Hermite subdivision on manifolds via parallel transport

Transcription

1 Adances in Comutational Mathematics manuscrit No. (will be inserted by the editor Hermite subdiision on manifolds ia arallel transort Caroline Moosmüller Receied: date / Acceted: date Abstract We roose a new adation of linear Hermite subdiision schemes to the manifold setting. Our construction is intrinsic, as it is based solely on geodesics and on the arallel transort oerator of the manifold. The resulting nonlinear Hermite subdiision schemes are analyzed with resect to conergence and C 1 smoothness. Similar to reious work on manifold-alued subdiision, this analysis is carried out by roing that a so-called roximity condition is fullfilled. This condition allows to conclude conergence and smoothness roerties of the manifold-alued scheme from its linear counterart, roided that the inut data are dense enough. Therefore the main art of this aer is concerned with showing that our nonlinear Hermite scheme is close enough, i.e., in roximity, to the linear scheme it is deried from. Keywords Hermite subdiision manifolds subdiision C 1 analysis roximity Mathematics Subect Classification ( A 65D17 53A99 1 Introduction Hermite subdiision is an iteratie method for constructing a cure together with its deriaties from discrete oint-ector data. It has mainly been studied in the linear setting, where many results concerning conergence and smoothness are aailable, such as Dyn and Lein (1995, 1999; Dubuc and Merrien (2005; Dubuc (2006; Dubuc and Merrien (2009; Merrien and Sauer (2012 and others. In a recent aer (Moosmüller 2016 we roose an analogue of linear Hermite schemes in manifolds which are equied with an exonential ma. This The author gratefully acknowledges suort by the doctoral rogram Discrete Mathematics, funded by the Austrian Science Fund FWF under grant agreement W1230. Caroline Moosmüller Institut für Geometrie, TU Graz Koernikusgasse 24, 8010 Graz, Austria Tel.: moosmueller@tugraz.at

2 2 Caroline Moosmüller construction works ia conersion of ector data to oint data, and makes use of the well-established methods of non-hermite subdiision in manifold, see Grohs (2010 for an oeriew. The resent aer inestigates manifold analogues of Hermite subdiision rules which work directly with ectors and emloy the arallel transort oerators aailable in Riemannian manifolds and also in Lie grous. In this way subdiision works directly on Hermite data in an intrinsic way. The C 1 conergence analysis of the nonlinear schemes we obtain by the arallel transort aroach is roided from their linear counterarts by means of a roximity condition for Hermite schemes introduced by Moosmüller (2016. This condition allows to conclude C 1 conergence of the manifold-alued scheme if it is close enough to a C 1 conergent linear one. Similar to most reious results on manifold subdiision, C 1 conergence can only be deduced if the inut data are dense enough. The aer is organized as follows: In Section 2 we recaitulate Hermite subdiision on both linear saces and manifolds. Section 3 discusses arallel transort and geodesics, which we use in Section 4 to define the arallel transort analogue of a linear Hermite scheme. The main art of this aer is concerned with roing that the roximity condition holds between the arallel transort Hermite scheme and the linear scheme it is deried from (Section 5. The results are then stated in Section 6. Throughout this aer we use as an instructie examle a certain non-interolatory Hermite scheme which is the de Rham transform (Dubuc and Merrien 2008 of a scheme roosed by Merrien ( Hermite subdiision: Basic concets In this section we recall some known facts about linear Hermite subdiision and introduce a generalized concet of Hermite subdiision for manifold-alued data. 2.1 Linear Hermite subdiision The data to be refined by a linear Hermite subdiision scheme consists of a ointector sequence, where we assume that both oint and ector sequence take alues in the same finite dimensional ector sace V. The sace of all such oint-ector sequences is denoted by l(v 2, and an element of this sace is written as (. A linear subdiision oerator S A is a ma l(v 2 l(v 2, which is defined by S A ( i = Z A i 2 (, i Z, ( l(v 2, (1 where the finitely suorted sequence A l(l(v 2 2 is called mask. A linear Hermite subdiision scheme is the rocedure of constructing ( 1, ( 1 2 (,... from inut data 0 2 l(v 2 by the rule 0 D n( n n = S n A ( 0 0,

