Smoothness Equivalence Properties of General Manifold-Valued

Data Subdivision Schemes

Gang Xie ∗ Thomas P.-Y. Yu †

February 19, 2008

Abstract:Based on a vector-bundle formulation, we introduce a new family of nonlinear subdivision schemes for

manifold-valued data. Any such nonlinear subdivision scheme is based on an underlying linear subdivisionscheme. We show that if the underlying linear subdivision scheme reproduces Πk, then the nonlinear schemesatisfies an order k proximity condition with the linear scheme. We also develop a new “proximity ⇒smoothness” theorem, improving the one in [12]. Combining the two results, we can conclude that if theunderlying linear scheme is Ck and stable, the nonlinear scheme is also Ck.

The family of manifold-valued data subdivision scheme introduced in this paper includes a variant of thelog-exp scheme, proposed in [10], as a special case, but not the original log-exp scheme when the underlyinglinear scheme is non-interpolatory. The original log-exp scheme uses the same tangent plane for both theodd and the even rules, while the variant uses two different, judiciously chosen, tangent planes. We alsopresent computational experiments that indicate that the original smoothness equivalence conjecture postedin [10] is unlikely to be true.

Our result also generalizes the recent results in [17, 16, 5, 6]. It uses only the intrinsic smoothness structureof the manifold and (hence) does not rely on any embedding or Lie group or symmetric space or Riemannianstructure. In particular, concepts such as geodesics, log and exp maps, or projection from ambient spaceplay no explicit role in the theorem. Also, the underlying linear scheme needs not be interpolatory.

Acknowledgments. The work of this research was partially supported by the the National Science Foun-dation grant DMS 0512673. The main result in this paper was first announced, based on an extrinsicformulation, in the MAIA 2007 conference (Alesund, Norway, August 22-26, 2007.) (The formulation wasthen revised into the current intrinsic setup.) The second named author would like to thank the organizersMichael Floater and Tom Lyche for their invitation, while the first named author would like to thank thehospitality of the Drexel University mathematics department during his postdoctoral visit at Drexel in 2007.

1 Introduction

In [10], the following log-exp scheme was proposed for the subdivision of data taking values on a Riemannianmanifold or a Lie group:

yj+1,2i = (Syj)2i = expyj,i

( ∑

`

a` logyj,i(yj,i+`)

), yj+1,2i+1 = (Syj)2i+1 = expyj,i

( ∑

`

b` logyj,i(yj,i+`)

).

Here (a`) and (b`) comes from the mask of a linear subdivision scheme T . It was conjectured by Donoho thatthis scheme satisfies a so-called smoothness equivalence property: if T produces Ck smooth curves, thenso does S. While this conjecture had stimulated a number of studies and partial solutions, e.g. [17, 16, 5, 6]and the references therein, the conjecture remains unsolved.

∗Department of Mathematics, East China University of Science and Technology, Shanghai, China, 200237. Email:gangxie2006@gmail.com.

†Department of Mathematics, Drexel University, 3141 Chestnut Street, 206 Korman Center, Philadelphia, PA 19104, U.S.A..Email: yut@drexel.edu. URL: http://www.math.drexel.edu/∼tyu.

1

Geodesic-reproducing property. This log-exp scheme enjoys a geodesic-reproducing property. Ifthe initial data consists of uniform samples of an arclength parameterized geodesics, then the log-exp schemereproduces the geodesics. Related to this, it is clear that if the underlying Riemannian metric is flat, thenthe smoothness equivalence property is obviously satisfied. For example, applying the log-exp scheme S on acylinder is essentially the same as using the corresponding linear T on the plane. This suggests that perhapsDonoho’s conjecture is true because (a) log and exp try to ‘locally flatten’ the manifold, and (b) subdivisionis a local process. If one buys this intuition, one would speculate that the conjectured smoothness equivalenceproperty is somewhat linked to the geodesic-reproducing property.

A 2005 Experiment. In an attempt to justify this intuition, we considered the case when M = Sn andexplored what happened when logx(y) above was replaced by

gx(y) := orthogonal projection of y to TxM .

Just like logx, gx is a local (C∞) diffeomorphism and has an inverse which we call fx. And we considered

xy

logx(y) x

y

gx(y)

Figure 1: logx(y) and gx(y) on Sn

the following scheme

yj+1,2i = (Syj)2i = fyj,i

( ∑

`

a` gyj,i(yj,i+`)), yj+1,2i+1 = (Syj)2i+1 = fyj,i

( ∑

`

b` gyj,i(yj,i+`)).

Unlike the log-exp scheme, this g-f scheme is not geodesic-reproducing. In the 2005 SIAM Conference onGeometric Design and Computing held at Pheonix, the first named author presented computational evidencesthat suggested the followings: When the underlying linear scheme is the C3 quartic B-Spline scheme (with(a−1, a0, a1) = (5/16, 5/8, 1/16), (b−1, b0, b1) = (1/16, 5/8, 1/16)), then

• the corresponding log-exp scheme is C3, but

• the corresponding g-f scheme is only C2 but not C3.

For more details of this experiment, see [15, Chapter 4]. It was very evident from this experiment that theg-f scheme suffered from a breakdown of smoothness equivalence, while Donoho’s log-exp scheme, in thiscase, seemed to be doing fine in terms of smoothness equivalence. At the time of the conference, the authorswere inclined to blame the breakdown to the fact that the g-f scheme was not geodesic-reproducing.

Surprisingly, all these turned out to be rather misleading.

1. First of all, Donoho’s original smoothness equivalence conjecture occurs to be not true. We shall presenta set of computations that shows that the original log-exp scheme generally suffers a breakdown ofsmoothness equivalence at smoothness order 4 or 5. These computations require the use of variableprecision arithmetic (vpa) in Maple. At the time of the 2005 SIAM conference, we were only usingstandard IEEE754 floating point computation in Matlab and could not investigate smoothness equiv-alence breakdown at smoothness order higher than 3, as numerical differentiation of order 4 or higheris highly sensitive to floating point errors.

2

2. A simple twist of the arguments in [16] shows that Donoho’s conjecture is true when the underlyinglinear subdivision scheme is interpolatory. When the underlying linear subdivision scheme is non-interpolatory, we show in this article that a modified scheme of the following form can be proved tosatisfy smoothness equivalence:

yj+1,2i = (Syj)2i = fyj,i

(∑

`

a` gyj,i(yj,i+`)),

yj+1,2i+1 = (Syj)2i+1 = fyj,i+1/2

( ∑

`

b` gyj,i+1/2(yj,i+`)),

where yj,i+1/2 is a judiciously chosen point on the manifold. In fact this yj,i+1/2 can be convenientlycomputed using an auxiliary interpolatory subdivision scheme ; see Section 3. The only propertyof f and g used in the proof is the smoothness of (x, y) 7→ fx(y) and (x, y) 7→ gx(y). Therefore,contrary to what the 2005 experiment suggested, smoothness equivalence property has little to do withgeodesics or the log or exp maps, it only has to do with the smoothness of logx(y) and expx(y) in xand y.1 In particular, we can change logx and expx to the gx and fx above, then, as long as we changethe base point for the odd rule from yj,i to yj,i+1/2, we get full smoothness equivalence.

1.1 Empirical breakdown of smoothness equivalence for Donoho’s log-exp scheme

An empirical way to check if a subdivision scheme S appears to be Ck is to plot the k-th order divideddifferences of the subdivision data vj := Sjv0, i.e. 2jk∆kvj , and inspect visually if the plot looks continuous.Notice that, when the scheme is non-interpolatory, we only have vj,i ≈ (S∞v0)(2−ji), but under suitablestability conditions the following strong convergence can be shown:

(2jk∆kvj)i → (S∞v0)(k)(2−ji), j →∞. (1.1)

(When S is a linear scheme, this is well-known, see [11, 4]. When S is nonlinear, it can be shown via aproximity condition with a stable linear scheme. The latter forms part of the proof of Theorem 2.4.)

