and grid X = {xn,k | n 0, k ∈ Z } reproduces constants. We set dn, j = xn, j +1 − xn, j and
Q˜ n (i , j ) =
(1,1)
Sn
(i , j ) dn, j
(0,1)
∞
k =1
dn, j
Sn
(i , j ) dn+1,i
(1,0)
Sn
∞
(i , j + k )
(0,0)
Sn dn+1,i
k =1
(i , j +k)
(5)
.
Moreover, let { Q n } is a sequence of matrices defined by
Q n (2i , 2 j ) = Q˜ n
(0,0)
Q n (2i , 2 j + 1) = Q˜ n
(0,1)
(i , j ),
Q n (2i + 1, 2 j ) = Q˜ n
(1,0)
(i , j ), (1,1) Q n (2i + 1, 2 j + 1) = Q˜ n (i , j ),
(i , j ),
(6)
where Q˜ n (i , j ) is the (k, l)-entry of matrix Q˜ n (i , j ). Then, the scalar subdivision with { Q n } is local. Moreover, un , n = 1, 2, . . . , given by (k,l)
(1 )
( 0)
un (2i ) = f n (i ),
un (2i + 1) = f n (i )/dn,i ,
(7) (0)
are the refinements of the scalar subdivision scheme with { Q n }, where f n = ( f n , scheme H.
(1) f n ) T are the refinements of the Hermite subdivision
Proof. By the locality of H, there exits an integer N such that condition of (1) of Definition 4 holds. This together with the reproducibility of { S n } for constants gives Q˜ n (i , j ) = 0 when |i − 2 j | N. So the subdivision scheme with { Q n } is local. We now establish un+1 = Q n un , n = 1, 2, . . . . First we set Qˆ n (i , j ) as follows.
Qˆ n (i , j ) =
(1,0) (i , j ) ( k∞=1 S n (i , j + k)) . (0,1) (0,0) S n (i , j ) ( k∞=1 S n (i , j + k)) (1,1)
Sn
It is easily seen by (5) that
Qˆ n (i , j ) = Qˆ n
Q˜ n
Q˜ n
(0,0)
(0,1)
(i , j ) (1,0) ˜ Qn (i , j )dn+1,i
(i , j )/dn, j (1,1) ˜ Qn (i , j )dn+1,i /dn, j
Qˆ n
.
(9)
(i , j ) denotes the (k, l)-entry of matrix Qˆ n (i , j ), 0 k, l 1. By definition of Qˆ n(0,1) (i , j ) in (8) we have (0,1) (1,0) (i , j − 1) − Qˆ n (i , j ) = S n (i , j ). Summation by parts yields
(k,l)
(0,1)
(8)
j
(1,0)
Sn
(i , j ) f n(0) ( j ) =
j
Qˆ n
(0,1)
(i , j ) f n(0) ( j ).
Y. Zhao, D.-R. Chen / Computer Aided Geometric Design 27 (2010) 372–381
This together with f n+1 = S n f n gives (1 )
f n +1 ( i ) =
Qˆ n
(0,1)
( 0)
(i , j ) f n ( j ) +
j
(1,1)
375
(1 )
Sn
(i , j ) f n ( j ).
j
(0,1) (0,1) (1,1) (0,0) By Qˆ n (i , j ) = Q˜ n (i , j )/dn, j and S n (i , j ) = Q˜ n (i , j ) we have
(1 )
f n +1 ( i ) =
Q˜ n
(0,1)
( 0)
(i , j )
f n ( j) dn, j
j
+
Q˜ n
(0,0)
(1 )
(i , j ) f n ( j ).
j
Taking (6) and (7) into account we obtain
un+1 (2i ) =
D (2i , 2 j )un (2 j ) +
j
=
Q n (2i , j )un ( j ).
j
It remains to establish un+1 (2i + 1) = of f n+1 = S n f n . This implies ( 0)
f n +1 ( i ) =
D (2i , 2 j + 1)un (2 j + 1)
j
(0,0)
Sn
( 0)
j
(i , j ) f n ( j ) +
Q n (2i + 1, j )un ( j ). To this end, we take the difference operator on both sides
j
(0,0)
Qˆ n
(1,1)
(1 )
(i , j ) f n ( j ).
(10)
j
On the other hand, we have S n right hand side of (10) reads
(0,1)
Sn
(i , j ) = Qˆ n
(1,1)
(i , j − 1) − Qˆ n
(1,1)
(i , j ) by (8). Summation by parts, the first sum on the
(1,1) (i , j − 1) − Qˆ n(1,1) (i , j ) f n(0) ( j ) = Qˆ n (i , j ) f n(0) ( j ).
j
j
Consequently, divided by dn+1,i , (10) becomes ( 0)
f n +1 ( i ) dn+1,i
=
Q˜ n
(1,1)
(i , j )
f n(0) ( j )
j
=
dn, j
+
Q˜ n
(1,0)
(i , j ) f n(1) ( j )
j
Q n (2i + 1, j )un ( j ).
j
The proof is complete.
2
For a scalar subdivision scheme with { Q n }, if the sum of every entry in the same row of Q n equals to 1: then the scalar subdivision scheme with { Q n } reproduces constants.
j
Q n (i , j ) = 1,
Lemma 2. Suppose that the Hermite subdivision scheme H = {{ S n }, X} reproduces linear functions. Let Q n , n = 1, 2, . . . , be the matrices given as in (6). Then the scalar subdivision scheme with { Q n } reproduces constants. Proof. By the construction of Q n in (6) and (5), the sum of the entries of 2i-th row in Q n is
(1,1)
Sn
(i , j ) +
j
∞ j
=
(1,1)
Sn
(i , j ) +
=
j
∞ (1,0) (1,0) ( S n (i , j − 1 + k) − S n (i , j + k))xn, j j
(1,1)
Sn
(i , j + k)(xn, j +1 − xn, j )
k =1
j
(1,0)
Sn
(i , j ) +
k =1
(1,0)
Sn
(i , j )xn, j = 1,
j
where the last equality follows from the conditions in Definition 4. Similarly we have complete. 2
j
Q n (2i + 1, j ) = 1. The proof is
With the above results we can give a sufficient condition for the convergence of Hermite subdivision scheme H in terms of the condition for its associated scalar subdivision. For a grid X = {xn,k | n 0, k ∈ Z}, we have a grid X = {xn ,k , n 0, k ∈ Z} is given by xn ,2k = xn+1,2k , xn ,2k+1 = xn+1,2k+1 .
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Y. Zhao, D.-R. Chen / Computer Aided Geometric Design 27 (2010) 372–381
Theorem 3. Let X = {xn,k | n 0, k ∈ Z} be the quasi-regular grid and H = {{ S n }, X} the Hermite subdivision scheme reproduces linear functions. Suppose that the associated scalar subdivision scheme ({ Q n }, X ) converges. If, for every initial vector sequence f 0 , (0) the sequence { f n (0)} of numbers converges, then H converges. Proof. Although the proof is elementary we include it for the completeness. For simplicity we assume x0,0 = 0, so that xn,0 = 0 for any n 0. Suppose that g , gn , are continuous functions on R and that gn are linear on [xn,i , xn,i +1 ], n 0, i ∈ Z. We note that the sequence { gn } converges uniformly to g in any finite interval iff for any positive integer L,
lim
sup
n→∞ −2n L i 2n L
gn (xn,i ) − g (xn,i ) = 0.
(11)
Let { F n } with F n = ( F n , F n ) T , and F˜ n be generated by H and ({ Q n }, X ) respectively. Denote φ1 the limit of { F˜ n (x)}. Recall the definition of {{ Q n }, X }. By (11), the convergence of subdivision scheme {{ Q n }, X } is equivalent to (0)
lim
sup
n→∞ −2n L i 2n L
and
lim
sup
n→∞ −2n L i 2n L
(1)
f n(0) (i )
= 0,
− φ ( x ) 1 n + 1 , 2i + 1
d n ,i
(1 )
F n (i ) − φ1 (xn,i ) = 0, (1)
∀L > 0
(12)
∀ L > 0. (1)
The latter is just (11) for φ1 , F n . Therefore, { F n } converges uniformly to φ1 in any finite interval. x (0) (0) We now establish (11) for φ0 , F n , where φ0 (x) = y 0 + 0 φ1 (t ) dt with y 0 being the limit of { f n (0)}. For any
> 0,
L > 0, using (12) for φ1 , F˜ n and X , we obtained an integer N such that
f n(0) (i )
− φ ( x ) 1 n+1,2i +1 < ,
d n ,i
n > N , −2n L i 2n L .
