Smoothness equivalence properties of univariate subdivision schemes and their projection analogues

Transcription

1 Numerische Mathematik manuscript No. (wi be inserted by the editor) Smoothness equivaence properties of univariate subdivision schemes and their projection anaogues Phiipp Grohs TU Graz Institute of Geometry Kopernikusgasse Graz Austria Received: date / Revised version: date Summary We study the foowing modification of a inear subdivision scheme S: Let M be a surface embedded in Eucidean space, and P a smooth projection mapping onto M. Then the P -projection anaogue of S is defined as T := P S. As it turns out, the smoothness of the scheme T is aways at east as high as the smoothness of the underying scheme S or the smoothness of P minus 1, whichever is ower. To prove this we use the method of proximity as introduced by Waner et. a. [10, 9]. Whie smoothness equivaence resuts are aready avaiabe for interpoatory schemes S, this is the first resut that confirms smoothness equivaence properties of arbitrary order for genera non-interpoatory schemes. 1 Introduction The anaysis of noninear subdivision schemes derived from inear schemes has been an active topic in the past few years. The main idea is to modify a inear subdivision scheme which is defined in Eucidean space so as to operate in noninear geometries. In [10,9] it is shown that if the difference between the inear scheme and its noninear modification is sma enough, then the noninear scheme enjoys the same smoothness as the inear scheme. The condition for the noninear scheme to be cose enough to the inear scheme is referred to

2 Phiipp Grohs as proximity condition. It is aso known that for some natura noninear anaogues of inear schemes that operate in noninear geometries, the noninear schemes satisfy proximity conditions with the inear schemes that ensure C smoothness of the noninear scheme, provided the inear scheme is C and some other technica conditions are met [11,6,9]. Up to now this was the strongest resut concerning smoothness equivaence properties of genera noninear schemes. If the underying inear subdivision scheme is interpoatory, then it can be shown that for a very genera famiy of noninear anaogues, the corresponding noninear scheme enjoys the same smoothness as the inear scheme [6,5,14,17]. In the present work we consider the foowing noninear perturbation of a inear subdivision scheme S that operates on a surface M embedded in some Eucidean space: First perform inear subdivision in Eucidean space, and then map the obtained data back onto M via a suitabe projection mapping P. This yieds the so caed P -projection anaogue T := P S of S. We show that if S is of C k smoothness, then so is T. Our resuts confirm a genera smoothness equivaence conjecture [3] for noninterpoatory subdivision schemes. 1.1 Contents The outine of this paper is as foows: After briefy introducing some we known definitions and resuts on inear subdivision schemes in Section, in Section 3 we define the projection anaogue of a inear subdivision scheme and discuss some appications. Section 4 contains the main resut, namey the (rather ong) proof of a genera proximity condition that is satisfied between a inear scheme and its projection anaogue. Finay in Section 5 we use a resut from [9] to show the impication proximity smoothness of the noninear scheme. Linear subdivision We briefy review some we known facts from the theory of inear subdivision schemes. A the materia in this section can be found in [1,4], where ony the case of diation factor is considered. The extension to arbitrary diation factors is easy. Basicay a convergent subdivision scheme takes a sequence of points as input and produces another, more dense, sequence of points. Iterative appication of the subdivision operator yieds a continuous