3 Hermite subdiision on manifolds ia arallel transort 3 where D L(V 2 2 is the block-diagonal dilation oerator ( 1 0 D =. Here a constant c is to be understood as c id V. A linear Hermite subdiision oerator or scheme is called interolatory if its mask satisfies A 0 = D and A 2i = 0 for all i Z\0. We always assume a linear Hermite scheme to satisfy the sectral condition, which is a useful assumtion for the analysis of linear Hermite schemes (Dubuc and Merrien 2005; Dubuc 2006; Dubuc and Merrien 2009; Merrien and Sauer We require that u to a arameter shift the subdiision oerator reroduces a degree 1 olynomial and its deriatie ( + iw for, w V. w i Z To be recise, we require that there exists ϕ R such that ( ( + (i + ϕw + i+ϕ S A = 2 w 1 w i Z 2 w, i Z for all, w V. This condition is equialent to the requirement that there exists ϕ R such that the constant sequence c i = ( 0 and the linear sequence li = for i Z, V resectiely obey the rules ( (i+ϕ S A c = c and S A l = 1 l. (2 2 The sectral condition can also be exressed by means of the mask A = ( a b c d. It is equialent to a i 2 = 1, c i 2 = 0, (3 Z Z Z a i 2 + b i 2 = 1 (i ϕ, 2 Z c i 2 + d i 2 = 1 2, (4 for all i Z and some ϕ R, which indicates the arameter transform. Equation (3 is equialent to the reroduction of constants, whereas (4 exresses the reroduction of linear functions. 2.2 Hermite subdiision on manifolds We would like to consider Hermite subdiision in the more general setting of manifolds. In this context, tangent ectors sere as oint-ector inut data for Hermite subdiision. Therefore, the inut data is samled from the tangent bundle T M = x M T xm of a manifold M. Its associated sequence sace is denoted by l(t M. In order to retain the analogy to the linear case, an element of l(t M is written as a air ( consisting of an M-alued oint sequence and a ector sequence which takes alues in the aroriate tangent sace, i.e., i T i M for i Z. A subdiision oerator U on T M is a ma that takes arguments in l(t M and roduces again a oint-ector sequence. It must satisfy

4 4 Caroline Moosmüller (i L 2 U = UL, where L is the left shift oerator, and (ii U has comact suort, that is, there exists N N such that both U ( 2i and U ( only deend on ( i N ( i+n 2i+1 i N,..., i+n for all i Z and sequences (. Let D : l(t M l(t M be the dilation oerator ( ( 1 2, which is an analogue of the block-diagonal oerator D defined in Section 2.1. An Hermite subdiision scheme is the rocedure of constructing ( ( 1, 2 1,... 2 from inut data ( 0 l(t M by the rule 0 D n( n n = U n ( 0 0. An Hermite subdiision oerator or scheme is called interolatory if U ( 2i = D ( i for all ( and i Z. Note that these definitions are direct generalizations of the concets introduced in Section 2.1: Eery linear subdiision oerator satisfies conditions (i and (ii from aboe. If U is linear then the definition of (interolatory Hermite subdiision scheme is equialent to the one gien in Section C 1 conergence To a sequence n of oints in a ector sace we associate a cure F n ( n, which is the iecewise linear interolant of n on the grid 2 n Z. We say that a oint-ector sequence ( n is C 1 conergent, if F n n ( n res. F n ( n conerge uniformly on comact interals to a continuously differentiable cure res. its deriatie. If the oint-ector sequence is manifold-alued, then we require that the aboe is true in a chart. A Hermite scheme defined by the subdiision oerator U is said to be C 1 conergent, if the oint-ector sequence ( n is C 1 conergent. n constructed ia D n ( n n = U n ( 0 Examle 1 We consider the de Rham transform (Dubuc and Merrien 2008 of one of the interolatory linear Hermite schemes introduced by Merrien (1992. It is a non-interolatory scheme with mask A 2 = 1 8 A 0 = 1 8 ( ( , A 1 = 1 8, A 1 = 1 8 ( ( In Dubuc and Merrien (2008 it is shown that the sectral condition is satisfied and that this scheme is C 1 conergent , 0