Suppose i > 0. By summing up the above inequalities over 0 j i − 1 we have, for n > N ,
i −1
( 0)
( 0) φ1 (xn+1,2 j +1 )dn, j < x0, L .
f n (i ) − f n (0) −
(13)
j =0
On the other hand, by the integrability of φ1 in [0, x0, L ] and dn, j = O (1/2n ), we can assume without loss of generality for N as above, the function φ0 (x) satisfies,
i −1
φ1 (xn+1,2 j +1 ) dn, j < ,
φ0 (xn,i ) − y 0 −
n > N , −2n L i 2n L
j =0
together with (13) yields for n > N , 0 < i 2n L
( 0)
f n (i ) − φ0 (xn,i ) < x0, L + f n(0) (0) − y 0 < |x0, L | + 1 (0)
(0)
where we have assumed that | f n (0) − y 0 | < for n > N. For −2n L i < 0, by the same argument, | f n (i ) − φ0 (xn,i )| < (0)
(|x0,L | + 1) when n is large enough. This establishes (11) for φ0 , F n and X. The proof of the theorem is completed. 2 We now turn to the convergence of scalar subdivision schemes. For a local and bounded scalar subdivision scheme with
{ Q n }, if it reproduces constants, then it is known from, e.g. Daubechies et al. (1999), that there is a local and bounded subdivision scheme with { D n }, referred to as the difference subdivision scheme of { Q n }, satisfying the so-called commutation formula Q n = D n , n 0. Lemma 4. (See Maxim and Mazure, 2004, Theorem 3.8.) Assume that a local and bounded scalar subdivision scheme with { Q n } reproduces constants. Denote { D n } the difference subdivision scheme of { Q n }. Suppose that
∃ N , K 0 s.t. sup D n+ K . . . D n+2 D n+1 ∞ < 1.
(14)
n> N
Then with any grid X, the scalar subdivision schemes with { Q n } converges, where the matrix norm A p is defined by
A p = sup Ax p | x ∈ p (Z), x p 1 .
(15)
Lemma 5. (See Maxim and Mazure, 2004, Theorem 4.2.) Suppose that S and S are local and bounded scalar subdivision schemes with { Q n } and { Q n } respectively, and that both of them produce constants. Assume that the following conditions hold.
Y. Zhao, D.-R. Chen / Computer Aided Geometric Design 27 (2010) 372–381
377
(1) They are equivalent in the sense that Q n − Q n α 2−β n , n 0, where α and β are some positive constants. (2) The difference subdivision scheme { D n } of { Q n } satisfies (14) with some N , K . Then the difference subdivision scheme { D n } of { Q n } also satisfies (14) with some N , K . 3. Interpolatory Hermite subdivision schemes In this section we apply the results of previous section to discuss the convergence of (2N + 2)-point interpolatory Hermite subdivision schemes with grids which are very close to regular grid in a suitable sense. The convergence will be established once the (2N + 2)-point interpolatory Hermite subdivision scheme with regular grid converges. We first construct the sequence { S n } of matrices decided by a grid X. Let N be a nonnegative integer. For any k ∈ Z and l ∈ {k − N , K − N + 1, . . . , k + N + 1}, let Λk,l = {k − N , k − N + 1, . . . , k + N + 1}\{l}. Then
L n,k,l (x) =
x − xn,i xn,l − xn,i
i ∈Λk,l
,
k − N l k + N + 1,
are Lagrange polynomials based on points {xn,l | k − N l k + N + 1}. Furthermore, the Hermite polynomials are
2(x − xn,l ) gn,k,l (x) = 1 − L n2,k,l (x), xn,l − xn,m
hn,k,l (x) = (x − xn,l ) L n2,k,l (x).
m∈Λk,l
Then let { S n } be given by
gn,k,l (xn+1,2k+s ) hn,k,l (xn+1,2k+s ) S n (2k + s, l) = , k − N l k + N + 1 , s = 0, 1 , gn ,k,l (xn+1,2k+s ) hn ,k,l (xn+1,2k+s ) S n (2k + s, l) = 0, l < k − N or l > k + N + 1, s = 0, 1.
(16)
We denote this subdivision scheme {{ S n }, X} as HX and call it (2N + 2)-point Hermite interpolatory subdivision scheme. By above construction, we have
S n (2k, l) = δk,l I ,
k, l ∈ Z,
where I denotes the 2 × 2 identity matrix. Consequently, the subdivision scheme HX is interpolatory, i.e., f n+1 (2k) = f n (k), k ∈ Z. We also can obtain an explicit expression of f n+1 (2k + 1) in terms of Hermite polynomial P n,k of degree not larger that 4N + 3, which interpolates the data f n (l) at 2N + 2 points xn,l , k − N l k + N + 1:
T
P n,k (xn,l ), P n ,k (xn,l )
= f n (l),
k − N l k + N + 1,
where P n ,k is the derivative of P n,k . Indeed, it follows from (16) that
T
f n+1 (2k + 1) = P n,k (xn+1,2k+1 ), P n ,k (xn+1,2k+1 )
,
k ∈ Z.
(17)
It is well known that P n,k is represented by
P n,k (x) =
k+ N +1
( 0)
(1 )
f n (xn,l ) gn,k,l (x) + f n (xn,l )hn,k,l (x) .
(18)
l=k− N
We specify X∗ = {k/2n |n 0, k ∈ Z} the regular grid. The convergence of (2N + 2)-point Hermite interpolatory subdivision scheme HX∗ as above has been studied by many authors. In the simplest case, i.e., N = 0, the matrices are given as the following, S n∗ (2i , j ) = δi , j I , S n∗ (2i + 1, j ) = 0 for j = i or i + 1, and
S n∗ (2i + 1, i ) =
1 2 −3 n 2 +1
1 2n+3 −1 4
,
S n∗ (2i + 1, i + 1) =
1 2 3
2n+1
−1 2n+3 −1 4
.
It is known from Merrien (1992) that the interpolatory Hermite subdivision HX∗ converges. Moreover, the convergence of the interpolatory Hermite subdivision HX∗ has been established for N = 1 and in Dubuc and Merrien (2001), and for N = 2, 3, 4 in Yu (2005). To consider the case N = 5, 6, we note that the matrices S n∗ in HX∗ satisfies (4) (Dubuc, 2006). Therefore, by construction, the associated scalar subdivision scheme { Q n∗ } is uniformed in the sense that { Q n∗ } are independent of n, which in turn implies that the difference subdivision scheme { D n∗ } of { Q n∗ } is also uniformed. We set D ∗ = D n∗ , n 0. By computation we get D ∗2 ∞ = 0.8675, 0.8968 respectively. The convergence of HX∗ then follows from Lemma 4. However, in practice, we need to consider the Hermite interpolatory subdivision scheme HX with irregular grids X. We restrict ourselves to the grids equivalent to X∗ in the following sense.
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Y. Zhao, D.-R. Chen / Computer Aided Geometric Design 27 (2010) 372–381
Definition 5. We define that a grid X = {xn,k | n 0, k ∈ Z} is equivalent to the regular grid X∗ if, for any positive integer K , there exist two positive numbers γ , η such that
− K 2l − k K − 1
⇒
dn,l
γ 2−ηn , − 2
d
n 0,
n+1,k
(19)
where, as before, dn,k = xn,k+1 − xn,k . From (19), the ratio dn,l /dn+1,k with X is very close to 2, the ratio with regular grid X∗ . In this sense, we may regard X as a grid resulting from slight perturbation on X∗ . Lemma 6. (See Maxim and Mazure, 2004.) Suppose that G : R → R is a function meeting the following two requirements: (1) G is Hölder of index α > 1. (2) There exist two positive numbers m, M such that its derivative G satisfies 0 < m G (x) M, x ∈ R. Let xn,k := G −1 (xn∗,k ), n 0, k ∈ Z. Then X = {xn,k | n 0, k ∈ Z} is quasi-regular and it is equivalent to the regular grid X∗ . For example, G (x) = μx + ρ sin(η x) with μ > 0 and constants ρ , η subject to |ρη| < μ. Fig. 1 shows a regular and an irregular (obtained, via G (x) which μ = 1, ρ = 0.5, η = 1) Hermite scheme in action. Starting from a set of Hermite data and grid, subdivision scheme successively refines Hermite data successively. The distinction between a regular and an irregular schemes should be evident from the figures. Lemma 7. Let X = {xn,k | n 0, k ∈ Z} be given as in Lemma 6. Suppose that { Q n } and { Q n∗ } are the associated (scalar) subdivision schemes with HX and HX∗ respectively, as in (6). Then { Q n } is equivalent to { Q n∗ }. Proof. We adopted the notations S n∗ and Q˜ n∗ for the regular grid X∗ . Recall that Q˜ n , associated with HX , are given as in (5).
For our purpose, it suffices to establish that there exist positive numbers α , β , such that Q˜ n k, l = 0, 1. We only consider the case k = l = 1. The arguments for other cases are similar.
(k,l)
By constructions of Q˜ n
(1,1)
I=
dn, j dn+1,i
II = 2
α 2−β n , n 0,
(1,1) (1,1) (i , j ), we rewrite Q˜ n (i , j ) − Q˜ ∗ n (i , j ) = I − II with
∞ (0,0) −2 S n (i , j + k ) ,
∞
k =1
(0,0)
Sn
(i , j + k ) −
k =1
where the equality
∗(k,l)
− Q˜ n
∞
∗(0,0)
Sn
(i , j + k ) ,
k =1 dn∗, j
dn∗+1,i
= 2 for regular grid X∗ is used. (0,0)
(0,0)
It follows from (16) that | S n (i , j )| is up bounded uniformly by some constant. Moreover, S n (2i , j ) = 0 iff i = j, ∞ (0,0) (0,0) and S n (2i + 1, j ) = 0 only if i , j satisfy j − ( N + 1) i j + N. Consequently, | k=1 S n (i , j + k)| is up bounded uniformly by a constant, and is nonzero only if − K 2i − j K − 1, where K is an integer independent of n, i , j. As ∞ (0,0) known in Lemma 6, X and X∗ are equivalent. Therefore (19) holds for those i , j which satisfy k=1 S n (i , j + k) = 0.