3 Tite Suppressed Due to Excessive Length 3 curve in the imit. If this subdivision operator is a inear mapping, we speak of inear subdivision. We make this more precise: Let (a i ) i Z be a sequence of rea numbers, ony finitey many of them nonzero. Then we define the inear subdivision operator S associated with this sequence by (Sp) i = a i Nj p j, (1) where p i R m are the input data. The sequence (a i ) i Z is caed the mask of S, the number N > 1 is caed diation factor. S is caed of finite mask, since the support of the sequence (a i ) is finite. A (inear or noninear) subdivision scheme is caed convergent, if the sequences p := S p converge to a continuous curve. This can be made precise as foows: Denote by F (p ) the piecewise inear interpoation of the sequence p on the grid 1 Z. Then S is said to N be convergent, if the sequence F (p ) converges uniformy to a curve F (p)(t). F (p) is caed the imit function of S corresponding to the input data p. If the imit function is C n for a p, we say that S is C n. A natura estimate for the n-th derivative of the function F (p) samped on the grid 1 Z is the eft hand derivative of F N (p ). It is given by N n n S p, where Note that n = n j=0 ( p) i := p i+1 p i. ( ) n ( 1) j E n j = (E I) n, () j with E as the shift operator that maps a sequence (p i ) i Z to the sequence (p i+1 ) i Z. The derived schemes (provided they exist) satisfy the foowing reations: S [0] := S, S [n] := N S [n 1], n 1. Consider the generating function a(x) := a ix i. The Laurent poynomia a(x) is caed the symbo of S. It is not hard to see that the existence of derived schemes up to order n + 1 is equivaent to the fact that a(1) = N and a () (ζ) = 0, = 0,..., n, ζ N = 1 ζ (3) These conditions aso impy that a i Nj = 1 for a i.

4 4 Phiipp Grohs The above property is caed reproduction of constants. In what foows we wi aways assume that (3) hods and thus the derived schemes up to order n + 1 exist. This assumption is actuay not a big restriction: Theorem 1 If a inear subdivision scheme S produces C k imit functions, then the derived schemes S [1],..., S [k+1] exist, if S is nonsinguar, i.e. nonzero sequences get mapped to nonzero functions. A (inear or noninear) subdivision operator operates on the space of sequences p = (p i R m ) i Z. We define a norm on this space via p := sup p i, i Z where refers to any norm on the finite dimensiona vector space R m. Note that this norm is finite ony for a bounded sequence of points. However, we are ony interested in smoothness properties of F (p)(t), which, near a given parameter vaue, depend ony on a finite number of points of the initia sequence p. This is we known and foows from the fact that S is of finite mask. Therefore we can without oss of generaity assume that p is bounded. We can even assume that p is ony a finite sequence. The foowing theorem is we known. Theorem Let S be a inear subdivision scheme such that the derived schemes up to order n + 1 exist. S produces C n functions if and ony if µ j := 1 N ρ(s[j+1] ) < 1 for a j = 0,..., n. (4) Here ρ(s [j+1] ) means the spectra radius of the operator S [j+1] with respect to the norm. Actuay for Theorem to be vaid it is necessary and sufficient that (4) hods ony for j = n. If the subdivision scheme produces smooth imit functions, there is a better estimate for the constants µ j : Theorem 3 Under the assumptions of Theorem µ j = 1 N for j = 0,..., n 1. Proof It is easy to show that under the assumptions of Theorem, the derived schemes S [j] converge to the j-th derivative of the imit function of S for j = 1,..., n (c.f. [4]). Therefore there exists a constant A such that (S [j] ) A for Z + and a j = 1,..., n. It

5 Tite Suppressed Due to Excessive Length 5 foows that the spectra radius ρ(s [j+1] ) 1 for j = 1,..., n. Since Sp = p for a constant sequences p, we get ρ(s [j+1] ) = 1. We give an exampe: The Lane-Riesenfed subdivision scheme of order n, which produces B-spine curves of degree n, is given by the symbo a(x) = (1 + x xn 1 ) n+1 (Nx) n. It satisfies (4) with µ j = 1 N, j = 0,..., n. By Theorem, the Lane- Riesenfed scheme is C n. It can be shown that µ j = 1 N is the best possibe decay rate. 3 The Projection anaogue The projection anaogue of a inear subdivision scheme S is defined by first appying a inear subdivision scheme S foowed by a projection mapping P. If inear subdivision is appied to data contained in a surface in some vector space R m, the resut is in genera not contained in that surface. If we combine inear subdivision with a projection-type mapping onto that surface, we obtain a refinement operator which creates data contained in the surface again. Such a projection mapping does not have to be gobay defined, because we can reasonaby assume that inear subdivision wi create data which ie not too far away from the surface. Definition 1 Let P : O R m R m, O open in R m, be a smooth mapping with the property that P ran(p ) = id and there exists an ε- neighborhood of ran(p ) contained in P s domain of definition, O. Then P is caed a Projection mapping. The P -projection anaogue is defined by first appying the subdivision operator S to the points (p i ran(p )) i Z and then projecting onto the range of P via a projection mapping P. In other words, the P -projection anaogue T of S is the subdivision scheme defined by P (Sp), p = (p i ran(p )) i Z. (5) We woud ike to give some exampes of what the mapping P might ook ike (compare aso [1, 10]):