When both k and j are sufficiently large, then the computed values of (2jk∆kvj)i are very inaccurate dueto floating point errors. Fortunately, we can use vpa provided by Maple to improve accuracy. In Figure 2,we plot (29·6∆6S9δ)i on 2−9Z, computed using vpa with 16, 18, . . . , 21 digits. Here, S is the linear degree8 B-spline subdivision scheme. This scheme is C7, so the 6-th order divided differences is continuouslydifferentiable. The plots clearly illustrate how vpa is crucial for illustrating the smoothness of the 6-th orderdivided differences.

16 digits 18 digits 19 digits 20 digits 21 digits

Figure 2: Plots of the 6th derivative of the degree 8 B-Spline based on taking the 6th order divided differencesof the level 9 subdivision data, computed using variable precision arithmetic (vpa) in Maple with 16, 18, . . . , 21decimal digits. Note: vpa with 16 digits is comparable to standard IEEE 754 floating point computation,and is insufficient in this particular case to give an accurate plot of the 6th order divided differences. Withvpa, one can compute with higher accuracy at a reasonable computational speed.

Using this naive divided difference approach, we now compare Donoho’s log-exp scheme with a modifiedlog-exp scheme, the latter is a special case of the general scheme to be proposed in Section 3. For now, all we

1Since expx(y) is defined by geodesics, and geodesics are described by ODEs, the smoothness of (x, y) 7→ expx(y) comesfrom the smoothness of the solutions of ODEs and the smooth dependence of the solutions on initial data. See [8, ChapterXIV] for a modern functional analytic proof of this nontrivial classical result.

3

need to know about this modified log-exp scheme is that it has a provable smoothness equivalence property.In this computational experiment, the manifold is the 2-sphere with the Riemannian metric inherited fromR3; the linear scheme is the degree 7 B-spline subdivision scheme.

It can be shown, based on the arguments in [12], that the original log-exp scheme is C2 in this case.The same is true for the modified log-exp scheme. These are illustrated by the first row of Figure 3. Themodified log-exp scheme is proved to be C6, a fact that is also evident from the plots. Also very evident isthat the original log-exp scheme fails to be C5; in fact the plots also suggest that the scheme may not evenbe C4 (see the 3rd row.)

As the control experiment in Figure 2 shows, when a divided difference plot appears to be not smooth, itcan be due to roundoff errors and/or the non-smoothness of the underlying function. However, by generatingthe same plot with increasing precision using vpa, the former effect goes away and we can reliably judgewhether the underlying function is smooth or not. For instance, look at the leftmost panel of the 4th row ofFigure 3. The 5th order divided difference plots (of level j = 8 subdivision data) for both schemes, computedusing only 8 digits, exhibit no smoothness whatsoever, but as the precision increases, it becomes evident thatthe modified scheme is C5 smooth but the original scheme is not. The roughness observed in the originalscheme is not due to roundoffs.

A Maple spreadsheet that reproduces this experiment can be found in

http://www.math.drexel.edu/~tyu/DonohoConjectureBreakDown .

2 Proximity Conditions and Smoothness

In this section, we offer a new “Proximity ⇒ Smoothness” theorem. This kind of theorems was first proposedin [13, 12]. Compared to a corresponding theorem in [12], the new version here gets rid of certain unnaturalassumptions pertaining to the underlying linear scheme. Other related papers that use the same notion ofproximity conditions are [14, 7].

Throughout this section, we let Z+ := 0 ∪ N. For any sequence x = (xi)i in an Euclidian space, let|x|∞ := sup

i‖xi‖2.

For any M ⊆ RN and δ > 0, let

XM :=x : Z→ M

∣∣ |∆x|∞ < ∞and XM,δ :=

x : Z→ M

∣∣ |∆x|∞ < δ

.

For j ∈ N, let

Γj :=

γ = (γ1, · · · , γj)

∣∣∣ γi ∈ Z+,

j∑

i=1

γi > 2,

j∑

i=1

i γi = j + 1

.

Note that 2 6 |γ| := γ1 + · · ·+ γj 6 j + 1 for any γ ∈ Γj . Obviously, Γj is a finite set for every j ∈ N. Forany x ∈ XM , let

Ωj(x) :=∑

γ∈Γj

j∏

i=1

|∆ix|γi∞.

For example, Ω1(x) = |∆x|2∞ and Ω2(x) = |∆x|3∞ + |∆x|∞|∆2x|∞.Since |∆kx|∞ 6 2|∆k−1x|∞ for any k ∈ N, it follows that for any j ∈ N, there exists a constant αj such

that for any x ∈ XM,δ,Ωj+1(x) 6 αjΩj(x). (2.1)

Lemma 2.1. Let S : XM,δ → XM . Let S be a convergent linear subdivision operator with dilation factorD. Suppose there exists C1 > 0 such that

|Sx− Sx|∞ 6 C1Ω1(x), ∀ x ∈ XM,δ,

then there exist δ′ > 0, C > 0 and β > 0 such that Sjx is well-defined and |∆Sjx|∞ 6 CD−jβ |∆x|∞ forany j ∈ N and x ∈ XM,δ′ .

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Proof: See Section A.1.For any sequence x : Z → RN , n ∈ Z+, and D > 1, D ∈ N, we define Fn

D(x) to be the piecewise linearfunction with Fn

D(x)(iD−n) = xi. For fixed D and n, FnD can be viewed as a linear map from the space of

sequences x to the space of piecewise linear functions; moreover, we have

|FnD(x)−Fn

D(x)|∞ = |x− x|∞, ∀ x, x. (2.2)

For a subdivision operator S : XM,δ → XM with dilation factor D, if there exists δ′ > 0 such thatFn

D(Snx) converges to a Ck (k ∈ Z+) function as n →∞ for any x ∈ XM,δ′ , then we say S is Ck.To prove S is Ck (k ∈ Z+), it suffices to show that there exists a δ′ > 0 such that for any x ∈ XM,δ′ and

j = 0, 1, · · · , k, FnD(Djn∆jSnx) converges uniformly as n →∞. See [3, Theorem 4.1].

We need the following lemma. Its proof can be found in [2] or [12, Lemma 1].

Lemma 2.2. Let S be a convergent linear subdivision operator with dilation factor D. Then there exists aconstant C > 0 such that for any sequence x and n ∈ N,

|FnD(Sx)−Fn−1

D (x)|∞ 6 C|∆x|∞.

We first prove the following basic “proximity ⇒ continuity” result.

Theorem 2.3. Let S : XM,δ → XM be a subdivision operator and S be a convergent linear subdivisionoperator. Suppose S and S have the same dilation factor and there exists C1 > 0 such that

|Sx− Sx|∞ 6 C1Ω1(x), ∀x ∈ XM,δ. (2.3)

Then S is C0.

Proof: Suppose S and S have dilation factor D. It follows from Lemma 2.1 that there exists C > 0, β > 0and δ′ > 0 such that Snx is well-defined and |∆Snx|∞ 6 CD−nβ |∆x|∞ for any n ∈ N and x ∈ XM,δ′ .Combining with Lemma 2.2, (2.2) and (2.3), we have for any n ∈ N and x ∈ XM,δ′ ,

|FnD(Snx)−Fn+1

D (Sn+1x)|∞ 6 |FnD(Snx)−Fn+1

D (SSnx)|∞ + |Fn+1D (SSnx)−Fn+1

D (Sn+1x)|∞6 C|∆Snx|∞ + |Sn+1x− SSnx|∞6 C|∆Snx|∞ + C1|∆Snx|2∞6 CCD−nβ |∆x|∞ + C1C

2D−2nβ |∆x|2∞6 CCD−nβδ′ + C1C

2D−2nβδ′2.

Therefore FnD(Snx) (n ∈ N) is a Cauchy sequence. Note Fn

D(Snx) is a C0 function for any n ∈ N. HenceFn

D(Snx) converges to a C0 function as n →∞. This means S is C0.

The final goal of this section is to extend Theorem 2.3 to higher order smoothness. To prove this theoremwe need to first develop a few auxiliary lemmas, see Appendix A.2.

Theorem 2.4. Let S be a subdivision operator defined on XM,δ and S be a Ck (k > 1) linear L∞-stablesubdivision operator. Suppose S and S have the same dilation factor. If there exists C1 > 0 such that forany x ∈ XM,δ and j = 1, · · · , k

|∆j−1Sx−∆j−1Sx|∞ 6 C1Ωj(x).