Consequently, | I | γ 2−η n , n 0, i , j ∈ Z, for some positive numbers γ , η . As for II, it is not difficult to deduce from the equivalence between X and X∗ that
(0,0)
S n (i , j ) − S n∗(0,0) (i , j ) γ 2−ηn ,
n 0, i , j ∈ Z.
(i , j ) = 0 we have |II| γ 2−η n , n 0, i , j ∈ Z. (k,l) ˜ For any i, there are only finite number of j’s such that Q n (i , j ) = 0. Therefore, the estimations above for I and II, (1,1) ∗(1,1) Q˜ n − Q˜ n α 2−β n , n 0, are as desired. The proof is complete. 2 (0,0)
By the fact that, for any j, there are only finite i such that S n
Theorem 8. Let N 0 be an integer. If the (2N + 2)-point interpolatory Hermite subdivision scheme HX∗ converges, then the (2N + 2)-point Hermite interpolatory subdivision scheme HX also converges, where X is given as in Lemma 6. Proof. We prove first that the associated subdivision scheme {{ Q n }, X } of HX converges. Since X is quasi-regular, we have by (16) a constant C such that
(0,0)
n (0,1)
S n (i , j ) C ,
2 S n (i , j ) C ,
(1,0)
(1,1)
S n (i , j )/2n C ,
S n (i , j ) C ,
n 0, i , j ∈ Z.
(20)
Y. Zhao, D.-R. Chen / Computer Aided Geometric Design 27 (2010) 372–381
379
Fig. 1. Regular and irregular Hermite subdivision.
This gives the boundedness of { Q n } of the associated scalar subdivision scheme of HX . As known, for regular grid, the corresponding difference subdivision scheme { D n∗ } of { Q n∗ } is uniform, i.e., all D n∗ , n 0, are the same and are denoted by D ∗ . If the (2N + 2)-point interpolatory Hermite subdivision scheme HX∗ converges, then it follows from Dubuc (2006), Theorem 14 that D ∗m ∞ < 1 for some m. This is just (14) with K = m and any N 1. It follows from Lemmas 5 and 7 that condition (14) holds for { D n }, the difference subdivision scheme of { Q n }. Consequently, the subdivision scheme {{ Q n }, X } converges by Lemma 4. (0) Second we establish the convergence of { f n (0)}. The convergence of {{ Q n }, X } implies, for any j, both sequences (0) (1) { f n ( j )/dn, j } and { f n ( j )} are bounded. Therefore the equality ( 0)
( 0)
f n+1 (0) − f n (0) = (0)
NS
(0,0)
Sn
( 0) ( 0) (0,1) (1 ) (0, j ) f n ( j ) − f n (0) + S n (0, j ) f n ( j )
j =− N S
(0)
tells f n+1 (0) − f n (0) = O (1/2n ), n → ∞. The proof of the theorem is complete by Theorem 3.
2
(21)
380
Y. Zhao, D.-R. Chen / Computer Aided Geometric Design 27 (2010) 372–381
4. L p -convergence We have only considered so far the uniform convergence of subdivision schemes. In this section we will mention briefly their analogy in L p setting. A subdivision scheme {{ S n }, X} converges in L p , if for any f 0 ∈ ( p (Z))2 , there is a vector valued function F ∈ ( L p (R))2 , such that R F n − F p dx → 0 as n → ∞. It is not difficult to see that the condition is equivalent to
p xn, j +1 1 / p 1 F (x) dx dn, j = 0. f n ( j) − dn, j
lim
n→∞
j ∈Z
xn, j
It is essentially the same as in Jia (1995), Micchelli and Sauer (1998), Chen (2003) and Liu and Chen (2005), if X is regular. Clearly the L p -convergence of the associated scalar subdivision scheme { Q n } implies the following condition
lim
n→∞
1 2n
n L −1 2
j =−2n L
p 1 / p xn, j +1
f ( 0) ( j )
1
n
− φ1 (x) dx
= 0,
dn, j
dn, j
∀ L > 0,
(22)
xn, j
provided that a2−n dn, j . This is a generalized version of (12) for p 1. (0)
It is worth pointing out that, with (12) replaced by (22) in the proof of Theorem 3, we can also establish | f n (i ) −
φ0 (xn,i )| → 0. Indeed, the desired result follows from dn, j b2−n and the inequality
xn,i
( 0)
( 0)
f n (i ) − f n (i ) − φ1 (x) dx
0
p 1/ p 2n L −1 xn, j +1 1/q 2n L −1
f n(0) q 1
− φ1 (x) dx
dn, j .
dn, j
dn, j n n j =−2 L
j =−2 L
xn, j
As known in previous results, condition (14) plays an important role. In the study of L p -convergence we’ll use a generalized version of (14) for p ∈ [1, ∞]
∃ integers N˜ , K˜ 0,
˜
s.t. sup D n+ K˜ . . . D n+2 D n+1 p < 2 K / p ,
(23)
˜ n> N
where, as before, { D n } is the difference subdivision scheme of { Q n }. Indeed, it is weaker than (14) because of the following theorem. Theorem 9. Suppose that a scalar subdivision S with { D j } is local and bounded. If (14) holds for some two integers N , K 0, then, ˜ , K˜ 0. for any p 1, (23) also holds for some integers N Proof. Recall that N S and M S are given in Definition 3 for { D j }. For any integers m, n 0, let A m,n = D n+m . . . D n+2 D n+1 and denote A m,n = (am,n (i , j ))i , j ∈Z . It is easily seen by the method of induction that
am,n (i , j ) = 0
⇒
i − 2m j 2m − 1 N S .
(24)
Based on the assumption made in (14), for some K , N, A K ,n ∞ < 1, ∀n > N. A standard argument (Jia, 1995; Maxim and Mazure, 2004) yields that there are a positive number μ < 1 and an integer K 0 such that A m,n ∞ μm , ∀m K , n > N , and, consequently, |am,n (i , j )| μm , ∀m K , n > N, i , j ∈ Z. Since for any j ∈ Z, the number of nonzero entries am,n (i , j ) is less than 2m+1 N S by (24), we have
Am,n 1 = sup
am,n (i , j ) < μm 2m+1 N S ,
j
∀m K , n > N .
i
˜ < 2 and integer K˜ such that A K˜ ,n 1 μ ˜ K˜ , ∀n > N. Appealing to Schur lemma we Therefore, there are positive number μ conclude ˜
˜
˜
˜ K /p , ˜ K /p < μ A K˜ ,n p μ K (1−1/ p ) μ The proof is complete.
∀n > N .
2
Now we establish the analogy of Lemmas 4 and 5 in L p setting.
Y. Zhao, D.-R. Chen / Computer Aided Geometric Design 27 (2010) 372–381
381
Theorem 10. For p ∈ [1, ∞), both Lemmas 4 and 5 hold with (14) replaced by (23). Proof. Following the arguments of Maxim and Mazure (2004), Theorem 3.8, we can easily prove that, the conditions in Lemma 4, with (14) replaced by (23), imply the L p -convergence of { Q n } provided dn, j b2−n . That is the analogy of Lemma 4 in L p setting. For the analogy of Lemma 5, let
B m,n = bm,n (i , j ) i , j ∈Z := D n+m . . . D n+2 D n+1 − D n +m . . . D n +2 D n +1 . Under assumption (1) of Lemma 5, it is obtained in the proof of Maxim and Mazure (2004), Theorem 4.2 that, for any m, there exists a number C m satisfying B m,n ∞ C m 2−β n , ∀n 0. Note that bm,n (i , j ) satisfies (24) with N S re satisfying, placed with max{ N S , N S }. As in the proof Theorem 9, we conclude that, for any m, there exists a number C m −β n ∀n 0, B m,n p C m 2 , which can be arbitrarily small as n → ∞. It together with (23) and D n+m . . . D n+2 D n +1 p D n+m . . . D n+2 D n+1 p + B m,n p implies that there exists N such that
sup D
n> N
n+ K˜
˜ . . . D n +2 D n +1 p < 2 K / p ,
as desired. The proof is complete.