6 6 Phiipp Grohs 3.1 Subdivision on surfaces If S R m is a surface embedded into Eucidean space R m, then it is possibe to define a subdivision scheme that operates in S: First we need a projection mapping P : O R m S, where O is an open set with S O. This mapping can be reaized e.g. as a cosest point projection, or if the surface is given as a eve set f(x 1,..., x n ) = c, we can et P equa the gradient fow. Then the P -projection anaogue of any inear subdivision scheme operates in S. 3. Avoiding obstaces Another appication is the foowing: Assume that we are subdividing in Eucidean space R m. For certain appications it may be necessary for the subdivided curve to avoid an obstace O R m. We define P as foows: If a point p R m is not in O, then P is just the identity mapping. If p O, then P projects p onto the boundary of O. O can for exampe be the interior of a cosed surface S R m. Then the mapping P acts as the projection on S on the points in the interior of S and as the identity mapping on a the other points. Note that the mapping P is ony continuous. Nevertheess the P - projection anaogue T fufis the owest order proximity condition and imit curves enjoy C 1 smoothness [1]. 3.3 Lie groups We show how to define a projection mapping onto a compact subgroup G of the orthogona group O m. The scaar product x, y := trace(x T y) corresponding to the Frobenius norm is both right- and eft invariant []. We can thus define P as the cosest-point projection onto G with respect to the Frobenius norm. We can extend this construction to the case that G is any compact matrix group, because after a suitabe change of coordinates, we can think of G as a compact subgroup of O m []. Now et us define a semidirect product G R m, where G is a compact matrix group. After choosing suitabe coordinates, we can assume that G is a subgroup of O m. We can denote eements of G R m by (m + 1) (m + 1)-matrices of the form ( 1 0 t g ),

7 Tite Suppressed Due to Excessive Length 7 with t R m and g G. We consider the group operation ( ) ( ) ( ) :=. t 1 g 1 t g t 1 + g 1 t g 1 g If G is equa to SO m, then G R m is just the Eucidean motion group SE m. A short computation shows that the Frobenius norm on G R m induced by the Frobenius norm on R (m+1) (m+1) is right invariant, but not eft invariant. However, this does not prevent us from defining P as the cosest-point projection onto G R m with respect to the Frobenius norm. If a matrix of the form ( 1 0 t h ), h GL m is given, then its cosest-point projection onto G R m is given by ( ) 1 0, g = P (h), t g where P is now the cosest point projection onto G. If G = O m, we can compute the cosest point projection expicity: et h GL m, with the singuar vaue decomposition h = v T Σw, where v, w O m. The cosest point projection onto O m of h is now given by P (h) = v T w [8]. A different kind of projection anaogue has been studied in [1, 10,9], where smoothness equivaence resuts up to C have been obtained. In [14] smoothness equivaence has been proven for interpoatory subdivision schemes if P is the cosest point projection onto the sphere and reated manifods. In [6] this resut is extended to arbitrary mappings P and mutivariate subdivision schemes. For noninterpoatory subdivision schemes, however, the resut in the present paper is the first one that estabishes smoothness equivaence for arbitrary smoothness order. 4 Estabishing a proximity condition Our anaysis considers the noninear scheme T as a perturbation of the inear scheme S. In this spirit we define Definition A inear scheme S and its P -projection anaogue T satisfy a oca proximity condition of order n if for every compact set K ran(p ) there exist constants δ > 0, C such that for a initia data p K Z with p < δ we have the estimate n 1 (T p Sp) C p α 1... n p αn. (6) α 1,...,αn Z + 0 α 1 +α +...+nα n=n+1