Then S is Ck.

Proof: Suppose the dilation factor of S and S is D. Since S is Ck (k > 1), linear and L∞-stable, it follows(e.g. [11, 4]) that there exist C1, · · · , Ck+1 > 1 and µ ∈ [0, 1) such that

|∆jSnx|∞ 6 CjD−jn|∆jx|∞, j = 1, · · · , k; (2.4)

|∆k+1Snx|∞ 6 Ck+1D(−j−1+µ)n|∆k+1x|∞. (2.5)

5

We can artificially modify the above constants such that

Cj+1j < Cj+1, j = 1, · · · , k. (2.6)

Since µ < 1, there exists m ∈ N such that

µk+1 := µ +logD Ck+1

m< 1.

Define µj = logD Cj

m for j = 1, · · · , k. Then it follows from (2.6) that

µj <µj+1

j + 1, j = 1, · · · , k.

Let T := Sm and T := Sm be two new subdivision operators. Then both of them have dilation factorD := Dm and T is Ck, linear and L∞-stable. Hence T has derived subdivision operators T1, · · · , Tk. Itfollows from Lemma A.3 that there exists Cm > 0 such that

|∆j−1Tx−∆j−1T x|∞ 6 CmΩj(x), ∀x ∈ XM,δ, j = 1, · · · , k.

It follows from (2.4)(2.5) that

|∆j T x|∞ 6 D−j+µj |∆jx|∞, j = 1, · · · , k + 1.

Choose

0 < ε < min(

1− µ1, min16j6k

(µj+1

j + 1− µj

)).

Define µ1 = µ1 + ε, µ2 = µ2, · · · , µk+1 = µk+1. Then µj ∈ [0, 1) for j = 1, · · · , k + 1 and

µj <µj+1

j + 1, j = 1, · · · , k.

It follows from Lemma A.1 that there exist 0 < δ′ 6 δ and polynomials P1, · · · , Pk+1 such that for any n ∈ Nand x ∈ XM,δ′ ,

|∆jTnx|∞ 6 D(−j+µj)nPj(n)|∆x|∞, j = 1, · · · , k + 1.

It follows from Lemma A.2 that T = Sm is Ck. Therefore S is Ck.

Remark: Theorem 2.3 is similar to [13, Theorem 3] while Theorem 2.4 is similar to [12, Theorem 6]. But ineach case our theorem is more general. For example, [12, Theorem 6] applies when S is a B-spline subdivisionoperator but not when S is a Deslauriers-Dubuc subdivision operator.

3 A New Nonlinear Subdivision Scheme for Manifold-Valued Data

In this section we introduce the general subdivision scheme for manifold-valued data as promised in Section 1.The definition assumes some familiarity with basic manifold theory. Those who do not want to deal withmanifold concepts can basically skip this section and directly study the localized version in (4.1)-(4.3).

Let M be a smooth manifold of dimension n. We let T be a linear, not necessarily interpolatory,subdivision operator defined by

(Ty)2i =∑

`

a` yi+`, (Ty)2i+1 =∑

`

b` yi+`. (3.1)

Our general nonlinear subdivision scheme S for M -valued data will be derived from this linear scheme T aswell as an auxiliary linear interpolatory subdivision scheme T defined by

(T y)2i = yi, (T y)2i+1 =∑

`

c` yi+` . (3.2)

6

To prepare for our definition of S, we need also two pairs of smooth maps (f, g) and (f , g) defined as follows.Let π : V → M and π : V → M be two vector bundles over M , with ranks N and N respectively. Write

V (x) := π−1(x) and V (x) := π−1(x)

as the fibres of V and V at x, respectively. Recall that V has by definition a differentiable structure ofdimension n + N . Elements in V will be denoted by a tuple of the form (x, v), where x ∈ M and v ∈ V (x),Similar comments apply to V .

Assume that, for every x ∈ M , there are open neighborhoods Ux and Ux of x, and associated smoothmappings

gx : Ux → V (x), gx : Ux → V (x).

It is crucial for us to also assume that the mappings (x, y) 7→ gx(y) and (x, y) 7→ gx(y) are jointly smoothin x and y. For this purpose, we need to first assume that

U := (x, y) : x ∈ M, y ∈ Ux and U := (x, y) : x ∈ M, y ∈ Ux (3.3)

are open in M ×M . This openness assumption allows us to sensibly talk about the smoothness of the mapsg and g below. It also implies that Ux cannot get arbitrarily small locally in a sense to be made precise inthe proof of Lemma 3.2.

Letg : U → V, g : U → V , (3.4)

be defined byg(x, y) = (x, gx(y)), and g(x, y) = (x, gx(y));

we assume that both maps are smooth. Without loss of generality, we can also assume that

gx(x) = 0 and gx(x) = 0. (3.5)

Let Ex (resp. Ex) be an open set in V (x) (resp. V (x)) that contains gx(Ux) (resp. gx(Ux)) such that

E := (x, v) : x ∈ M,v ∈ Ex (3.6)

is open in V (resp. E := (x, v) : x ∈ M, v ∈ Ex is open in V .) Then, let

fx : Ex → M, fx : Ex → M, (3.7)

be two smooth maps that satisfy

fx(gx(y)) = y, ∀ x ∈ M, y ∈ Ux, fx(gx(y)) = y, ∀ x ∈ M, y ∈ Ux. (3.8)

We then assume also thatf : E → M and f : E → M

are smooth; this means fx(y) and fx(y) are jointly smooth in x and y.We are now ready to define S.

Definition 3.1. Given T , T , (f, g) and (f , g) defined above, a nonlinear subdivision scheme S is defined by:

(Sx)2i = fxi

( ∑

`

a` gxi(xi+`)), (Sx)2i+1 = fxi+1/2

(∑

`

b` gxi+1/2(xi+`))

(3.9)

wherexi+1/2 = fxi

(∑

`

c` gxi(xi+`)). (3.10)

We can, and we will, view (3.10) as an interpolatory subdivision scheme for M -valued data:

(Sx)2i = xi, (Sx)2i+1 = fxi

( ∑

`

c` gxi(xi+`)). (3.11)

7

Examples of this abstract scheme include:

• V = TM (the tangent bundle.) M is a Riemannian manifold or a Lie group, gx = logx and fx = expx.In this case S is a variant of the log-exp scheme proposed in [10]. Note that (V , f , g) can be chosenarbitrarily as long as they satisfy the basic assumptions. See Section 1.1 for a comparison of the originallog-exp scheme with this variant.

• V = M × Rn (a trivial bundle.) gx(y) = (x, i(y)), where i : M → Rn is an embedding of M intoRn, fx(v) = i−1(the point in i(M) closest to v). In this case, this is the closest point projectionscheme studied in [13, 17, 6]. Note: Mapping a point from the ambient space of a regular surface tothe closest point on the surface is well-defined on a tubular neighborhood of the surface (see, e.g., [1].)This tubular neighborhood of i(M) furnishes the natural domain of fx. In this case, both gx and fx

are essentially independent of x; therefore the auxiliary interpolatory scheme T does not actually playa role in the definition of S.

We shall prove that if the initial sequence (x`)` is dense enough, then the subdivision processes Sjx, Sjx,j = 0, 1, 2, . . ., are well-defined. In virtue of Lemma 2.1, this can be established by proving (a) S and S arewell-defined in one step (i.e. Sx and Sx are well-defined for dense enough sequences x), and (b) proximityconditions between S and T and between S and T .

Proximity conditions will be the main theme of the next section. To prove the one-step well-definedness,we need the following.

Lemma 3.2. Let M be a topological manifold, and V be a vector bundle over M . Let (W,φ) be any chartof M such that the vector bundle V is trivial on W . Let K be any compact subset of W . Let f and gbe defined as above, except that they are only assumed to be continuous (instead of smooth, as we do notassume in this lemma that there is a differentiable structure on M .) Let (w`) be any finitely supportedsequence. (Assume, without loss of generality, that 0 ∈ support(w).) Then for every ε > 0, there exists

δ := δ(W,φ,K, w`, f, g) > 0

such that for any (x`)`∈support(w), x` ∈ K, with

max`∈support(w)

‖φ(x`)− φ(x0)‖ < δ, (3.12)

fx0(∑

` w` gx0(x`)) is well-defined and falls inside the chart W , moreover

‖φ(fx0(∑

`

w` gx0(x`)))− φ(x0)‖ < ε.