2
At the end, we mention some results concerning the scalar, local and stationary subdivision schemes { Q n }. In this case, there is a finitely supported sequence a = (a(i ))i ∈Z of numbers, called the mask of { Q n }, such that Q n (i , j ) = a(i − 2 j ), n 0, i , j ∈ Z. The difference subdivision scheme { D n } is also local and stationary, i.e., D n (i , j ) = b(i − 2 j ), n 0, i , j ∈ Z, with b a finitely supported sequence of numbers. We’d like list some refined results of Theorem 10 for stationary subdivision schemes with regular grid. 1. Clearly, (23) becomes D m p < 2m/ p for some m. This condition is not only sufficient but also necessary for the L p -convergence (Jia, 1995). 2. The perturbation of a stationary subdivision scheme has been considered in Chen and Plonka (2002); Daubechies and Huang (1995). It is proved in Chen and Plonka (2002) that, for a local, stationary and L p -convergent scheme { Q n } with and stationary subdivision scheme { Q n } with mask a mask a, there is a small positive number δ such that, for any local (2i ) = satisfying a − a p δ , { D n } also converges in L p provided a i i a (2i + 1) = 1. For any f 0 ∈ (Z), denote F a and F a the limit of sequences of functions generated by { Q n } and { Q n } respectively. An interesting estimation
F a − F a p C a − a p f 0 p is established in Chen and Plonka (2002), where C is some constant independent of f 0 . For p = ∞, a weaker version of the estimation is obtained in Daubechies and Huang (1995), where masks are not necessarily finitely supported. The refined results above are actually true in Rs with s 1. However, it is not clear how to generalize Theorem 10 to Rs . References Cavaretta, A.S., Dahmen, W., Micchelli, C.A., 1991. Stationary subdivision. Mem. Amer. Math. Soc. 93 (453). Chen, D.R., 2003. Spectral radii and eigenvalues of subdivision operators. Proc. Amer. Math. Soc. 132, 1113–1123. Chen, D.R., Plonka, G., 2002. Convergence of cascade algorithms in Sobolev spaces for perturbed refinement masks. J. Approx. Theory 119, 133–155. Daubechies, I., Guskov, I., Sweldens, W., 1999. Regularity of irregular subdivision. Constr. Approx. 15, 381–426. Daubechies, I., Guskov, I., Sweldens, W., 2001. Commutation for irregular subdivision. Constr. Approx. 17, 479–514. Daubechies, I., Huang, Y., 1995. How does truncation of the mask affect a refinable function? Constr. Approx. 11, 365–380. Dubuc, S., 2006. Scalar and Hermite subdivision schemes. Appl. Comput. Harmonic Anal. 21, 376–394. Dubuc, S., Merrien, J.-L., 2001. A 4-point Hermite subdivision scheme. In: Mathematical Methods for Curves and Surfaces. Vanderbilt University Press, Nashville, TN, pp. 113–122. Dubuc, S., Merrien, J.-L., 2006. Convergent vector and Hermite subdivision schemes. Constr. Approx. 23, 1–22. Dyn, N., Levin, D., 1995. Analysis of asymptotically equivalentbinary subdivision scheme for curve design. J. Math. Anal. Appl. 193, 594–621. Dyn, N., Levin, D., Gregory, J.A., 1987. A 4-point interpolatory subdivision scheme for curve design. Comput. Aided Geom. Design 4, 257–268. Dyn, N., Levin, D., Gregory, J.A., 1991. Analysis of linear binary subdivision schemes for curve design. Constr. Approx. 7, 127–147. Han, B., Yu, T.P.-Y., Piper, B., 2004. Multivariate refinable Hermite interpolants. Math. Comp. 73, 1913–1935. Han, B., Yu, T.P.-Y., Xue, Y., 2005. Noninterpolatory Hermite subdivision schemes. Math. Comp. 74, 1345–1367. Jia, R.Q., 1995. Subdivision schemes in L p spaces. Adv. Comp. Math. 3, 309–341. Liu, H.Y., Chen, D.R., 2005. Convergence of subdivision schemes and smoothness of limit functions. J. Math. Anal. Appl. 306, 740–751. Maxim, V., Mazure, M.L., 2004. Subdivision schemes and irregular grids. Numer. Algorithms 35, 1–28. Merrien, J.L., 1992. A family of Hermite interpolants by bisection algorithms. Numer. Algorithms 2, 187–200. Micchelli, C.A., Sauer, T., 1998. On vector subdivision. Math. Z. 229, 621–674. Yu, T.P.-Y., 2005. On the regularity analysis of interpolatory Hermite subdivision schemes. J. Math. Anal. Appl. 302, 201–216. Zhou, D.X., 2000. Multiple refinable Hermite interpolants. J. Approx. Theory 102, 46–71.
Contents lists available at ScienceDirect
Computer Aided Geometric Design www.elsevier.com/locate/cagd
Convergence of irregular Hermite subdivision schemes ✩ Yao Zhao, Di-Rong Chen ∗ Department of Mathematics, LMIB, Beijing University of Aeronautics and Astronautics, Beijing 100083, PR China
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 13 December 2008 Received in revised form 13 December 2009 Accepted 12 February 2010 Available online 19 February 2010
We consider in this paper the convergence of Hermite subdivision schemes related to the irregular grids X. It is reduced to the convergence of a scalar subdivision scheme, so the results concerning perturbation of convergent scalar subdivision schemes are applied. For interpolatory subdivision schemes HX , with the irregular grid X which is equivalent to the regular grid X∗ in a suitable sense, we proved that the convergence of HX is implied by that of HX∗ . Our arguments work in the settings of L p -convergence and uniform convergence. © 2010 Elsevier B.V. All rights reserved.
Keywords: Hermite subdivision scheme Associated subdivision scheme Difference subdivision scheme Irregular grid Interpolatory subdivision scheme
1. Introduction A Hermite subdivision scheme recursively computes a function φ and its derivative φ from the initial state of the scheme, sequence of vectors in R2 . In this paper, we consider the convergence of Hermite subdivision schemes with respect to irregular grids. In order to introduce the Hermite subdivision schemes, we need two components. One is a sequence of matrices, which shows how the data on new level is computed; the other is the grid, which specifies where to put the data. Let F = {( f (i ))i ∈Z | f (i ) ∈ R2 } be the set of all sequences of vectors in R2 . Suppose that, for n = 1, 2, . . . , S n = ( S n (i , j ))i , j ∈Z is a bi-infinite matrix with its entry S n (i , j ) being 2 × 2 matrix. With an initial sequence f 0 = ( f 0 (k)) ∈ F , we can construct recursively f n = S n f n−1 , for n = 1, 2, . . . , by { S n }, which
S n f (i ) =
S n (i , j ) f ( j ),
i ∈ Z.
(1)
j ∈Z
Definition 1. A grid is a set X = {xn,k | n 0, k ∈ Z} of real numbers satisfying the following conditions. (1) At the coarsest level, we have limk→−∞ x0,k = −∞, limk→∞ x0,k = ∞. (2) At any level n 0, the sequence xn,k is strictly increasing. (3) The set of points of a given level is contained in the set of points of the next level n + 1, in the sense that xn+1,2k = xn,k , for any n 0, k ∈ Z. (4) limn→∞ supk dn,k = 0, where dn,k = xn,k+1 − xn,k . ✩ Research supported in part by NSF of China under grants 10871015 and 10972018, and National Basic Research Program of China under grants 9732010CB731900 and 973-2006CB303102. Corresponding author. Tel.: +86 010 82317934; fax: +86 010 82317933. E-mail addresses: [email protected] (Y. Zhao), [email protected] (D.-R. Chen).
*
0167-8396/$ – see front matter doi:10.1016/j.cagd.2010.02.004
© 2010
Elsevier B.V. All rights reserved.
Y. Zhao, D.-R. Chen / Computer Aided Geometric Design 27 (2010) 372–381
373
A grid X is quasi-regular if there exist positive numbers a, b such that a2−n dn,k b2−n for any n 0, k ∈ Z. In particular, the grid X∗ = {k/2n | n 0, k ∈ Z} is said to be regular. A Hermite subdivision scheme H is consisted of { S n } and X , which generates a sequence { F n } of vector valued functions (0) (1) F n = ( F n , F n ) T : R −→ R2 , satisfying ( 0)
(1 )
F n (xn, j ) = f n ( j ), F n , F n
are affine on [xn, j , xn, j +1 ],
n 0, j ∈ Z.
(2)
We say that a Hermite subdivision scheme is interpolatory if in each subdivision step the values at the even gridpoints are kept, i.e.,
f n+1 (2k) = f n (k),
n 0, k ∈ Z.
(3)
Definition 2. We define that the Hermite subdivision scheme H = {{ S n }, X} converges if, for every sequence ( f 0 (i ))i ∈Z , the sequence { F n } of vector valued functions, given as above, converges uniformly in any finite interval to a vector valued function F = ( F (0) , F (1) ) : R −→ R2 , with property
dF (0) (x) dx
= F (1) (x), i.e. F (1) (x) is just the derivative of F (0) (x).