5 Hermite subdiision on manifolds ia arallel transort 5 Fig. 1: The linear non-interolatory Hermite subdiision scheme of Examle 1. Left: Inut data and second iteration ste. Right: Inut data and limit cure. 3 Parallel transort and geodesics Using arallel transort and geodesics, we are going to adat linear Hermite subdiision to work on manifold data. We here discuss these concets for submanifolds of Euclidean sace (i.e., surfaces and for matrix grous, een though they belong to the more general classes of Riemannian manifolds res. Lie grous. The reason is that the conergence and smoothness analysis resented in Section 5 can be reduced to the secial cases of surfaces and matrix grous. 3.1 Surfaces On a surface M in R n we consider ector fields V (t along a cure g(t, i.e., we require that V (t T g(t M for all t. We say that such a ector field V is arallel along g if its deriatie is orthogonal to M. Equialently, the roection of V to the tangent sace T g(t M anishes for all t, i.e. DV dt := ( V tang = 0. (5 Therefore, arallel ector fields are the solutions of the linear differential equation (5. Let the cure g connect the oints and m on M, i.e., g(0 = and g(1 = m. The arallel transort along g, denoted by P m : T M T m M, is defined as follows: P m ( means V (1, where V is the arallel ector field along g with initial alue V (0 =. Parallel transort along g satisfies P m q P q = P m, (6 where q is any oint on the cure. Furthermore, it is an isometry, that is P m ( =. This is not difficult to show: For two ector fields V, W along g a roduct rule holds: d DV V, W = dt dt, W + V, DW. (7 dt

6 6 Caroline Moosmüller If V is arallel along g, then (7 imlies that d dt V (t, V (t = 0, i.e., V (t is constant for all t. So P m is an isometry. In addition to arallel transort, we need the concet of geodesics. A geodesic is a cure g on M such that ġ is arallel along g, i.e., a cure which satisfies the differential equation Dġ dt = 0. It is useful to exress geodesics by means of the exonential maing, which is defined as follows: ex ( means g(1, where g is the geodesic starting at the oint with tangent ector. A geodesic g can then be written as g(t = ex (t. We mention that D dt, arallel transort, geodesics and exonential maing are actually concets of Riemannian geometry. Here they are described only for the secial case of surfaces in Euclidean sace. For details we refer to textbooks on differential geometry, e.g. do Carmo ( Matrix grous This section discusses arallel transort and geodesics in matrix grous, i.e., subgrous of GL(n, R. We use the matrix exonential function ex( = 1 k=0 k! k to define an exonential maing by ex ( = ex( 1. Then a geodesic 1 g starting at the oint and tangent ector is defined by g(t = ex (t. (8 The cure g(t is a left translate of the 1-arameter subgrou ex(t 1, and it is also a right translate, since ex( 1 = ex( 1. We define three different arallel transorts + P m, P m and 0 P m on G, which are maings of T G to T m G. The first two are gien by left res. right multilication, that is + m P ( = m 1 and P m ( = 1 m. Let g(t = ex (t be the geodesic connecting and m. Denote by µ,m the geodesic midoint of and m, i.e., µ,m = g( 1 2. Then the third kind of arallel transort is defined by 0 m P ( = µ,m 1 1 µ,m. (9 Therefore, as in the Riemannian case, an exonential maing, geodesics and arallel transort can be defined in matrix grous. 2 1 We call these cures geodesics to emhasize the analogy to the Riemannian case. Note that in the grou case we define geodesics ia the exonential ma, but in the Riemannian case, we define the exonential ma ia geodesics. 2 In fact a more general statement is true, which also gies a connection to the case of D Riemannian manifolds: On Lie grous, three oerators +, D 0 D and can be defined, dt dt dt which ma a ector field along a cure to another ector field along the same cure. They all define the same geodesics, namely (8 and induce the three arallel transorts from aboe. While + P m m and P are indeendent of the cure connecting and m, Definition (9 is only alid if the cure under consideration is the geodesic connecting and m. For details see e.g. Postniko (2001. Furthermore, if the grou G carries a bi-inariant metric, then the Riemannian coariant deriatie D dt on G coincides with 0 D (Kobayashi and Nomizu 1969, dt Chater X.