Proof: By local triviality, the chart (W,φ) induces an isomorphism

i : (x, v) ∈ V : x ∈ W, v ∈ V (x) → φ(W )× RN .

We also write ix := i(x, ·) : V (x) → RN . Note that i(x, v) = (φ(x), ix(v)). Without loss of generality, we canassume that ix(0) = 0.

1 The openness assumption on U = (x, y) : x ∈ M,y ∈ Ux implies that (φ(x), φ(y)) : x ∈ K, y ∈ Ux∩Wis open in φ(K) × Rn. Since x ∈ Ux for all x, (φ(x), φ(y) − φ(x)) : x ∈ K, y ∈ Ux ∩W is an open set inφ(K)×Rn which contains the ‘line’ φ(K)×0, and therefore also contains, in virtue of the compactness ofK and the tube lemma [9], a ‘tube’ φ(K)×BRn(0, r).2 Hence, for every x ∈ K, φ(Ux ∩W ) contains an openball of radius r > 0. Of course, this r can be chosen such that the closure of BRn(φ(x), r) is also containedin φ(Ux ∩W ).

So we can write g in local coordinates as a map of the form

g(W,φ) : φ(K)×BRn(0, r) → φ(K)× RN , (x, y) → i(g

(φ−1(x), φ−1(x + y)

)). (3.13)

2We denote by BRn (x, r), or simply B(x, r) when the dimension is clear from the context, an open ball in Rn centered atx ∈ Rn with radius r > 0.

8

Since g(W,φ)(x, y) = (x, v), the only information in this map is the part that maps (x, y) to v. We denotethis map by

g : φ(K)×BRn(0, r) → RN . (3.14)

Note that g(x, 0) = 0 by (3.5). We also write

gx := g(x, ·) : BRn(0, r) → RN .

2 We claim that there exists a r′ > 0 such that

BRN (0, r′) ⊆ ix(f−1

x (φ−1(B(φ(x), r)))), ∀x ∈ K. (3.15)

This is again proved using the tube lemma. Consider the continuous F : E → M ×M defined by F (x, v) =(x, fx(v)). We have

(x, y) : x ∈ K, y ∈ φ−1(B(φ(x), r)) open⊂ K ×W ⊂ M ×M,

F−1((x, y) : x ∈ K, y ∈ φ−1(B(φ(x), r))) open⊂⋃

x∈K

Ex

open⊂⋃

x∈K

V (x),

thereforei(F−1((x, y) : x ∈ K, y ∈ φ−1(B(φ(x), r)))) open⊂ φ(K)× RN .

But the set i(F−1((x, y) : x ∈ K, y ∈ φ−1(B(φ(x), r)))) contains the ‘line’ φ(K) × 0 and hence also a

‘tube’ φ(K)×BRN (0, r′). This is equivalent to our claim (3.15).Using (3.15), we can write f in the local coordinates as:

f : φ(K)×BRN (0, r′) → BRn(0, r), (x, v) 7→ φ(f

(i−1(x, v)

))− x. (3.16)

Define alsofx := f(x, ·) : BRN (0, r′) → BRn(0, r). (3.17)

3 Let ε > 0 be given. Since φ(K) × BRN (0, r′) is compact, f is uniformly continuous, so there exists aδ′ ∈ (0, r′] such that (x, y) ∈ φ(K)×BRN (0, δ′) ⇒ ‖f(x, v)‖ < ε. Similarly, there exists a δ ∈ (0, r] such that(x, y) ∈ φ(K)×BRn(0, δ) ⇒ ‖gx(y)‖ < δ′/

∑` |w`|.

Therefore if x` ∈ φ(K) satisfies ‖x`−x0‖ < δ, then ‖gx0(x`)‖ < δ′/∑ |w`|, which implies ‖∑

` w`gx0(x`)‖ <δ′, which then implies ‖fx0(

∑w` gx0(x`))‖ < ε.

4 Establishing Proximity Conditions

Since our subdivision schemes T and T act locally to the data, there is a minimal set of indices, which wedenote here by N (0, i), such that the level zero data x0,k, k ∈ N (0, i), is enough to determine the limitfunctions T∞y and T∞y at the whole interval [i− 1, i + 1]. In order to analyze the scheme in Definition 3.1is smooth, it suffices to show that it produces a smooth limit on [i − 1, i + 1] for every i ∈ Z. We shallassume that for every i ∈ Z there is a chart (W,φ) of the manifold that trivializes both vector bundles Vand V and also contains the data x0,k, k ∈ N (0, i). This forms part of our ‘dense enough’ assumption onthe initial manifold-valued data. With this assumption, we can rewrite the scheme locally in terms of localcoordinates. By shift invariance, it suffices to analyze only the case of i = 0.

Following the proof of Lemma 3.2, given a chart (W,φ) on which both V and V are trivial, we can thenwrite g, g, f and f in local coordinates as

g : K ×BRn(0, r) → RN , f : K ×BRN (0, r′) → BRn(0, r), (4.1)

g : K ×BRn(0, r) → RN , f : K ×BRN (0, r′) → BRn(0, r). (4.2)

9

Here K ⊂ φ(W ) is a compact set in Rn; this K is the φ(K) in Lemma 3.2. Notice that we abuse notationshere by using the same symbols to denote the localized version of g, g, f and f . By construction, g(x, 0) = 0,g(x, 0) = 0, f(x, 0) = 0, f(x, 0) = 0, and

f(x, g(x, y)) = y = f(x, g(x, y)) ∀x ∈ K, y ∈ BRn(0, r).

By Lemma 3.2, if x` ∈ K, ` ∈ N (0, 0) satisfies ‖x` − x0‖ < ε for a sufficiently small ε > 0 then for allrelevant indices h,

(Sx)2h = xh + f(xh,

∑

`

a` g(xh, xh+` − xh)),

xh+1/2 = (Sx)2h+1 = xh + f(xh,

∑

`

c` g(xh, xh+` − xh)),

(Sx)2h+1 = xh+1/2 + f(xh+1/2,

∑

`

b` g(xh+1/2, xh+` − xh+1/2))

(4.3)

are well-defined and all fall within the ball B(x0, r). Notice that (4.3) is our original subdivision schemes inDefinition 3.1 written in local coordinates.

The goal of this section is to prove the following two proximity conditions:

Theorem 4.1. Assume that the polynomial reproduction orders of T and T are at least k. Then thereexists a constant C > 0 such that for any sufficient dense sequence (x`)`∈N (0,0),

‖Sx− T x‖ 6 C Ωk′ ((x`)`) (4.4)

and‖∆k′−1(Sx− Tx)‖ 6 C Ωk′ ((x`)`) (4.5)

for all k′ = 1, . . . , k.

Proof: We prove the theorem for k′ = k, the proof for any smaller k′ is identical. We divide the proof intoa number of steps. The first four steps serve as a preparation for the proof of both parts of the theorem. In1-4 below, (w`)` is any finitely supported sequence with

∑w` = 1 and, in particular, can be any one of

the (a`),(b`) or (c`) from the masks of T and T , and f and g may refer to either the f and g in (4.1) or elsethe f and g in (4.2). Step 5 finishes the proof of (4.4). Steps 6-9 deal with (4.5).

The assumption that T reproduces Πk (= the space of polynomials with degree 6 k) is equivalent to∑

`

c` π(`) = π(1/2), ∀π ∈ Πk; (4.6)

that T reproduces Πk is equivalent to∑

`

a` π(` + 1/2) =∑

`

b` π(`), ∀π ∈ Πk. (4.7)

1 WriteF (m) :=

1m!

dmf |(x0,0), G(m) :=1m!

dmg|(x0,0). (4.8)

These are multi-linear maps with

‖F (m)(u1, . . . , um)‖ . ‖u1‖ · · · ‖um‖, ‖G(m)(u1, . . . , um)‖ . ‖u1‖ · · · ‖um‖. (4.9)

The hidden constants can be chosen independent of x0 but dependent on the compact set K.By Taylor’s theorem,

xh + f(xh,∑

`

w` g(xh, xh+` − xh)) = xh +k∑

m=1

F (m)(ω, . . . , ω) + O(‖ω‖k+1),

10

where

ω := ωh :=

(xh − x0,

∑

`

w` g(xh, xh+` − xh)

).