In the scalar case, i.e., S n (i , j ) are real numbers instead of matrices, the subdivision scheme generates a sequence of polygonal lines F n : R −→ R, n 0. Correspondingly, in this case, the convergence of subdivision schemes is simplified to the uniform convergence of the sequence { F n } in any finite interval. Subdivision schemes are tools for constructing curves and surfaces in computer aided geometric design. There are extensive study on the subdivision schemes related to the regular grid. We have only mentioned Cavaretta et al. (1991), Dyn and Levin (1995), Dyn et al. (1987), Dyn et al. (1991) and Jia (1995). However, the limitation to regular grid is too restrictive. Subdivision schemes related to irregular grids are necessary for construction of spline curves with non-regularly spaced knots. Another application that calls for the irregular setting is wavelets and multiresolution analysis for irregular samples. In Daubechies et al. (1999), Daubechies et al. (2001) and Maxim and Mazure (2004), the authors provided a sufficient condition for the convergence of the scalar subdivision schemes with irregular grids. Moreover, by comparing subdivision schemes, the convergence of a given subdivision scheme may be derived from a convergent one (Maxim and Mazure, 2004). In particular, the authors proved that if the Lagrange interpolatory subdivision schemes with respect to regular grid converges meanwhile an irregular grid is very close to the regular one, then Lagrange interpolatory subdivision scheme with respect to this irregular grid also converges. Hermite subdivision schemes, which combine the values of the function and its derivatives, are a natural extension of scalar subdivision schemes. This has drawn many researchers’ attention (Dubuc, 2006; Dubuc and Merrien, 2006, 2001; Han et al., 2005; Merrien, 1992; Zhou, 2000). In Dubuc and Merrien (2006), Merrien (1992), convergent interpolatory Hermite subdivision schemes are constructed. In practice, we need to consider the Hermite interpolatory subdivision scheme HX with the irregular grids. In this paper, we studied the convergence of Hermite subdivision schemes related to irregular grids. One of the purposes is to extend the results of Maxim and Mazure (2004) for scalar subdivision schemes to the setting of Hermite subdivision schemes mentioned above. For our purpose, the associated (scalar) subdivision schemes are introduced with respect to irregular grids. Together with a weak condition, the convergence of this scalar subdivision scheme implies the convergence of the Hermite subdivision scheme. The implicational relation enables us to work in scalar case, not necessarily in vector case, where the Hermite subdivision schemes are considered originally. Such results for subdivision schemes with regular grids satisfying
S n (i , j ) = diag 1, 2n+1 A (i − 2 j ) diag 1, 2−n ,
i , j ∈ Z,
(4)
have been established in Dubuc (2006) and Dubuc and Merrien (2006), where { A (i )} is a sequence of 2 × 2 matrices. Our method is somehow different from theirs, but the above implicational relation still holds in a more general setting, the L p -convergence setting. The convergence of subdivision schemes in L p (Jia, 1995), particularly in L 2 , had been one of the focuses in wavelet analysis. Hermite subdivision schemes in L p setting are considered in Han et al. (2004, 2005), Zhou (2000), etc. For interpolatory subdivision schemes, HX is determined by X. If an irregular grid X is equivalent to the regular grid X∗ in a suitable sense, then the slightly perturbing HX∗ results in HX . With the results for perturbation on convergent subdivision schemes, we proved that the convergence of HX is implied by the convergence of HX∗ . Our paper is organized as follows. In Section 2, the notion of associated (scalar) subdivision scheme is introduced. We then turn the problem of convergence of a Hermite subdivision scheme into convergence of its associated subdivision scheme. In Section 3, we proved that if an interpolatory Hermite subdivision scheme with regular grid converges, then interpolatory Hermite subdivision schemes with irregular girds which are very close to regular grid also converge. In Section 4, we mentioned briefly the generalizations in L p setting. Some refined results in stationary case are listed. We end this section by introducing some terminologies.
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Y. Zhao, D.-R. Chen / Computer Aided Geometric Design 27 (2010) 372–381
Definition 3. (1) A subdivision scheme is local if there exists an integer n S such that for any n 0, i , j 0, S n (i , j ) = 0 ⇒ |i − 2 j | N S . (2) A subdivision scheme is bounded if there exists a number M S < +∞ such that, for all n 0, S n := sup{| S n (i , j )|, i , j ∈ Z} M S . Definition 4. (1) A Hermite subdivision scheme with { S n }n0 reproduces constants if
1 1 = , S n (i , j ) 0
j ∈Z
n 0, i ∈ Z.
0
(2) A Hermite subdivision scheme with a sequence of matrices { S n }n0 and grid X = {xn,k | n 0, k ∈ Z} reproduces linear functions if it satisfies above and if there exists a number c such that
S n (i , j )
xn, j
1
j ∈Z
=
xn+1,i + c
,
1
n 0, i ∈ Z.
2. Associated scalar subdivision schemes Given a Hermite subdivision scheme, we can construct a scalar subdivision. Then the convergence of a Hermite subdivision scheme may be deduced to the convergence of its associated (scalar) subdivision scheme. The difference operator f as usual: f (i ) = f (i + 1) − f (i ) for any sequence f = f (i ), with one index i, of vectors or matrices. Moreover, for convenience, if f = ( f (i , j ))i , j ∈Z is a sequence with two indices i and j, we use the same notation f to denote the partial difference operator with respect to the first index, i.e., f = ( f (i + 1, j ) − f (i , j ))i , j∈Z . (k,l)
Theorem 1. Denote S n (i , j ) = ( S n (i , j ))0k,l1 and S n = ( S n (i , j ))i , j ∈Z . Suppose that the local Hermite subdivision scheme H with { S n } and grid X = {xn,k | n 0, k ∈ Z } reproduces constants. We set dn, j = xn, j +1 − xn, j and
Q˜ n (i , j ) =
(1,1)
Sn
(i , j ) dn, j
(0,1)
∞
k =1
dn, j
Sn
(i , j ) dn+1,i
(1,0)
Sn
∞
(i , j + k )
(0,0)
Sn dn+1,i
k =1
(i , j +k)
(5)
.
Moreover, let { Q n } is a sequence of matrices defined by
Q n (2i , 2 j ) = Q˜ n
(0,0)
Q n (2i , 2 j + 1) = Q˜ n
(0,1)
(i , j ),
Q n (2i + 1, 2 j ) = Q˜ n
(1,0)
(i , j ), (1,1) Q n (2i + 1, 2 j + 1) = Q˜ n (i , j ),
(i , j ),
(6)
where Q˜ n (i , j ) is the (k, l)-entry of matrix Q˜ n (i , j ). Then, the scalar subdivision with { Q n } is local. Moreover, un , n = 1, 2, . . . , given by (k,l)
(1 )
( 0)
un (2i ) = f n (i ),
un (2i + 1) = f n (i )/dn,i ,
(7) (0)
are the refinements of the scalar subdivision scheme with { Q n }, where f n = ( f n , scheme H.
(1) f n ) T are the refinements of the Hermite subdivision
Proof. By the locality of H, there exits an integer N such that condition of (1) of Definition 4 holds. This together with the reproducibility of { S n } for constants gives Q˜ n (i , j ) = 0 when |i − 2 j | N. So the subdivision scheme with { Q n } is local. We now establish un+1 = Q n un , n = 1, 2, . . . . First we set Qˆ n (i , j ) as follows.
Qˆ n (i , j ) =
(1,0) (i , j ) ( k∞=1 S n (i , j + k)) . (0,1) (0,0) S n (i , j ) ( k∞=1 S n (i , j + k)) (1,1)
Sn
It is easily seen by (5) that
Qˆ n (i , j ) = Qˆ n
Q˜ n
Q˜ n
(0,0)
(0,1)
(i , j ) (1,0) ˜ Qn (i , j )dn+1,i
(i , j )/dn, j (1,1) ˜ Qn (i , j )dn+1,i /dn, j
Qˆ n
.
(9)
(i , j ) denotes the (k, l)-entry of matrix Qˆ n (i , j ), 0 k, l 1. By definition of Qˆ n(0,1) (i , j ) in (8) we have (0,1) (1,0) (i , j − 1) − Qˆ n (i , j ) = S n (i , j ). Summation by parts yields
(k,l)
(0,1)
(8)
j
(1,0)
Sn
(i , j ) f n(0) ( j ) =
j
Qˆ n
(0,1)
(i , j ) f n(0) ( j ).
Y. Zhao, D.-R. Chen / Computer Aided Geometric Design 27 (2010) 372–381
This together with f n+1 = S n f n gives (1 )
f n +1 ( i ) =
Qˆ n
(0,1)
( 0)
(i , j ) f n ( j ) +
j
(1,1)
375
(1 )
Sn
(i , j ) f n ( j ).
j
(0,1) (0,1) (1,1) (0,0) By Qˆ n (i , j ) = Q˜ n (i , j )/dn, j and S n (i , j ) = Q˜ n (i , j ) we have
(1 )
f n +1 ( i ) =
Q˜ n
(0,1)
( 0)
(i , j )
f n ( j) dn, j
j
+
Q˜ n
(0,0)
(1 )
(i , j ) f n ( j ).
j
Taking (6) and (7) into account we obtain
un+1 (2i ) =
D (2i , 2 j )un (2 j ) +
j
=
Q n (2i , j )un ( j ).
j
It remains to establish un+1 (2i + 1) = of f n+1 = S n f n . This implies ( 0)
f n +1 ( i ) =
D (2i , 2 j + 1)un (2 j + 1)
j
(0,0)
Sn
( 0)
j
(i , j ) f n ( j ) +
Q n (2i + 1, j )un ( j ). To this end, we take the difference operator on both sides
j
(0,0)
Qˆ n
(1,1)
(1 )
(i , j ) f n ( j ).