8 8 Phiipp Grohs The goa of the present section is to prove the foowing theorem: Theorem 4 Let S be a inear subdivision scheme such that the derived schemes S [j], j = 1,..., n + 1 exist, and et P : O R m be a C n+1 projection mapping. Then S satisfies a oca proximity condition of order n with its P -projection anaogue T The noninear scheme T is ony defined if the subdivided data Sp ies within O. This can aways be achieved by choosing initia data p with p sma: Lemma 1 Let O be such that an ε neighborhood of ran(p ) is sti contained in O. Then there exists δ > 0 such that for initia data p with p δ, the subdivided data Sp remains within O. The constant δ depends on S and on ε but not on p. Proof It is we known (compare [4]) that for a inear subdivision scheme S there exists a constant C > 0 such that This impies our statement. sup inf Sp i p j C p. i Z The proof of Theorem 4 takes up the whoe section. In the first subsection we derive a tractabe condition for a proximity condition to hod. In the second subsection we prove that this condition is aways fufied if the derived schemes up to order n + 1 exist. 4.1 Expressing the differences through a generating function In order to obtain a proximity condition as required by Theorem 4, we first ook for a way to rewrite the difference n 1 (Sp T p). At the end of this section we wi see that a proximity condition hods if a certain generating function has many vanishing derivatives (compare aso [5,7,6]). The checking of this second fact then is the topic of Section 4.3. We start with the foowing emma: Lemma Let S be a subdivision scheme with mask (a i ) i Z and T its P -projection anaogue. Let K ran(p ) be a compact set. Then there exists δ > 0 (depending ony on K) such that for a initia data (p i ) i Z K Z with p < δ and v j := p j p i n n 1 1 (T p Sp) i =! B,i + O( p n+1 ), (7) =

9 Tite Suppressed Due to Excessive Length 9 where and b,j = B,i = j=(j 1,...,j ) Z ( n 1 b,j ) 0 d P pi (v j1,..., v j ) (8) { (a i+h Nj1 a i+h Nj1 ) h Z if j 1 =... = j, (a i+h Nj1... a i+h Nj ) h Z ese. The n 1 operator acts on the index h. Proof The proof is done by Tayor expansion. Let a(x) be the symbo of S and T its projection anaogue, i.e. T p = P (Sp) with p ran(p ). Ceary, Sp = S(P (p)). Therefore we can write (9) (T p Sp) i+h = (P (Sp) S(P (p))) i+h (10) = P ( a i+h Nj p j ) a i+h Nj P (p j ), h = 0,..., N 1. By compactness of K, there exists δ > 0 such that for a v with v δ and a p K we can write P in its Tayor expansion (reca that we assumed that P C n+1 ) as P (p + v) = n =0 1! d P p (v,..., v) + O( v n+1 ). (11) Since the mask of S is finite, ony a finite number (depending ony on the support of the mask) of initia data points contributes to the computation of (Sp) i+h and (T p) i+h, h = 0,..., N 1. Therefore we may assume that p is ony a finite sequence and thus sup j p j p i = O( p ) and aso max h=0,...,n 1 j a i+h Nj(p i p j ) = O( p ). It foows that there exists δ > 0 such that for a sequences p K Z with p < δ we can write p j = P (p j ) = and thus n =0 (Sp) i+h = a i+h Nj p j = n = 1! d P pi (p j p i,..., p j p i ) + O( p j p i n+1 ), n a i+h Nj =0 =0 1! d P pi (p j p i,..., p j p i ) + O( v n+1 ) a i+h Nj 1! d P pi (v j,..., v j ) + O( v n+1 ), (1)