Using∑

` w` = 1, ω can be rewritten as ω =∑

` w`κ`, where

κ` := κ`,h :=(xh − x0, g(xh, xh+` − xh)

). (4.10)

Since K is compact and g is smooth, in particular Lipschitz, there is a constant A > 0 such that

‖g(x, y1)− g(x, y2)‖ 6 A ‖y1 − y2‖, ∀x ∈ K, y1, y2 ∈ B(0, r). (4.11)

Therefore, max`,h ‖κ`,h‖ = O(‖∆x‖) and maxh ‖ωh‖ = O(‖∆x‖). Here ‖∆x‖ simply means max` ‖x`+1−x`‖.This, together with the multi-linearity of F (m), yields:

f(xh,∑

`

w` g(xh, xh+` − xh)) =k∑

m=1

∑

`1,...,`m

w`1 · · ·w`mF (m)(κ`1 , . . . , κ`m

) + O(‖∆x‖k+1). (4.12)

On the other hand,∑

`

w` xh+` = xh +∑

`

w` f(xh, g(xh, xh+` − xh)

)

= xh +∑

`

w`

k∑m=1

F (m)(κ`, . . . , κ`) + O(‖∆x‖k+1)

= xh +k∑

m=1

∑

`

w` F (m)(κ`, . . . , κ`) + O(‖∆x‖k+1).

(4.13)

2 Assume N (0, 0) = −L, . . . ,−L + p. Recall from the definition of N (i, 0) that p is a finite number thatonly depends on the finitely supported (a`), (b`) and (c`). Moreover, since the polynomial reproductionorders of T and T are at least k, p > k.

For ` > 0, let

D` := the `-th order difference of x−L, . . . , x−L+` =∑

j=0

(−1)`−j

(`

j

)x−L+j . (4.14)

We can recover x−L+` from D0, . . . , D` by

x−L+` =∑

j=0

(`

j

)Dj . (4.15)

For any −L 6 ` 6 −L + p,

x` − x0 =p∑

j=1

A`j Dj , (4.16)

where

A`j :=

(L + `

j

)−

(L

j

). (4.17)

Note that

A•j :=(L + •)(L + • − 1) · · · (L + • − j + 1)

j!− L(L− 1) · · · (L− j + 1)

j!∈ Πj , (4.18)

and A`j is the evaluation of this polynomial at `, and Ax

j can be thought of as a polynomial function in thevariable x. In the sequel, a lot of polynomials will be formed out of these polynomials.

11

Since p > k, we can also rewrite (4.16) as

x` − x0 =k∑

j=1

A`j Dj + O(Ωk(x`)) (4.19)

3 The first component of κ`,h is exactly (4.19); the second component of κ`,h can be written as

g(xh, xh+` − xh) =k∑

n=1

G(n)(η`,h, . . . , η`,h) + O(‖∆x‖k+1),

where

η`,h = (xh − x0, xh+` − xh)

=( k∑

j=1

Ahj Dj ,

k∑

j=1

(Ah+`j −Ah

j )Dj

)+ O(Ωk(x`))

=k∑

j=1

Ahj (Dj , 0) + (Ah+`

j −Ahj )(0, Dj) + O(Ωk(x`)).

If we define

Dεj :=

(Dj , 0), ε = 0(0, Dj), ε = 1 , (4.20)

then

η`,h =1∑

ε=0

k∑

j=1

(Ah+ε`j − εAh

j )Dεj + O(Ωk(x`)). (4.21)

We introduce the shorthand notations

Ek := I = (ε1, . . . , εn) : 1 6 n 6 k, εi = 0, 1,Jk := J = (j1, . . . , jn) : 1 6 n 6 k, 1 6 ji 6 k.

For I ∈ Ek, [I] := n, |I| := ε1 + · · ·+ εn. Similarly, for J ∈ Jk, [J ] := n, |J | := j1 + · · ·+ jn. Define also

Ek ⊗ Jk := (I, J) ∈ Ek × Jk, : [I] = [J ] (4.22)

By multi-linearity of G(n) and also (4.9),

G(n)(η`,h, . . . , η`,h) =∑I∈Ek[I]=n

∑J∈Jk[J]=n

Qh,`I,J G(n)(DI,J ) + O(Ωk(x`)),

where DI,J := (Dε1j1

, . . . , Dεnjn

) and Qh,`I,J :=

∏ni=1(A

h+εi`ji

− εiAhji

). By the comment below (4.18), we canview, for any fixed I = (ε1, . . . , εn) and J = (j1, . . . , jn),

Qx,yI,J :=

n∏

i=1

(Ax+εiyji

− εiAxji

) (4.23)

as a bivariate polynomial in x and y of total degree |J | = j1 + · · ·+ jn.So we have

κ`,h =( k∑

j=1

Ahj Dj ,

k∑n=1

∑

(I,J)∈Ek⊗Jk[I]=[J]=n

Qh,`I,J G(n)(DI,J)

)+ O(‖∆x‖k+1) + O(Ωk(x`))︸ ︷︷ ︸

=O(Ωk(x`))

.

12

To simplify this expression, note that Ahj is actually the Qh,`

J,I with J = (j), I = (0) ([J ] = [I] = 1). Therefore,κ`,h can be written as

κ`,h =∑

(I,J)∈Ek⊗Jk

Qh,`I,J U I

J + O(Ωk(x`)), (4.24)

where

U IJ :=

(0, G(n)(DI

J))

if [I] = [J ] = n > 1, or I = (1), J = (j)(Dj , G

(1)(D(0)(j))

)if I = (0), J = (j)

. (4.25)

4 We now combine (4.12) with (4.24) and (4.9).

f(xh,∑

`

w` g(xh, xh+` − xh))

=k∑

m=1

∑

`1,...,`m

w`1 · · ·w`m

∑

(Ii,Ji)∈Ek⊗Jki=1,...,m

Qh,`1I1,J1 · · ·Qh,`m

Im,Jm F (m)(U I1

J1 , . . . , U Im

Jm) + O(Ωk(x`))

=k∑

m=1

∑

(Ii,Ji)∈Ek⊗Jki=1,...,m

[ ∑

`1,...,`m

w`1 · · ·w`mQh,`1I1,J1 · · ·Qh,`m

Im,Jm

]F (m)(U I1

J1 , . . . , U Im

Jm) + O(Ωk(x`))

=k∑

m=1

∑

(Ii,Ji)∈Ek⊗Jki=1,...,m

[m∏

i=1

∑

`

w` Qh,`Ii,Ji

]F (m)(U I1

J1 , . . . , U Im

Jm) + O(Ωk(x`)).

(4.26)

Next, we combine (4.13) with (4.24) and (4.9).

∑

`

w` xh+` = xh +k∑

m=1

∑

`

w` F (m)(κ`,h, . . . , κ`,h) + O(‖∆x‖k+1)

= xh +k∑

m=1

∑

`

w`

∑

(Ii,Ji)∈Ek⊗Jki=1,...,m

Qh,`I1,J1 · · ·Qh,`

Im,Jm F (m)(U I1

J1 , . . . , U Im

Jm) + O(Ωk(x`))

= xh +k∑

m=1

∑

(Ii,Ji)∈Ek⊗Jki=1,...,m

[ ∑

`

w`

m∏

i=1

Qh,`Ii,Ji

]F (m)(U I1

J1 , . . . , U Im

Jm) + O(Ωk(x`))

(4.27)

By (4.26) and (4.27), we have now quantify the difference between xh + f(xh,∑

` w` g(xh, xh+` − xh)) and∑` w` xh+` by

xh + f(xh,∑

`

w` g(xh, xh+` − xh))−∑

`

w` xh+`

=k∑

m=1

∑

(Ii,Ji)∈Ek⊗Jki=1,...,m

Υh(I1,J1),...,(Im,Jm) F (m)(U I1

J1 , . . . , U Im

Jm) + O(Ωk(x`)),(4.28)

where

Υh(I1,J1),...,(Im,Jm) =

m∏

i=1

∑

`

w` Qh,`Ii,Ji −

∑

`

w`

m∏

i=1

Qh,`Ii,Ji . (4.29)

Note: The data (x`) does not enter the definition of Υh(I1,J1),...,(Im,Jm), whereas the mask (w`) does not

enter the definition of F (m)(U I1

J1 , . . . , U Im

Jm).