(10)
j
On the other hand, we have S n right hand side of (10) reads
(0,1)
Sn
(i , j ) = Qˆ n
(1,1)
(i , j − 1) − Qˆ n
(1,1)
(i , j ) by (8). Summation by parts, the first sum on the
(1,1) (i , j − 1) − Qˆ n(1,1) (i , j ) f n(0) ( j ) = Qˆ n (i , j ) f n(0) ( j ).
j
j
Consequently, divided by dn+1,i , (10) becomes ( 0)
f n +1 ( i ) dn+1,i
=
Q˜ n
(1,1)
(i , j )
f n(0) ( j )
j
=
dn, j
+
Q˜ n
(1,0)
(i , j ) f n(1) ( j )
j
Q n (2i + 1, j )un ( j ).
j
The proof is complete.
2
For a scalar subdivision scheme with { Q n }, if the sum of every entry in the same row of Q n equals to 1: then the scalar subdivision scheme with { Q n } reproduces constants.
j
Q n (i , j ) = 1,
Lemma 2. Suppose that the Hermite subdivision scheme H = {{ S n }, X} reproduces linear functions. Let Q n , n = 1, 2, . . . , be the matrices given as in (6). Then the scalar subdivision scheme with { Q n } reproduces constants. Proof. By the construction of Q n in (6) and (5), the sum of the entries of 2i-th row in Q n is
(1,1)
Sn
(i , j ) +
j
∞ j
=
(1,1)
Sn
(i , j ) +
=
j
∞ (1,0) (1,0) ( S n (i , j − 1 + k) − S n (i , j + k))xn, j j
(1,1)
Sn
(i , j + k)(xn, j +1 − xn, j )
k =1
j
(1,0)
Sn
(i , j ) +
k =1
(1,0)
Sn
(i , j )xn, j = 1,
j
where the last equality follows from the conditions in Definition 4. Similarly we have complete. 2
j
Q n (2i + 1, j ) = 1. The proof is
With the above results we can give a sufficient condition for the convergence of Hermite subdivision scheme H in terms of the condition for its associated scalar subdivision. For a grid X = {xn,k | n 0, k ∈ Z}, we have a grid X = {xn ,k , n 0, k ∈ Z} is given by xn ,2k = xn+1,2k , xn ,2k+1 = xn+1,2k+1 .
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Y. Zhao, D.-R. Chen / Computer Aided Geometric Design 27 (2010) 372–381
Theorem 3. Let X = {xn,k | n 0, k ∈ Z} be the quasi-regular grid and H = {{ S n }, X} the Hermite subdivision scheme reproduces linear functions. Suppose that the associated scalar subdivision scheme ({ Q n }, X ) converges. If, for every initial vector sequence f 0 , (0) the sequence { f n (0)} of numbers converges, then H converges. Proof. Although the proof is elementary we include it for the completeness. For simplicity we assume x0,0 = 0, so that xn,0 = 0 for any n 0. Suppose that g , gn , are continuous functions on R and that gn are linear on [xn,i , xn,i +1 ], n 0, i ∈ Z. We note that the sequence { gn } converges uniformly to g in any finite interval iff for any positive integer L,
lim
sup
n→∞ −2n L i 2n L
gn (xn,i ) − g (xn,i ) = 0.
(11)
Let { F n } with F n = ( F n , F n ) T , and F˜ n be generated by H and ({ Q n }, X ) respectively. Denote φ1 the limit of { F˜ n (x)}. Recall the definition of {{ Q n }, X }. By (11), the convergence of subdivision scheme {{ Q n }, X } is equivalent to (0)
lim
sup
n→∞ −2n L i 2n L
and
lim
sup
n→∞ −2n L i 2n L
(1)
f n(0) (i )
= 0,
− φ ( x ) 1 n + 1 , 2i + 1
d n ,i
(1 )
F n (i ) − φ1 (xn,i ) = 0, (1)
∀L > 0
(12)
∀ L > 0. (1)
The latter is just (11) for φ1 , F n . Therefore, { F n } converges uniformly to φ1 in any finite interval. x (0) (0) We now establish (11) for φ0 , F n , where φ0 (x) = y 0 + 0 φ1 (t ) dt with y 0 being the limit of { f n (0)}. For any
> 0,
L > 0, using (12) for φ1 , F˜ n and X , we obtained an integer N such that
f n(0) (i )
− φ ( x ) 1 n+1,2i +1 < ,
d n ,i
n > N , −2n L i 2n L .
Suppose i > 0. By summing up the above inequalities over 0 j i − 1 we have, for n > N ,
i −1
( 0)
( 0) φ1 (xn+1,2 j +1 )dn, j < x0, L .
f n (i ) − f n (0) −
(13)
j =0
On the other hand, by the integrability of φ1 in [0, x0, L ] and dn, j = O (1/2n ), we can assume without loss of generality for N as above, the function φ0 (x) satisfies,
i −1
φ1 (xn+1,2 j +1 ) dn, j < ,
φ0 (xn,i ) − y 0 −
n > N , −2n L i 2n L
j =0
together with (13) yields for n > N , 0 < i 2n L
( 0)
f n (i ) − φ0 (xn,i ) < x0, L + f n(0) (0) − y 0 < |x0, L | + 1 (0)
(0)
where we have assumed that | f n (0) − y 0 | < for n > N. For −2n L i < 0, by the same argument, | f n (i ) − φ0 (xn,i )| < (0)
(|x0,L | + 1) when n is large enough. This establishes (11) for φ0 , F n and X. The proof of the theorem is completed. 2 We now turn to the convergence of scalar subdivision schemes. For a local and bounded scalar subdivision scheme with
{ Q n }, if it reproduces constants, then it is known from, e.g. Daubechies et al. (1999), that there is a local and bounded subdivision scheme with { D n }, referred to as the difference subdivision scheme of { Q n }, satisfying the so-called commutation formula Q n = D n , n 0. Lemma 4. (See Maxim and Mazure, 2004, Theorem 3.8.) Assume that a local and bounded scalar subdivision scheme with { Q n } reproduces constants. Denote { D n } the difference subdivision scheme of { Q n }. Suppose that
∃ N , K 0 s.t. sup D n+ K . . . D n+2 D n+1 ∞ < 1.
(14)
n> N
Then with any grid X, the scalar subdivision schemes with { Q n } converges, where the matrix norm A p is defined by
A p = sup Ax p | x ∈ p (Z), x p 1 .
(15)
Lemma 5. (See Maxim and Mazure, 2004, Theorem 4.2.) Suppose that S and S are local and bounded scalar subdivision schemes with { Q n } and { Q n } respectively, and that both of them produce constants. Assume that the following conditions hold.
Y. Zhao, D.-R. Chen / Computer Aided Geometric Design 27 (2010) 372–381
377
(1) They are equivalent in the sense that Q n − Q n α 2−β n , n 0, where α and β are some positive constants. (2) The difference subdivision scheme { D n } of { Q n } satisfies (14) with some N , K . Then the difference subdivision scheme { D n } of { Q n } also satisfies (14) with some N , K . 3. Interpolatory Hermite subdivision schemes In this section we apply the results of previous section to discuss the convergence of (2N + 2)-point interpolatory Hermite subdivision schemes with grids which are very close to regular grid in a suitable sense. The convergence will be established once the (2N + 2)-point interpolatory Hermite subdivision scheme with regular grid converges. We first construct the sequence { S n } of matrices decided by a grid X. Let N be a nonnegative integer. For any k ∈ Z and l ∈ {k − N , K − N + 1, . . . , k + N + 1}, let Λk,l = {k − N , k − N + 1, . . . , k + N + 1}\{l}. Then
L n,k,l (x) =
x − xn,i xn,l − xn,i
i ∈Λk,l
,
k − N l k + N + 1,
are Lagrange polynomials based on points {xn,l | k − N l k + N + 1}. Furthermore, the Hermite polynomials are
2(x − xn,l ) gn,k,l (x) = 1 − L n2,k,l (x), xn,l − xn,m
hn,k,l (x) = (x − xn,l ) L n2,k,l (x).
m∈Λk,l
Then let { S n } be given by
gn,k,l (xn+1,2k+s ) hn,k,l (xn+1,2k+s ) S n (2k + s, l) = , k − N l k + N + 1 , s = 0, 1 , gn ,k,l (xn+1,2k+s ) hn ,k,l (xn+1,2k+s ) S n (2k + s, l) = 0, l < k − N or l > k + N + 1, s = 0, 1.
(16)
We denote this subdivision scheme {{ S n }, X} as HX and call it (2N + 2)-point Hermite interpolatory subdivision scheme. By above construction, we have
S n (2k, l) = δk,l I ,
k, l ∈ Z,
where I denotes the 2 × 2 identity matrix. Consequently, the subdivision scheme HX is interpolatory, i.e., f n+1 (2k) = f n (k), k ∈ Z. We also can obtain an explicit expression of f n+1 (2k + 1) in terms of Hermite polynomial P n,k of degree not larger that 4N + 3, which interpolates the data f n (l) at 2N + 2 points xn,l , k − N l k + N + 1:
T
P n,k (xn,l ), P n ,k (xn,l )
= f n (l),
k − N l k + N + 1,
where P n ,k is the derivative of P n,k . Indeed, it follows from (16) that
T
f n+1 (2k + 1) = P n,k (xn+1,2k+1 ), P n ,k (xn+1,2k+1 )
,
k ∈ Z.