10 10 Phiipp Grohs with v j := p j p i and v = (v j ). The noninear scheme can be written as (T p) i+h = P ( = n =0 a i+h Nj p j ) = P (p i + a i+h Nj v j ) 1! d P pi ( a i+h Nj v j,..., a i+h Nj v j ) (13) (reca that a i+h Nj = 1 and use (11)). For = 0,..., n we consider the expression d P pi ( a i+h Nj v j,..., a i+h Nj v j ) a i+h Nj d P pi (v j,..., v j ). (14) Observe that by the mutiinearity of d P pi the expression (14) equas (b,j ) h d P pi (v j1,..., v j ). With (1) and (13), (10) becomes (T p Sp) i+h = n =0 1 (b,j ) h d P pi (v j1,..., v j ) + O( p n+1 ).! It is easiy seen from the reproduction of constants that the terms corresponding to = 0, 1 vanish. We therefore have (T p Sp) i+h = n = 1 (b,j ) h d P pi (v j1,..., v j ) + O( p n+1 ).! It remains to appy the inear operator n 1 acting on the index h to arrive at the desired resut. We show how to estabish a proximity condition by rewriting the expressions B,i. Lemma 3 Let the assumptions be as in Lemma. If for any =,..., n, i Z it is possibe to express B,i in the form c j,β d P pi ( β1 v j1,..., β v j ) with (15) j=(j 1,...,j ) Z β Γ Γ := {(β 1,..., β ) Z + : i=1 β i = n + 1} and mutivariate, finite sequences c j,β, then the P -projection anaogue T of S and S satisfy a oca proximity condition of order n in the sense of Definition.

11 Tite Suppressed Due to Excessive Length 11 Proof Since the sequence p is K-vaued and K is compact, there exists a uniform constant C 1 such that d P pi (v 1,..., v ) C 1 v 1... v for a vectors v 1,..., v and i Z. With it foows that C,β := {j : c j,β 0} max c j,β j (15) β Γ C 1 C,β β 1 v... β v, v being the sequence (p j p i ). Since, v = O( p ), r v = O( r p ) and n+1 p n p, there exists a constant C 3 such that (15) C 3 β 1 p... β p. β Γ,β k n k By setting α r := {j : β j = r}, we obtain an estimate of the form (15) α1 C 4... n p αn. n+1 r=1 rαr=n+1 p By Lemma 11 we get for any i a constant such that n 1 p i α1 C i... n p αn. n+1 r=1 rαr=n+1 p By shift-invariance it suffices to ony consider the vaues i = 0,..., N 1. This proves the statement. Our goa is to rewrite B,i in the form (15). Note that this is a purey agebraic probem. We first derive a necessary and sufficient condition for such a rewriting rue to exist. To do this we make some definitions. With v j := p j p i et P := { e j d P pi (v j1,..., v j ) : e j = 0 for amost a j}. P is the set of a forma expressions of the form (8). Of course B,i is in P. Define a mapping { P R[x 1, x , x, x 1 Φ : ] e jd P pi (v j1,..., v j ) e jx j x j.