13

5 We are now ready to prove the first part of the theorem, namely (4.4). For this purpose, we use (4.28)-(4.29) with (w`) = (c`) and we replace (f, g) by the (f , g) from (4.2). Since (c`) comes from the mask of aΠk reproducing subdivision scheme,

∑` c` π(`) = π(1/2) for any π ∈ Πk. By (4.23), Qh,`

Ii,Ji = πIi,Ji,h(`) fora πIi,Ji,h ∈ Π|Ji|. Therefore, when |J1|+ · · ·+ |Jm| 6 k, πIi,Ji,h ∈ Πk and

∏mi=1 πIi,Ji,h ∈ Πk, so

Υh(I1,J1),...,(Im,Jm) =

m∏

i=1

∑

`

c` Q0,`Ii,Ji −

∑

`

c`

m∏

i=1

Q0,`Ii,Ji

=m∏

i=1

(πIi,Ji,h(1/2)

)−( m∏

i=1

πIi,Ji,h

)(1/2) = 0.

(4.30)

By (4.28),

‖(Sx)2h+1 − (T x)2h+1‖ 6k∑

m=1

∑

(Ii,Ji)∈Ek⊗Jki=1,...,m

|Υh(I1,J1),...,(Im,Jm)| ‖F (m)(U I1

J1 , . . . , U Im

Jm)‖+ O(Ωk(x`)).

By (4.30), the only terms that survive on the right-hand side are those with |J1| + · · · + |Jm| > k. On theother hand,

‖F (m)(U I1

J1 , . . . , U Im

Jm)‖(4.9)

. ‖U I1

J1‖ · · · ‖U Im

Jm‖(4.25)+(4.9)

. ‖Dj1‖ · · · ‖DjPi |Ji|

‖, (4.31)

where (j1, . . . , jPi |Ji|) is the concatenation of J1, . . . , Jm. By recalling the definition of Ωk((x`)`) and Dj ,

we finish the proof of (4.4).

6 By (4.28)-(4.29),

(Sx)2h − (Tx)2h =k∑

m=1

∑

(Ii,Ji)∈Ek⊗Jki=1,...,m

Υh(I1,J1),...,(Im,Jm)F

(m)(U I1

J1 , . . . , U Im

Jm) + O(Ωk(x`)), (4.32)

where Υh(I1,J1),...,(Im,Jm) =

∏mi=1

∑` a` Qh,`

Ii,Ji −∑

` a`

∏mi=1 Qh,`

Ii,Ji .

We now show that

Υ•(I1,J1),...,(Im,Jm) :=m∏

i=1

∑

`

a` Q•,`Ii,Ji −

∑

`

a`

m∏

i=1

Q•,`Ii,Ji ∈ Π|J1|+···+|Jm|−2. (4.33)

If J = (j1, . . . , jn) and I = (ε1, . . . , εn), by (4.18) and (4.23),

Qx,yJ,I =

1j1! · · · jn!

x|J| + BJ,I(y)x|J|−1 +∑

i<|J|−1

BiJ,I(y)xi.

where BJ,I(y) and BiJ,I(y) are polynomials of y dependent on I, J . From this,

∑

`

a` Qx,`J,I =

1J !

(x|J| +

∑

`

a` BJ,I(`) · x|J|−1 + · · ·).

In above and below, J ! := j1! · · · jn! and the · · · involves terms of xi with i < |J | − 1. Therefore

m∏

i=1

∑

`

a` Qx,`Ii,Ji =

1J1! · · · Jm!

(x|J

1|+···+|Jm| +m∑

i=1

∑

`

a` BIi,Ji(`) · x|J1|+···+|Jm|−1

)+ · · · .

14

On the other hand

m∏

i=1

Qx,`Ii,Ji =

1J1! · · · Jm!

(x|J

1|+···+|Jm| +m∑

i=1

BIi,Ji(`) · x|J1|+···+|Jm|−1 + · · ·)

,

so∑

`

a`

m∏

i=1

Qx,`Ii,Ji =

1J1! · · · Jm!

(x|J

1|+···+|Jm| +∑

`

a`

m∑

i=1

BIi,Ji(`) · x|J1|+···+|Jm|−1 + · · ·)

.

And the claim (4.33) is proved.7 Since we have established (4.4) in 5, we can now write

xh+1/2 =∑

`

c` xh+` + O(Ωk((x`))) = x0 +∑

`

c` (xh+` − x0) + O(Ωk((x`)))

= x0 +∑

`

c`

p∑

j=1

Ah+`j Dj + O(Ωk((x`)))

= x0 +p∑

j=1

( ∑

`

c` Ah+`j

)Dj + O(Ωk((x`)))

= x0 +k∑

j=1

( ∑

`

c` Ah+`j

)Dj +

∑

j>k

( ∑

`

c` Ah+`j

)Dj + O(Ωk((x`)))

(4.6)+(4.18)= x0 +

k∑

j=1

Ah+1/2j Dj +

∑

j>k

(∑

`

c` Ah+`j

)Dj + O(Ωk((x`)))

= x0 +k∑

j=1

Ah+1/2j Dj + O(Ωk(x`)).

(4.34)

In the last equality,∑

j>k

( ∑` c` Ah+`

j

)Dj can be absorbed into O(Ωk((x`))) because ‖Dj‖∞ 6 |∆j(x`)|∞

and |∆j+1(x`)|∞ 6 2 |∆j(x`)|∞.

8 In this step, we basically need to repeat most of the derivations in 1, 2, 3 and 4 but for (Sx)2h+1 −(Tx)2h+1 instead of (Sx)2h − (Tx)2h. While the odd rule of S involves a change in the “base point” fromxh to xh+1/2, we shall continue to use the Taylor expansions of f and g at the same base points (4.8). Inthe process, we shall use the approximation of xh+1/2 derived in (4.34).

(Sx)2h+1 − (Tx)2h+1

=f(xh+1/2,∑

`

b` g(xh+1/2, xh+` − xh+1/2)−∑

`

b` f(xh+1/2, g(xh+1/2, xh+` − xh+1/2)

=k∑

m=1

F (m)( ∑

`

b` κ`,h, . . . ,∑

`

b` κ`,h

)−

∑

`

b`

k∑m=1

F (m)(κ`,h, . . . , κ`,h) + O(max`,h

‖κ`,h‖k+1),

(4.35)

where

κ`,h =(xh+1/2 − x0, g(xh+1/2, xh+` − xh+1/2)

)

=(xh+1/2 − x0,

k∑n=1

G(n)(η`,h, . . . , η`,h) + O(max`,h

‖η`,h‖k+1)),

(4.36)

15

η`,h = (xh+1/2 − x0, xh+` − xh+1/2)

(4.34)=

( k∑

j=1

Ah+1/2j Dj ,

k∑

j=1

(Ah+`j −A

h+1/2j )Dj

)+ O(Ωk(x`))

(4.20)=

1∑ε=0

k∑

j=1

(Ah+1/2+ε(`−1/2)j − εA

h+1/2j )Dε

j + O(Ωk(x`)).

(4.37)

The trick in the last equality, as we will see, will be crucial; also it is worth comparing (4.37) with (4.21).We can now write

max`,h

‖κ`,h‖k+1 = O(Ωk(x`)), max`,h

‖η`,h‖k+1 = O(Ωk(x`)),

and

κ`,h =( k∑

j=1

Ah+1/2j Dj ,

k∑n=1

G(n)(η`,h, . . . , η`,h))

+ O(Ωk(x`))

=( k∑

j=1

Ah+1/2j Dj ,

k∑n=1

G(n)(η`,h, . . . , η`,h))

+ O(Ωk(x`))

=( k∑

j=1

Ah+1/2j Dj ,

k∑n=1

∑

(Ii,Ji)∈Ek⊗Jki=1,...,n

Qh+1/2,`−1/2Ii,Ji G(n)(DIi,Ji)

)+ O(Ωk(x`))

=∑

(I,J)∈Ek⊗Jk

Qh+1/2,`−1/2I,J U I

J + O(Ωk(x`)).