(17)
It is well known that P n,k is represented by
P n,k (x) =
k+ N +1
( 0)
(1 )
f n (xn,l ) gn,k,l (x) + f n (xn,l )hn,k,l (x) .
(18)
l=k− N
We specify X∗ = {k/2n |n 0, k ∈ Z} the regular grid. The convergence of (2N + 2)-point Hermite interpolatory subdivision scheme HX∗ as above has been studied by many authors. In the simplest case, i.e., N = 0, the matrices are given as the following, S n∗ (2i , j ) = δi , j I , S n∗ (2i + 1, j ) = 0 for j = i or i + 1, and
S n∗ (2i + 1, i ) =
1 2 −3 n 2 +1
1 2n+3 −1 4
,
S n∗ (2i + 1, i + 1) =
1 2 3
2n+1
−1 2n+3 −1 4
.
It is known from Merrien (1992) that the interpolatory Hermite subdivision HX∗ converges. Moreover, the convergence of the interpolatory Hermite subdivision HX∗ has been established for N = 1 and in Dubuc and Merrien (2001), and for N = 2, 3, 4 in Yu (2005). To consider the case N = 5, 6, we note that the matrices S n∗ in HX∗ satisfies (4) (Dubuc, 2006). Therefore, by construction, the associated scalar subdivision scheme { Q n∗ } is uniformed in the sense that { Q n∗ } are independent of n, which in turn implies that the difference subdivision scheme { D n∗ } of { Q n∗ } is also uniformed. We set D ∗ = D n∗ , n 0. By computation we get D ∗2 ∞ = 0.8675, 0.8968 respectively. The convergence of HX∗ then follows from Lemma 4. However, in practice, we need to consider the Hermite interpolatory subdivision scheme HX with irregular grids X. We restrict ourselves to the grids equivalent to X∗ in the following sense.
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Definition 5. We define that a grid X = {xn,k | n 0, k ∈ Z} is equivalent to the regular grid X∗ if, for any positive integer K , there exist two positive numbers γ , η such that
− K 2l − k K − 1
⇒
dn,l
γ 2−ηn , − 2
d
n 0,
n+1,k
(19)
where, as before, dn,k = xn,k+1 − xn,k . From (19), the ratio dn,l /dn+1,k with X is very close to 2, the ratio with regular grid X∗ . In this sense, we may regard X as a grid resulting from slight perturbation on X∗ . Lemma 6. (See Maxim and Mazure, 2004.) Suppose that G : R → R is a function meeting the following two requirements: (1) G is Hölder of index α > 1. (2) There exist two positive numbers m, M such that its derivative G satisfies 0 < m G (x) M, x ∈ R. Let xn,k := G −1 (xn∗,k ), n 0, k ∈ Z. Then X = {xn,k | n 0, k ∈ Z} is quasi-regular and it is equivalent to the regular grid X∗ . For example, G (x) = μx + ρ sin(η x) with μ > 0 and constants ρ , η subject to |ρη| < μ. Fig. 1 shows a regular and an irregular (obtained, via G (x) which μ = 1, ρ = 0.5, η = 1) Hermite scheme in action. Starting from a set of Hermite data and grid, subdivision scheme successively refines Hermite data successively. The distinction between a regular and an irregular schemes should be evident from the figures. Lemma 7. Let X = {xn,k | n 0, k ∈ Z} be given as in Lemma 6. Suppose that { Q n } and { Q n∗ } are the associated (scalar) subdivision schemes with HX and HX∗ respectively, as in (6). Then { Q n } is equivalent to { Q n∗ }. Proof. We adopted the notations S n∗ and Q˜ n∗ for the regular grid X∗ . Recall that Q˜ n , associated with HX , are given as in (5).
For our purpose, it suffices to establish that there exist positive numbers α , β , such that Q˜ n k, l = 0, 1. We only consider the case k = l = 1. The arguments for other cases are similar.
(k,l)
By constructions of Q˜ n
(1,1)
I=
dn, j dn+1,i
II = 2
α 2−β n , n 0,
(1,1) (1,1) (i , j ), we rewrite Q˜ n (i , j ) − Q˜ ∗ n (i , j ) = I − II with
∞ (0,0) −2 S n (i , j + k ) ,
∞
k =1
(0,0)
Sn
(i , j + k ) −
k =1
where the equality
∗(k,l)
− Q˜ n
∞
∗(0,0)
Sn
(i , j + k ) ,
k =1 dn∗, j
dn∗+1,i
= 2 for regular grid X∗ is used. (0,0)
(0,0)
It follows from (16) that | S n (i , j )| is up bounded uniformly by some constant. Moreover, S n (2i , j ) = 0 iff i = j, ∞ (0,0) (0,0) and S n (2i + 1, j ) = 0 only if i , j satisfy j − ( N + 1) i j + N. Consequently, | k=1 S n (i , j + k)| is up bounded uniformly by a constant, and is nonzero only if − K 2i − j K − 1, where K is an integer independent of n, i , j. As ∞ (0,0) known in Lemma 6, X and X∗ are equivalent. Therefore (19) holds for those i , j which satisfy k=1 S n (i , j + k) = 0.
Consequently, | I | γ 2−η n , n 0, i , j ∈ Z, for some positive numbers γ , η . As for II, it is not difficult to deduce from the equivalence between X and X∗ that
(0,0)
S n (i , j ) − S n∗(0,0) (i , j ) γ 2−ηn ,
n 0, i , j ∈ Z.
(i , j ) = 0 we have |II| γ 2−η n , n 0, i , j ∈ Z. (k,l) ˜ For any i, there are only finite number of j’s such that Q n (i , j ) = 0. Therefore, the estimations above for I and II, (1,1) ∗(1,1) Q˜ n − Q˜ n α 2−β n , n 0, are as desired. The proof is complete. 2 (0,0)
By the fact that, for any j, there are only finite i such that S n
Theorem 8. Let N 0 be an integer. If the (2N + 2)-point interpolatory Hermite subdivision scheme HX∗ converges, then the (2N + 2)-point Hermite interpolatory subdivision scheme HX also converges, where X is given as in Lemma 6. Proof. We prove first that the associated subdivision scheme {{ Q n }, X } of HX converges. Since X is quasi-regular, we have by (16) a constant C such that
(0,0)
n (0,1)
S n (i , j ) C ,
2 S n (i , j ) C ,
(1,0)
(1,1)
S n (i , j )/2n C ,
S n (i , j ) C ,
n 0, i , j ∈ Z.
(20)
Y. Zhao, D.-R. Chen / Computer Aided Geometric Design 27 (2010) 372–381
379
Fig. 1. Regular and irregular Hermite subdivision.
This gives the boundedness of { Q n } of the associated scalar subdivision scheme of HX . As known, for regular grid, the corresponding difference subdivision scheme { D n∗ } of { Q n∗ } is uniform, i.e., all D n∗ , n 0, are the same and are denoted by D ∗ . If the (2N + 2)-point interpolatory Hermite subdivision scheme HX∗ converges, then it follows from Dubuc (2006), Theorem 14 that D ∗m ∞ < 1 for some m. This is just (14) with K = m and any N 1. It follows from Lemmas 5 and 7 that condition (14) holds for { D n }, the difference subdivision scheme of { Q n }. Consequently, the subdivision scheme {{ Q n }, X } converges by Lemma 4. (0) Second we establish the convergence of { f n (0)}. The convergence of {{ Q n }, X } implies, for any j, both sequences (0) (1) { f n ( j )/dn, j } and { f n ( j )} are bounded. Therefore the equality ( 0)
( 0)
f n+1 (0) − f n (0) = (0)
NS
(0,0)
Sn
( 0) ( 0) (0,1) (1 ) (0, j ) f n ( j ) − f n (0) + S n (0, j ) f n ( j )
j =− N S
(0)
tells f n+1 (0) − f n (0) = O (1/2n ), n → ∞. The proof of the theorem is complete by Theorem 3.