7 Hermite subdiision on manifolds ia arallel transort Unified notation The following sections treat surfaces and matrix grous simultaneously. Therefore we introduce a unified notation. M means either a surface or a matrix grou. The exonential maing of M is denoted by ex (. In the surface case, P m denotes the arallel tranort along the geodesic connecting and m. If M is a matrix grou, P m refers to one of the arallel transorts introduced in Section 3.2. Following Wallner et al. (2011, we introduce the symbols and which are analogues of oint-ector addition and difference. For, q M and T M, let = ex ( and q = ex 1 (q. (10 Note that in the matrix grou case the and oerations are inariant w.r.t. both left and right multilication. While is always smooth and often globally defined (this is the case in both matrix grous and comlete surfaces (Onishchik and Vinberg 1993; Helgason 1979, is in general only smooth in some neighbourhood of. Our results in Section 5 are based on Moosmüller (2016, which only considers dense enough inut data. We therefore assume that is always smooth. As in the matrix grou case, we define the midoint of two oints, q on M: If g is the geodesic connecting and q, then ( 1 µ,q = g = 1 ( q Hermite subdiision on manifolds ia arallel transort Starting with a linear Hermite subdiision oerator S A satisfying the sectral condition (2, we define a subdiision oerator U in a surface or a matrix grou M. Recall that we can write S A in the form ( S A = ( ( ( ai 2 b i 2 = c i i 2 d i 2 Z a i 2 + b i 2 Z Z c. (11 i 2 + d i 2 The reroduction of constants (3 is characterised by the conditions Z a i 2 = 1 and Z c i 2 = 0. This allows us to rewrite (11 as S A ( i = ( mi + Z a i 2( m i + b i 2 Z c, (12 i 2( m i + d i 2 for any base oint sequence m. We use (12 to define a subdiision oerator U that takes arguments in l(t M. Consider inut data ( l(t M. For the base oint sequence m l(m we either choose m i = i or m i = µ i, i+1 for i Z. In Wallner et al. (2011 these base oint sequences hae been used for the C 1 and C 2 analysis of manifold-alued subdiision rules. It was shown in Grohs (2010;

8 8 Caroline Moosmüller Fig. 2: The SO(3-alued Hermite subdiision scheme of Examle 2 with resect to the (0 arallel transort. Inut data are reresented by sherical triangles. Uer and lower left figures: Limit cures of oint-ector inut data and one triangle of the second iteration ste. Uer and lower right figures: second iteration ste (tangent ectors are omitted. Xie and Yu (2009, howeer, that base oint sequences hae to be chosen in a more sohisticated manner if one wants to obtain higher smoothness results. Based on (12 we now define the subdiision oerator U for manifold-alued data: ( ( ri U = P r, (13 i i m i (w i { r i = m i Z where a i 2( m i + b i 2 P m i (, w i = Z c i 2( m i + d i 2 P m i (. In Section 6 we show that the successiely generated data (, D 1 U (, D 2 U 2(,... conerge to a cure and its deriatie. Note that if M is a matrix grou, then U is inariant w.r.t. both left and right multilication. Furthermore, if the linear oerator S A is interolatory, then obiously so is U. We mention that U can be defined analogously in the more general cases of Riemannian manifolds and Lie grous.