(4.38)

In the second last equality, we used the notation DI,J and Qx,yI,J defined at and around (4.23), we also used the

observation made in the last step of (4.37); in the last equality, we used the U IJ defined in (4.25). Compare

(4.38) with (4.24).Similar to how we got (4.32), by combining (4.35) with (4.38), we have

(Sx)2h+1 − (Tx)2h+1 =k∑

m=1

∑

(Ii,Ji)∈Ek⊗Jki=1,...,m

Ξh(I1,J1),...,(Im,Jm)F

(m)(U I1

J1 , . . . , U Im

Jm) + O(Ωk(x`)), (4.39)

where

Ξh(I1,J1),...,(Im,Jm) =

m∏

i=1

∑

`

b` Qh+1/2,`−1/2Ii,Ji −

∑

`

b`

m∏

i=1

Qh+1/2,`−1/2Ii,Ji .

9 If we define the sequence χ(I1,J1),...,(Im,Jm) by

(χ(I1,J1),...,(Im,Jm))2h = Υh(I1,J1),...,(Im,Jm), (χ(I1,J1),...,(Im,Jm))2h+1 = Ξh

(I1,J1),...,(Im,Jm),

then we can write

(∆k−1Sx−∆k−1Tx

)2h

=k∑

m=1

∑

(Ii,Ji)∈Ek⊗Jki=1,...,m

(∆k−1χ(I1,J1),...,(Im,Jm))2hF (m)(U I1

J1 , . . . , U Im

Jm) + O(Ωk(x`)),

(∆k−1Sx−∆k−1Tx

)2h+1

=k∑

m=1

∑

(Ii,Ji)∈Ek⊗Jki=1,...,m

(∆k−1χ(I1,J1),...,(Im,Jm))2h+1F(m)(U I1

J1 , . . . , U Im

Jm) + O(Ωk(x`)).

(4.40)

16

By (4.7) and (4.23), when |J1|+ · · ·+ |Jm| 6 k,

Ξh(I1,J1),...,(Im,Jm) =

m∏

i=1

∑

`

b` Qh+1/2,`−1/2Ii,Ji −

∑

`

b`

m∏

i=1

Qh+1/2,`−1/2Ii,Ji

=m∏

i=1

∑

`

a` Qh+1/2,`Ii,Ji −

∑

`

a`

m∏

i=1

Qh+1/2,`Ii,Ji = Υh+1/2

(I1,J1),...,(Im,Jm).

(4.41)

Combining this with (4.33), when |J1| + · · · + |Jm| 6 k, the sequence χ(I1,J1),...,(Im,Jm) is a polynomial ofdegree 6 |J1|+ · · ·+ |Jm| − 2 sampled on a uniform grid, and hence its (k − 1)-th differences must vanish.In other words, all the terms in (4.40) with |J1|+ · · ·+ |Jm| 6 k vanish, and (4.5) is proved once we applyan argument akin to the one in (4.31).

Combining Theorem 2.4 and Theorem 4.1, we know that if we begin with a dense enough initial sequence,then locally all the subdivision points are well-defined and stay within a single chart3; moreover the smooth-ness of the limit curve is as high as that of the underlying linear subdivision scheme. In other words, wehave the following smoothness equivalence theorem.

Theorem 4.2. For any Ck interpolatory linear subdivision scheme T , if T is Ck, the corresponding nonlinearinterpolatory subdivision scheme S, defined by (3.11), is also Ck. For any Ck stable linear subdivision schemeT , if T is Ck, the corresponding nonlinear subdivision scheme S, defined by (3.9), is also Ck.

Remark. From the linear theory, if T is Ck, then T must reproduce Πk (i.e. condition (4.6).) Similarly, ifT is Ck, then T must reproduce Πk (i.e. condition (4.7).) Of course T (resp. T ) has to be Ck smooth beforewe can expect S (resp. S) to be Ck smooth. However, recall that the construction of S uses T , T needs notbe Ck smooth, it only needs to reproduce Πk in order for Theorem 4.2 to go through.

A Appendix

A.1 Proof of Lemma 2.1

(i) We prove that for each J ∈ N, there exists a δJ > 0 such that SJx is well-defined for any x ∈ XM,δJ . It isequivalent to showing that there exists a δJ > 0 such that |∆Sjx|∞ < δ for j = 0, · · · , J − 1 and x ∈ XM,δJ

.We use induction. When J = 1, we can choose δJ = δ. Suppose |∆Sjx|∞ < δ for j = 0, · · · , J − 1 and

x ∈ XM,δJ.

Since S is convergent, there exist α > 0 and C > 0 such that

|∆Sjx|∞ 6 CD−jα|∆x|∞for any x ∈ XM and j ∈ N.

It follows that

|∆SJx|∞ 6 |∆SSJ−1x|∞ + |∆SJx−∆SSJ−1x|∞6 CD−α|∆SJ−1x|∞ + 2|SJx− SSJ−1x|∞6 CD−α|∆SJ−1x|∞ + 2C1|∆SJ−1x|2∞6 (CD−α + 2C1δ)|∆SJ−1x|∞6 · · ·6 (CD−α + 2C1δ)J |∆x|∞.

(A.1)

Therefore we can choose

δJ+1 := min(

δJ ,δ

(CD−α + 2C1δ)J

).

3The ‘M ’ in Theorem 2.4 plays the role of a chart in the manifold setting.

17

(ii) We prove that for each K ∈ N, there exists a CK > 0 such that

|SKx− SKx|∞ 6 CKΩ1(x)

whenever SKx is well-defined.We use induction. The statement is true for K = 1 by assumption. Suppose that it is true for some

K ∈ N. Then it follows from (A.1) that

|SK+1x− SK+1x|∞ 6 |SK+1x− SSKx|∞ + |SSKx− SK+1x|∞6 C1|∆SKx|2∞ + |S|∞CK |∆x|2∞6 C1(CD−α + 2C1δ)2K |∆x|2∞ + |S|∞CK |∆x|2∞= (C1(CD−α + 2C1δ)2K + |S|∞CK)|∆x|2∞.

Therefore we can choose CK+1 := C1(CD−α + 2C1δ)2K + |S|∞CK .(iii) We prove that there exist C > 0, β > 0 and δ0 > 0 such that

|∆Sjx|∞ 6 CD−jβ |∆x|∞whenever Sjx is well-defined and |∆x|∞ < δ0.

Choose any 0 < β < α and K ∈ N such that K > logD Cα−β . Then CD−Kα < D−Kβ . It follows from (i)(ii)

that there exists CK > 0 and δK > 0 such that SKx exists for x ∈ XM,δKand

|SKx− SKx|∞ 6 CK |∆x|2∞.

Let δ′K := min(δK , D−Kβ−CD−Kα

2CK). Then for any x ∈ XM,δ′K , we have

|∆SKx|∞ 6 |∆SKx|∞ + |∆SKx−∆SKx|∞6 CD−Kα|∆x|∞ + 2CK |∆x|2∞= (CD−Kα + 2CK |∆x|∞)|∆x|∞6 D−Kβ |∆x|∞.

(A.2)

It follows from (A.1) that whenever Sjx is well-defined, we have

|∆Sjx|∞ 6 Cj |∆x|∞, (A.3)

where C = max(CD−α + 2C1δ, 1).Let δ0 := C−Kδ′K . Then it follows from (A.2)(A.3) that whenever Sjx is well-defined and |∆x|∞ < δ0,

|∆Sjx|∞ = |∆S`K+rx|∞ 6 D−`Kβ |∆Srx|∞ 6 D−`KβCr|∆x|∞ = D−jβDrβCr|∆x|∞ 6 CD−jβ |∆x|∞,

where C := max06r<K

DrβCr.