12 1 Phiipp Grohs We impose a ring structure on P such that Φ is a ring isomorphism. Mutipication by (1 x r ) α in the space R[x 1, x , x, x 1 ] corresponds to the mapping e j d P pi (v j1,..., v j ) e j d P pi (v j1,..., α v jr,..., v j ), as seen from (). It foows that a rewriting rue of the form (15) exists if and ony if the generating function B,i (x 1,..., x ) = Φ(B,i ) can be written in the form c β (x 1,..., x )(1 x 1 ) β 1... (1 x ) β, (16) β β =n+1 for some Laurent poynomias c β R[x 1, x 1 1,..., x, x 1 ]. Appying Tayor s formua yieds a necessary and sufficient condition for (15), namey that a derivatives of B,i (x 1,..., x ) of order n are equa to zero for (x 1,..., x ) = (1,..., 1). We summarize: Lemma 4 If a derivatives of order n of the Laurent poynomias B,i (x 1,..., x ) = Φ(B,i ), =,..., n vanish at the point (1,..., 1), then S and T satisfy a oca proximity condition of order n. We have B,i (x 1,..., x ) = (17) n 1 (( a i+h Nj x j 1 )... ( a i+h Nj x j ) ( a i+h Nj x j 1... xj )) 0. Proof By the previous discussion it foows that the vanishing of derivatives of the Laurent poynomias B,i (x 1,..., x ) impies that a rewriting rue of the form (15) for the expressions B,i exists. If we appy Lemma 3 we get the first caim that a oca proximity condition of order n hods between S and T. What remains to show is that (17) hods. Thus, we have to verify that Φ(B,i ) = n 1 (( a i+h Nj x j 1 )... ( a i+h Nj x j ) ( a i+h Nj x j 1... xj )) 0. This foows from (9) and the fact that Φ( E h ) = Φ(E h ) for every E h P. Again, we et act on the index h.

13 Tite Suppressed Due to Excessive Length Combinatorics of generating functions The goa of this section is to verify the condition given in Lemma 4. This wi be done by a series of emmas which are subsequenty used in the proof of Theorem 4 in Section 4.3. We woud ike to stress that this section may be read without any connection to noninear subdivision. It contains purey agebraic resuts for the symbos of inear schemes which are in our opinion interesting in their own right. It is easy to derive the foowing identity (ζ is a primitive N-th root of the unit): q k (x) := a k Nj x k Nj = 1 N 1 N (a(x) + r=1 ζ kr a(ζ r x)). (18) It wi turn out that the crux of the proof is to show that a derivatives of B,i (x 1,..., x ) vanish. First we ook at the moments of the sequences (a k+nj ). To do that, we use a we known fact from the theory of generating functions [13]: Lemma 5 Let q(x) := q jx j be the generating function of the sequence (q j ). Then the generating function of the sequence (j q j ) is given by (x x ) q(x). (19) From (18) and Lemma 5 we can deduce the foowing identities for the first two moments of (a k+nj ) : N N 3 ja k Nj = kn a (1) (1) (0) j a k Nj = k N (k 1)a (1) (1) + a () (1). (1) Lemma 6 Let a(x) be a Laurent poynomia satisfying (3) and define p r (k) := jr a k Nj. Then the foowing hods: p r (k) is a poynomia of degree r in k for 0 r n. There exist constants C r with r ( ) r ( N) n p r (k) = i ( N) r p r (k) + C r. () =1 The eading terms of p r (k) are given by 1 N r kr and r N r+1 a(1) (1)k r 1. (3)

14 14 Phiipp Grohs Proof In view of (18) and Lemma 5, the foowing hods: (k Nj) r a k Nj = (x x )r ( 1 N N (a(x) + ζ kr a(ζ r x))). (4) x=1 From the assumption that the derived schemes of orders up to n + 1 exist, it foows by (3) that the right hand side of (4) is a constant, i.e., does not depend on k. We denote it by C r. The binomia formua shows that r ( ) r ( N) r p r (k) = k ( N) r p r (k) + C r. (5) =1 It remains to show the statement about the two eading terms of p r. We use induction on r, the cases r = 1, foowing from (0), (1). 1 Let us first show that the eading coefficient of p r (k) equas N. By r the induction hypothesis, this hods for a p r (k), 1. Thus the eading coefficient of p r (k) is given by ( N) n r =1 r=1 ( ) r ( N) r N (r ) = 1 N r ( 1)r+1 =0 r =1 ( ) r ( 1) r. (6) It is not hard to see that (6) equas 1 N : Use the formua r r ( ) r x r = (x 1 + 1) r = (x 1) r, (7) and substitute x = 0. This gives r ( r =1 ) ( 1) r = ( 1) r = ( 1) r+1. Putting this into (6) gives the desired resut. It remains to prove that the term of order r 1 in p r (k) is given by r a (1) (1)k r 1 : N r+1 We start with (5) and extract the terms of order r 1. Using our induction assumption we come up with r ( ) r ( N) r a (1) r (r ) (1) ( N) N r +1. =1 If we differentiate (7) and et x = 0 (assuming that r ), we see that r ( ) r (r )( 1) r = r( 1) r+1. =1 Now the proof is compete.