9 Hermite subdiision on manifolds ia arallel transort 9 Examle 2 Consider the matrix grou SO(3 = { R 3 3 : is orthogonal and det( > 0}. The tangent sace at SO(3 is gien by T SO(3 = { R 3 3 : 1 is skew-symmetric}. We consider the arallel transort ersion of the linear Hermite scheme introduced in Examle 1. Recall from Section 3.3 that for, q SO(3 and T SO(3 the oerators, are gien by = ex( 1 and q = log( 1 q, where ex is the matrix exonential and log is the matrix logarithm. For inut data ( l(t SO(3 we choose the base oint sequence m as the midoints of consecutie oints of : m 2i = m 2i+1 = µ i+1, i = i ( i i+1. Furthermore, for i, Z we introduce the following sequences: i, = i m, w,i = P m i ( = The oerator U of (13 is gien by where m i 1 for the (+ arallel transort, 1 m i for the ( arallel transort, µ,m i 1 1 µ,m i for the (0 arallel transort. U ( ( ri = P r, i i m i (w i r 2i = m 2i 1 ( 48 8 i+1,2i i,2i 29 w i+1,2i + 31 w i,2i w 2i = 1 ( i+1,2i i,2i w i+1,2i w i,2i, r 2i+1 = m 2i w 2i+1 = 1 8, ( 152 i+1,2i i,2i+1 31 w i+1,2i ( i+1,2i i,2i w i+1,2i w i,2i+1 w i,2i+1. The coefficients are taken from Examle 1. We consider the bi-inariant inner roduct u, = trace(u T on SO(3. This bi-inariant inner roduct coincides with the standard inner roduct induced by R 9, since trace(u T = i, u i i. Therefore, SO(3 is a surface which carries a bi-inariant inner roduct. It is known that the (0 arallel tranort defined aboe coincides with the surface arallel transort (the same is true for the exonential maing. Therefore, the aboe calculations are also alid if SO(3 is iewed as a surface.,

10 10 Caroline Moosmüller 5 Proximity inequalities In order to conclude conergence and smoothness of ordinary manifold-alued subdiision rules, the roximity method was introduced, see Wallner and Dyn (2005; Wallner (2006 and others. This method requires to establish inequalities on the difference between linear subdiision rules and manifold-alued subdiision rules. Since we need a ariety of norms to state the roximity condition, we summarize all of them in the following section: 5.1 Different tyes of norms The notation, where is an element of V = R n, means that we use the Euclidean norm. On matrix grous we use the Frobenius norm g 2 = trace(gg T. As already mentioned in Examle 2, the Frobenius norm corresonds to the Euclidean norm, if the matrix entries are ut into a column ector. From this norm on V we induce the Euclidean norm ( 0 1 = ( on V 2. On the sace L(V 2 2 we use the oerator norm ( a b { ( (, ( a b 0 0 } = su where = 1, c d c d 1 where ( a b c d L(V 2 2 and ( 0 1 V 2. We equi the sace of sequences l(v 2 with the norm ( ( i = su and denote by l (V 2 the sace of all sequences which are bounded with resect to this norm. Similarly we define a norm for A l(l(v 2 2 : i Z i A = su A i i Z and denote by l (L(V 2 2 the sace of bounded sequences. A linear subdiision oerator S A as defined in (1 restricts to an oerator l (V 2 l (V 2 (. This follows from S A d A (, where d is a ositie integer such that the suort of A is contained in [ d, d]. Therefore S A has an induced oerator norm, which we denote by S A. We mention that for the roofs of the next section, the articular choices of the norms on V and V 2 are not imortant. We will only need the Euclidean norm in Examle 3. What we will use, howeer, are the following facts concerning the equialence of norms: Since in eery finite dimensional ector sace, any two norms are equialent, the Euclidean norm ( 0 1 on V 2 is equialent to ( 0 1 = max{ 0, 1 }. That is, there exist constants c 1, c 1 > 0 such that c 1 ( 0 1 ( 0 1 c2 ( 0 1. It follows immediately that also the norms ( = su i ( i i and ( on l(v 2 are equialent with the same constants: c 1 ( ( c 2 (. (14 1