2
(21)
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Y. Zhao, D.-R. Chen / Computer Aided Geometric Design 27 (2010) 372–381
4. L p -convergence We have only considered so far the uniform convergence of subdivision schemes. In this section we will mention briefly their analogy in L p setting. A subdivision scheme {{ S n }, X} converges in L p , if for any f 0 ∈ ( p (Z))2 , there is a vector valued function F ∈ ( L p (R))2 , such that R F n − F p dx → 0 as n → ∞. It is not difficult to see that the condition is equivalent to
p xn, j +1 1 / p 1 F (x) dx dn, j = 0. f n ( j) − dn, j
lim
n→∞
j ∈Z
xn, j
It is essentially the same as in Jia (1995), Micchelli and Sauer (1998), Chen (2003) and Liu and Chen (2005), if X is regular. Clearly the L p -convergence of the associated scalar subdivision scheme { Q n } implies the following condition
lim
n→∞
1 2n
n L −1 2
j =−2n L
p 1 / p xn, j +1
f ( 0) ( j )
1
n
− φ1 (x) dx
= 0,
dn, j
dn, j
∀ L > 0,
(22)
xn, j
provided that a2−n dn, j . This is a generalized version of (12) for p 1. (0)
It is worth pointing out that, with (12) replaced by (22) in the proof of Theorem 3, we can also establish | f n (i ) −
φ0 (xn,i )| → 0. Indeed, the desired result follows from dn, j b2−n and the inequality
xn,i
( 0)
( 0)
f n (i ) − f n (i ) − φ1 (x) dx
0
p 1/ p 2n L −1 xn, j +1 1/q 2n L −1
f n(0) q 1
− φ1 (x) dx
dn, j .
dn, j
dn, j n n j =−2 L
j =−2 L
xn, j
As known in previous results, condition (14) plays an important role. In the study of L p -convergence we’ll use a generalized version of (14) for p ∈ [1, ∞]
∃ integers N˜ , K˜ 0,
˜
s.t. sup D n+ K˜ . . . D n+2 D n+1 p < 2 K / p ,
(23)
˜ n> N
where, as before, { D n } is the difference subdivision scheme of { Q n }. Indeed, it is weaker than (14) because of the following theorem. Theorem 9. Suppose that a scalar subdivision S with { D j } is local and bounded. If (14) holds for some two integers N , K 0, then, ˜ , K˜ 0. for any p 1, (23) also holds for some integers N Proof. Recall that N S and M S are given in Definition 3 for { D j }. For any integers m, n 0, let A m,n = D n+m . . . D n+2 D n+1 and denote A m,n = (am,n (i , j ))i , j ∈Z . It is easily seen by the method of induction that
am,n (i , j ) = 0
⇒
i − 2m j 2m − 1 N S .
(24)
Based on the assumption made in (14), for some K , N, A K ,n ∞ < 1, ∀n > N. A standard argument (Jia, 1995; Maxim and Mazure, 2004) yields that there are a positive number μ < 1 and an integer K 0 such that A m,n ∞ μm , ∀m K , n > N , and, consequently, |am,n (i , j )| μm , ∀m K , n > N, i , j ∈ Z. Since for any j ∈ Z, the number of nonzero entries am,n (i , j ) is less than 2m+1 N S by (24), we have
Am,n 1 = sup
am,n (i , j ) < μm 2m+1 N S ,
j
∀m K , n > N .
i
˜ < 2 and integer K˜ such that A K˜ ,n 1 μ ˜ K˜ , ∀n > N. Appealing to Schur lemma we Therefore, there are positive number μ conclude ˜
˜
˜
˜ K /p , ˜ K /p < μ A K˜ ,n p μ K (1−1/ p ) μ The proof is complete.
∀n > N .
2
Now we establish the analogy of Lemmas 4 and 5 in L p setting.
Y. Zhao, D.-R. Chen / Computer Aided Geometric Design 27 (2010) 372–381
381
Theorem 10. For p ∈ [1, ∞), both Lemmas 4 and 5 hold with (14) replaced by (23). Proof. Following the arguments of Maxim and Mazure (2004), Theorem 3.8, we can easily prove that, the conditions in Lemma 4, with (14) replaced by (23), imply the L p -convergence of { Q n } provided dn, j b2−n . That is the analogy of Lemma 4 in L p setting. For the analogy of Lemma 5, let
B m,n = bm,n (i , j ) i , j ∈Z := D n+m . . . D n+2 D n+1 − D n +m . . . D n +2 D n +1 . Under assumption (1) of Lemma 5, it is obtained in the proof of Maxim and Mazure (2004), Theorem 4.2 that, for any m, there exists a number C m satisfying B m,n ∞ C m 2−β n , ∀n 0. Note that bm,n (i , j ) satisfies (24) with N S re satisfying, placed with max{ N S , N S }. As in the proof Theorem 9, we conclude that, for any m, there exists a number C m −β n ∀n 0, B m,n p C m 2 , which can be arbitrarily small as n → ∞. It together with (23) and D n+m . . . D n+2 D n +1 p D n+m . . . D n+2 D n+1 p + B m,n p implies that there exists N such that
sup D
n> N
n+ K˜
˜ . . . D n +2 D n +1 p < 2 K / p ,
as desired. The proof is complete.
2
At the end, we mention some results concerning the scalar, local and stationary subdivision schemes { Q n }. In this case, there is a finitely supported sequence a = (a(i ))i ∈Z of numbers, called the mask of { Q n }, such that Q n (i , j ) = a(i − 2 j ), n 0, i , j ∈ Z. The difference subdivision scheme { D n } is also local and stationary, i.e., D n (i , j ) = b(i − 2 j ), n 0, i , j ∈ Z, with b a finitely supported sequence of numbers. We’d like list some refined results of Theorem 10 for stationary subdivision schemes with regular grid. 1. Clearly, (23) becomes D m p < 2m/ p for some m. This condition is not only sufficient but also necessary for the L p -convergence (Jia, 1995). 2. The perturbation of a stationary subdivision scheme has been considered in Chen and Plonka (2002); Daubechies and Huang (1995). It is proved in Chen and Plonka (2002) that, for a local, stationary and L p -convergent scheme { Q n } with and stationary subdivision scheme { Q n } with mask a mask a, there is a small positive number δ such that, for any local (2i ) = satisfying a − a p δ , { D n } also converges in L p provided a i i a (2i + 1) = 1. For any f 0 ∈ (Z), denote F a and F a the limit of sequences of functions generated by { Q n } and { Q n } respectively. An interesting estimation
F a − F a p C a − a p f 0 p is established in Chen and Plonka (2002), where C is some constant independent of f 0 . For p = ∞, a weaker version of the estimation is obtained in Daubechies and Huang (1995), where masks are not necessarily finitely supported. The refined results above are actually true in Rs with s 1. However, it is not clear how to generalize Theorem 10 to Rs . References Cavaretta, A.S., Dahmen, W., Micchelli, C.A., 1991. Stationary subdivision. Mem. Amer. Math. Soc. 93 (453). Chen, D.R., 2003. Spectral radii and eigenvalues of subdivision operators. Proc. Amer. Math. Soc. 132, 1113–1123. Chen, D.R., Plonka, G., 2002. Convergence of cascade algorithms in Sobolev spaces for perturbed refinement masks. J. Approx. Theory 119, 133–155. Daubechies, I., Guskov, I., Sweldens, W., 1999. Regularity of irregular subdivision. Constr. Approx. 15, 381–426. Daubechies, I., Guskov, I., Sweldens, W., 2001. Commutation for irregular subdivision. Constr. Approx. 17, 479–514. Daubechies, I., Huang, Y., 1995. How does truncation of the mask affect a refinable function? Constr. Approx. 11, 365–380. Dubuc, S., 2006. Scalar and Hermite subdivision schemes. Appl. Comput. Harmonic Anal. 21, 376–394. Dubuc, S., Merrien, J.-L., 2001. A 4-point Hermite subdivision scheme. In: Mathematical Methods for Curves and Surfaces. Vanderbilt University Press, Nashville, TN, pp. 113–122. Dubuc, S., Merrien, J.-L., 2006. Convergent vector and Hermite subdivision schemes. Constr. Approx. 23, 1–22. Dyn, N., Levin, D., 1995. Analysis of asymptotically equivalentbinary subdivision scheme for curve design. J. Math. Anal. Appl. 193, 594–621. Dyn, N., Levin, D., Gregory, J.A., 1987. A 4-point interpolatory subdivision scheme for curve design. Comput. Aided Geom. Design 4, 257–268. Dyn, N., Levin, D., Gregory, J.A., 1991. Analysis of linear binary subdivision schemes for curve design. Constr. Approx. 7, 127–147. Han, B., Yu, T.P.-Y., Piper, B., 2004. Multivariate refinable Hermite interpolants. Math. Comp. 73, 1913–1935. Han, B., Yu, T.P.-Y., Xue, Y., 2005. Noninterpolatory Hermite subdivision schemes. Math. Comp. 74, 1345–1367. Jia, R.Q., 1995. Subdivision schemes in L p spaces. Adv. Comp. Math. 3, 309–341. Liu, H.Y., Chen, D.R., 2005. Convergence of subdivision schemes and smoothness of limit functions. J. Math. Anal. Appl. 306, 740–751. Maxim, V., Mazure, M.L., 2004. Subdivision schemes and irregular grids. Numer. Algorithms 35, 1–28. Merrien, J.L., 1992. A family of Hermite interpolants by bisection algorithms. Numer. Algorithms 2, 187–200. Micchelli, C.A., Sauer, T., 1998. On vector subdivision. Math. Z. 229, 621–674. Yu, T.P.-Y., 2005. On the regularity analysis of interpolatory Hermite subdivision schemes. J. Math. Anal. Appl. 302, 201–216. Zhou, D.X., 2000. Multiple refinable Hermite interpolants. J. Approx. Theory 102, 46–71.