15 Tite Suppressed Due to Excessive Length 15 Remark 1 The eft hand side of (3) is actuay we known: If the derived schemes up to order n + 1 exist, then the subdivision scheme S generates poynomias up to degree n. p r (k) can be viewed as taking the integer sampings of the poynomia k r as input and appying the inear subdivision scheme S. From the poynomia generation property of S it immediatey foows that p r (k) is a poynomia, and it is aso we known that the term of degree r is aso reproduced [1]. Remark It woud not be difficut to make the constants C r expicit using Stiring numbers (which describe the base change from the monomia basis to the basis consisting of poynomias Q u (x) := x(x 1)... (x u + 1) cf. [13]). We do not need this for our anaysis. The next we known emma describes some eementary properties of the Stiring numbers S j i. We wi use this resut ater. The proof is an easy exercise. Lemma 7 Let Q u (x) = x(x 1)... (x u + 1). Then Q u (x) = S u ux + S u u 1x u S u 0, with S u u = 1 and S u u 1 = u(u 1). We continue to coect emmas which contribute to our fina proximity resut. Lemma 8 Let a(x) be a Laurent poynomia with (3) and u u = n. Define E(k) := ( Q u1 (j)a k Nj )... ( Q u (j)a k Nj ) and (8) F (k) := Q u1 (j)... Q u (j)a k Nj. (9) Then E(k) and F (k) are poynomias of degree n in k. The terms of degree n and n 1 of E(k) and F (k) agree. Proof The fact that E(k) and F (k) are poynomias of degree n is a direct consequence of Lemma 6. From Lemma 7, we obtain F (k) = (30) ( j u 1 u 1(u 1 1) ) ( j u j u u (u 1) ) j u a k Nj,

16 16 Phiipp Grohs where the dots indicate terms of ower order which do not contribute to the two eading terms of F (k). Mutipying the poynomias in (30) yieds F (k) = ( ( j n u1 (u 1 1) It foows that ( u1 (u 1 1) F (k) = p n (k) u (u 1) ) ) j n a k Nj u (u 1) ) p n 1 (k) + s(k), where s(k) is a poynomia of degree n in k. Now we use Lemma 6 to concude that F (k) = 1 ( (u1 (u 1 1) u (u 1) N n kn (31) ) 1 N n 1 + n ) N n+1 a(1) (1) k n , where the dots indicate terms of order n. We have to check that E(k) is of the same form: Using Lemma 7, E(k) can be written as It foows that ( r=1 E(k) = j ur u r(u r 1) ( r=1 p ur (k) u r(u r 1) ) j ur a k Nj. ) p ur 1(k) If we use (3), we see that aso E(k) has the form (31) and the proof is compete. 4.3 Proof of the proximity condition Now we are ready to sum up the proof of Theorem 4. Proof (of Theorem 4) In view of Lemma 4 we need to verify that if we appy a differentia operator D := n ( x 1 ) u 1...( x ) u, u u = n to ( a i+h Nj x j 1 )... ( a i+h Nj x j ) ( a i+h Nj x j 1... xj ), (3) then appying the operator n 1 in h and evauating at (x 1,..., x ) = (1,..., 1) yieds zero for = 1,..., n. In the notation of Lemma 8, the