11 Hermite subdiision on manifolds ia arallel transort The roximity condition for Hermite schemes Consider a linear Hermite subdiision oerator S A and a manifold-alued Hermite subdiision oerator U. Then the roximity condition, introduced by Moosmüller (2016 for Hermite schemes, is gien by ( U S A ( c ( 2. (15 Here c is a constant and denotes the forward difference oerator i = i+1 i for i Z. To conclude C 1 conergence of U from conergence of S A, it is required that condition (15 is fullfilled wheneer ( is bounded and ( is small enough. In the following we roe that the roximity condition (15 holds between a linear oerator S A and the T M-alued oerator U constructed from S A (13, where M is a surface or matrix grou. Recall from Equation (13 that we defined sequences r, w by r i = m i a i 2 ( m i + b i 2 P m i (, (16 w i = c i 2 ( m i + d i 2 P m i (, for i Z. We also define r lin and w lin, which are the linear ersions of r and w. This means that and are relaced by + and resectiely and P m i ( is relaced by. Therefore, in order to roe (15, we hae to show the inequalities: r r lin c ( 2, (17 P r m(w w lin c ( 2. (18 The main ingredient in the roof is the following lemma: Lemma 1 Let M be a surface or matrix grou. Then for, m M and tangent ectors the following linearisations hold: = + + O( 2 as 0, (19 m = m + O( m 2 as m, (20 P m( = + O( m as m. (21 In the case that M is a surface, P m denotes the arallel transort along the geodesic connecting and m. If M is a matrix grou, then P m denotes one of the (+, (, or (0 arallel transorts. Proof In a chart of M, (19 and (20 are exactly the well-known linearization of the exonential ma. In order to roe (21, we first obsere that (m, P m( is smooth. On a surface, this can be deduced from the fact that the solution of an ordinary differential equation deends smoothly on the initial data. In the matrix grou case, the smoothness of this ma follows from the definition of the arallel

12 12 Caroline Moosmüller transort. Restricting to a unit ector and using Taylor exansion in a chart at m =, we obtain P m( = P ( + O( m = + O( m as m, = const. Since P m is a linear ma, for a general, we obtain P m( = + O( m as m. This comletes the roof. Corollary 1 (Proximity inequalities Let M be a surface or matrix grou. Consider bounded inut data ( on T M and a base oint sequence m, which is either gien by m i = i or m i = µ i, i+1 for i Z. Then the sequences r and w as defined in (16 satisfy r i = r lin i w i = w lin i P r i m i (w i = wi lin + O(su + O(su + O(su m i 2 + O(su m i su, m i 2 + O(su m i su m i su + O(su 2, + O(su 2, for m and 0 and i Z. In articular, the roximity inequalities (17 and (18 follow. Proof Using Lemma 1, the results for r and w immediately follow. Similarly, we can show that r i m i = O(su m i + O(su. This imlies P r i m i (w i = w i + O( r i m i w i = w lin i = w lin i + O(su + O(su m i su + O( r i m i w i m i 2 + O(su m i su + O(su 2, Furthermore, Lemma 1 imlies su m i c. Thus the aboe equations show that r r lin c max{ 2, 2 } and P r m(w w lin c max{ 2, 2 }. By the equialence of norms (14, the roximity inequality (17 and (18 are roed. This comletes the roof. 6 Results In the reious section we hae gathered all roximity inequalities we need to roe C 1 conergence of the manifold-alued Hermite scheme defined in Section 4. Our main theorem (Theorem 2 is analogous to Theorem 27 of Moosmüller (2016. Before we state the theorem, we hae to introduce the Taylor oerator. In linear Hermite subdiision, the Taylor oerator is the natural analogue to the forward difference oerator i = i+1 i for i Z, see Merrien and Sauer (2012. It acts on l(v 2 and is defined by ( 1 T =. 0 In Merrien and Sauer (2012 this oerator is called comlete Taylor oerator. We hae the following result:

13 Hermite subdiision on manifolds ia arallel transort 13 Theorem 1 (Merrien and Sauer, 2012 Let S A be a linear subdiision oerator which satisfies the sectral condition (2. Then we hae the following 1. There exists a linear subdiision oerator S B such that 2T S A = S B T. We call S B the Taylor scheme of S A. 2. If there exists N N such that S N B < 1, then the linear Hermite scheme associated to S A is C 1 conergent. Now we can state the main result of our aer: Theorem 2 Let S A be a linear subdiision oerator whose mask A satisfies the sectral condition (2, and let S B be the Taylor scheme of S A (Theorem 1. Let M be a surface or a matrix grou and let U be the manifold-alued analogue of S A gien by (13. Then we hae the following result: If there exists N N such that SB N < 1, then the Hermite scheme (, D 1 U (, D 2 U 2(,... is C 1 conergent wheneer ( are dense enough. The statement of the theorem remains true if surface is relaced by Riemannian manifold and matrix grou by Lie grou. Proof It is roed in Moosmüller (2016 that S N A < 1 for some integer N together with the roximity condition imlies C 1 conergence of the manifold-alued Hermite scheme. Therefore, the result follows from Section 5 and Moosmüller (2016. Note that the inut data does not hae to be bounded. This follows from the fact that on any comact interal the limit cure only deends on finitely many oints of the inut data. We can therefore w.l.o.g. assume that ( is bounded. The global embedding theorem states that any Riemannian manifold can be isometrically embedded as a surface into a Euclidean sace of sufficiently high dimension. The smoothness is resered by this embedding. Our result alies to this surface. Furthermore, by Ado s theorem, any Lie grou is locally isomorhic to a matrix grou. Therefore, the generalized statement is also true. This comletes the roof. Remark 1 We would like to remark on a ossible generalization of this result, which is a toic of future research. It would be natural to consider schemes which roduce more than one deriatie, i.e. schemes refining sequences with more than two comonents, with the kth comonent reresenting the (k 1st deriatie. This has been studied in the linear case, see e.g. Merrien and Sauer (2012. We beliee that such a generalization becomes quite technical: Aailable results from manifold subdiision suggest that the case of more than two deriaties is more inoled comared to the case of one deriatie (Grohs 2010; Xie and Yu Also, the data now hae to be samled from the et bundle of the manifold. Examle 3 We consider the linear subdiision oerator S A whose mask is defined in Examle 1. In Merrien and Sauer (2012 it is shown that the oerator S B satisfying 2T S A = S B T has the mask B 1 = 1 4 ( , B 0 = 1 4 ( , B 1 = 1 4 (

14 14 Caroline Moosmüller We roe S B < 1. The norm of a subdiision oerator is gien by It is well known that S B = su { S B ( : ( = 1 }. S B = max { Z B 2, Z B 2+1 }. Therefore, we hae to roe that max{ B 0, B 1 + B 1 } < 1. The oerator norm of a matrix w.r.t. to the Euclidean norm equals the sectral norm, therefore B i = λ max (Bi T B i, where λ max is the largest eigenalue of the matrix B T i B i for i = 1, 0, 1. This yields λ max (0 = λ max ( 1 = λ max (1 = < 1, < , < This imlies that S B < 1 and therefore the C 1 conergence of the linear Hermite scheme defined by S A. Furthermore, Theorem 2 shows that its arallel transort ersion on any Riemannian manifold or Lie grou is C 1 conergent for dense enough inut data. In articular this includes SO(3, i.e., our Examle Conclusion We hae studied a manifold-alued analogue of linear Hermite subdiision schemes which is defined by using the arallel transort oerator of the manifold. This construction is intrinsic and gies rise to a C 1 conergent nonlinear subdiision scheme, if the inut data are dense enough and the Taylor scheme is aroriately bounded (Theorem 2. Similar to most conergence and smoothness results of subdiision rules in general manifolds, the main ingredient of the roof is the method of roximity. Acknowledgements The author would like to thank Johannes Wallner for helful discussions on earlier ersions of this aer and gratefully acknowledges the suggestions of the anonymous reiewers. References do Carmo, M.P.: Riemannian geometry. Birkhäuser Verlag (1992 Dubuc, S.: Scalar and Hermite subdiision schemes. Alied and Comutational Harmonic Analysis 21(3, (2006 Dubuc, S., Merrien, J.L.: Conergent ector and Hermite subdiision schemes. Constructie Aroximation 23(1, 1 22 (2005

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