17 Tite Suppressed Due to Excessive Length 17 expression we get if we appy the operator D to (3), and evauate at (x 1,..., x ) = (1,..., 1) is T (i + h) := F (i + h) E(i + h). From Lemma 8 and the fact that the derived schemes of S of order n + 1 exist we know that T (i + h) is a poynomia of degree n in i + h. The operator n 1 annihiates a poynomias of degree ess than n 1, and thus ( n 1 T (i + h)) h=0 = 0. Therefore the condition (17) is satisfied and we can use Lemma 4. This competes the proof. 5 Smoothness equivaence Having estabished a genera proximity condition in the previous section, we now draw on resuts from [9] to show smoothness equivaence between a inear scheme and its projection anaogue. Theorem 5 Let S be a inear subdivision scheme with derived schemes S [1],..., S [n+1] and P a C n+1 projection mapping. Then the P -projection anaogue T of S produces C n imit curves for a initia data p = (p i ) i Z such that T p converges. Proof Since the mask of S is finite, the imit function F (p) ocay depends ony on a finite number of initia data points p i. It is therefore no restriction to assume that p takes its vaues in a compact set K. By the assumption that T is convergent, we may aso without oss of generaity assume that p < δ for any δ > 0. From Theorem 4 we get a proximity condition of the form n 1 (S T )p C α nα n=n+1 p α 1... n p αn which hods on compact sets and initia data p with p sma enough. Define µ i := 1 N ρ(s[i] ) < 1, i = 1,..., n + 1. The resuts of [9] say that if the contractivity constants µ 1,..., µ n < 1 satisfy µ n (µ 1 ) α 1 ( µ N )α... ( N n 1 )αn < 1 for a n-tupes (α 1,..., α n ) with α nα n = n+1, then T is C n. By Theorem 3 this is aways satisfied and that concudes the proof. N n

18 18 Phiipp Grohs 5.1 Discussion The statement such that T p converges in Theorem 5 means that in genera we have to start with a sequence p with δ := p sma. From the resuts in [10] it foows that we can choose δ so that 1+Cδ < 1, where C is chosen such that Sp T p C p. The constant δ is typicay rather sma (compare aso the discussion in [11, 10]). Ony for specia schemes and certain noninear anaogues (geodesic averaging) it is possibe to show that the noninear scheme converges for a initia data [15]. It has been reported [17] that it is surprising that the smoothness equivaence can be derived soey by agebraic manipuations of the mask coefficients, i.e. the smoothness of the noninear subdivision scheme does not depend on the amount of noninearity. For the present paper, this is not entirey true in the foowing sense: For any noninear anaogue smoothness equivaence is true for sequences p with δ := p such that 1 + Cδ < 1. The constant C comes from the proximity condition and is directy reated to the noninearity of the subdivision scheme. Even if this bound is in a probabiity far from sharp, it indicates that the more noninear our subdivision scheme is, the more dense our initia point sequence has to be. There exist resuts where the smoothness of a noninear subdivision scheme depends on the initia data [16]. These noninear schemes are however quite different from the ones considered here, in that they are not constructed from inear schemes. 6 Acknowedgments The author woud ike to thank Nira Dyn, Tomas Sauer and Johannes Waner for hepfu discussions. This research has been supported by the Austrian Science Fund (FWF) under grant P References 1. C. A. Micchei A. Cavaretta and W. Dahmen, Stationary subdivision, American Mathematica Society, Boston, MA, USA, D. Bump, Lie Groups, Springer, D. L. Donoho, Waveet-type representation of Lie-vaued data, 001, Tak at the IMI Approximation and Computation meeting, May 1 17, 001, Chareston, South Caroina. 4. N. Dyn, Subdivision schemes in computer-aided geometric design, Advances in Numerica Anaysis Vo. II (W. A. Light, ed.), Oxford University Press, Oxford, 199, pp

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