(iv) Last, we prove that there exists a δ′ > 0 such that Sjx is well-defined for any j ∈ N and x ∈ XM,δ′ .Let δ′ := min(δ, δ/C, δ0) where C and δ0 are from (iii). Then we use induction to prove that Sjx is

well-defined for any j ∈ N and x ∈ XM,δ′ . When j = 1, this is obviously true. Suppose Sjx is well-definedfor any x ∈ XM,δ′ . Then it follows from (iii) that

|∆Sjx|∞ 6 CD−jβ |∆x|∞ < C|∆x|∞ 6 δ,

for any x ∈ XM,δ′ . Therefore Sj+1x is well-defined for any x ∈ XM,δ′ .Combining (iii) and (iv), we have

|∆Sjx|∞ 6 CD−jβ |∆x|∞for any j ∈ N and x ∈ XM,δ′ .

18

A.2 Auxiliary lemmas for Theorem 2.4

Lemma A.1. Let S be a subdivision operator defined on XM,δ and S be a linear subdivision operator. Bothof them have the same dilation factor D. Suppose there exist µ1, · · · , µk+1 ∈ [0, 1) such that

|∆jSx|∞ 6 D−j+µj |∆jx|∞, ∀x, j = 1, · · · , k + 1,

andµj <

µj+1

j + 1, j = 1, · · · , k.

If there exists C > 0 such that for any x ∈ XM,δ and j = 1, · · · , k

|∆j−1Sx−∆j−1Sx|∞ 6 CΩj(x).

Then for any 0 < ε < min16j6k

(µj+1j+1 − µj

), there exist 0 < δ′ 6 δ and polynomials P2, · · · , Pk+1 such that for

any n ∈ N and any x ∈ XM,δ′

|∆Snx|∞ 6 D(−1+µ1+ε)n|∆x|∞,

|∆jSnx|∞ 6 D(−j+µj)nPj(n)|∆x|∞, j = 2, · · · , k + 1.

Proof: Since Ω1(x) = |∆x|2∞, j = 1 case has been proved in [13, Theorem 2], i.e. for any 0 < ε <

min16j6k

(µj+1j+1 − µj

), there exists 0 < δ′ 6 δ such that for any x ∈ XM,δ′ and any n ∈ N,

|∆Snx|∞ 6 D(−1+µ1+ε)n|∆x|∞.

We proceed by induction in j. Now assume the result is true for 1, · · · , j (j 6 k). Now for any n ∈ N,

|∆j+1Snx|∞ 6 |∆j+1SSn−1x|∞ + |∆j+1Snx−∆j+1SSn−1x|∞.

Since for any sequences y, z, we have |∆y −∆z|∞ = |∆(y − z)|∞ 6 2|y − z|∞. It follows that

|∆j+1Snx|∞ 6 |∆j+1SSn−1x|∞ + 4|∆j−1Snx−∆j−1SSn−1x|∞6 D−j−1+µj+1 |∆j+1Sn−1x|∞ + 4CΩj(Sn−1x).

Denote dn := |∆j+1Snx|∞. Then

dn 6 D−j−1+µj+1dn−1 + 4CΩj(Sn−1x)6 · · ·

6 D(−j−1+µj+1)nd0 + 4C

n−1∑

i=0

D(−j−1+µj+1)(n−1−i)Ωj(Six)

6 D(−j−1+µj+1)n

(d0 + 4CDj+1

n−1∑

i=0

Di(j+1−µj+1)Ωj(Six)

).

Since

Ωj(Six) =∑

γ∈Γj

|∆Six|γ1∞ · · · |∆jSix|γj∞

6∑

γ∈Γj

((D−1+µ1+ε)i|∆x|∞

)γ1 · · · ((D−j+µj )iPj(i)|∆x|∞)γj

=∑

γ∈Γj

|∆x||γ|∞P2(i)γ1 · · ·Pj(i)γj Di(µ1γ1+···+µjγj+εγ1−j−1)

6∑

γ∈Γj

|∆x||γ|∞P2(i)γ1 · · ·Pj(i)γj Di(µj |γ|+ε|γ|−j−1)

6∑

γ∈Γj

|∆x||γ|∞P2(i)γ1 · · ·Pj(i)γj Di(j+1)(µj+ε−1)

6 |∆x|∞∑

γ∈Γj

δ′|γ|−1P2(i)γ1 · · ·Pj(i)γj Di(j+1)(µj+ε−1)

19

andd0 = |∆j+1x|∞ 6 2j |∆x|∞.

It follows that

dn 6 D(−j−1+µj+1)n

2j + 4CDj+1

n−1∑

i=0

Di(j+1−µj+1)∑

γ∈Γj

δ′|γ|−1P2(i)γ1 · · ·Pj(i)γj Di(j+1)(µj+ε−1)

|∆x|∞

6 D(−j−1+µj+1)n

2j + 4CDj+1

n−1∑

i=0

∑

γ∈Γj

δ′|γ|−1P2(i)γ1 · · ·Pj(i)γj Di(j+1)(µj+ε−µj+1j+1 )

|∆x|∞

6 D(−j−1+µj+1)n

2j + 4CDj+1

n−1∑

i=0

∑

γ∈Γj

δ′|γ|−1P2(i)γ1 · · ·Pj(i)γj

|∆x|∞,

where the last inequality is due to the fact that µj + ε− µj+1j+1 < 0. Define

Pj+1(n) = 2j + 4CDj+1n−1∑

i=0

∑

γ∈Γj

δ′|γ|−1P2(i)γ1 · · ·Pj(i)γj .

Then|∆j+1Snx|∞ = dn 6 D(−j−1+µj+1)nPj+1(n)|∆x|∞.

Hence the lemma is proved.

Lemma A.2. Let S be a subdivision operator defined on XM,δ′ and S be a linear subdivision operator suchthat its derived subdivision operators S1, · · · , Sk (k > 1) all exist. S and S have the same dilation factor D.Suppose there exists C > 0 such that for any x ∈ XM,δ′ and j = 1, · · · , k

|∆j−1Sx−∆j−1Sx|∞ 6 CΩj(x).

If there exist µ1, · · · , µk+1 ∈ [0, 1) satisfying

µj <µj+1

j + 1, j = 1, · · · , k,

and polynomials P1, · · · , Pk+1 such that for any n ∈ N and any x ∈ XM,δ′

|∆jSnx|∞ 6 D(−j+µj)nPj(n)|∆x|∞, j = 1, · · · , k + 1.

Then S is Ck.

Proof: It follows from Theorem 2.3 that S is continuous. We only need to show for j = 1, · · · , k,Fn

D(Djn∆jSnx) converges uniformly as n →∞ for all x ∈ XM,δ′ . Consider

|Fn+1D (Dj(n+1)∆jSn+1x)−Fn

D(Djn∆jSnx)|∞6|Fn+1

D (Dj(n+1)∆jSn+1x)−Fn+1D (Dj(n+1)∆jSSnx)|∞ + |Fn+1

D (Dj(n+1)∆jSSnx)−FnD(Djn∆jSnx)|∞.

20

Using (2.2), we have

|Fn+1D (Dj(n+1)∆jSn+1x)−Fn+1

D (Dj(n+1)∆jSSnx)|∞=|Dj(n+1)∆jSn+1x−Dj(n+1)∆jSSnx|∞62Dj(n+1)|∆j−1Sn+1x−∆j−1SSnx|∞62CDj(n+1)Ωj(Snx)

=2CDj(n+1)∑

γ∈Γj

|∆Snx|γ1∞ · · · |∆jSnx|γj∞

62CDj(n+1)∑

γ∈Γj

(D(−1+µ1)nP1(n)|∆x|∞

)γ1 · · ·(D(−j+µj)nPj(n)|∆x|∞

)γj

=2CDj(n+1)∑

γ∈Γj

Dn(µ1γ1+···+µjγj−j−1)P1(n)γ1 · · ·Pj(n)γj |∆x||γ|∞

62CDj(n+1)∑

γ∈Γj

Dn(µj(j+1)−j−1)P1(n)γ1 · · ·Pj(n)γj δ′|γ|

62CDj(n+1)∑

γ∈Γj

Dn(µj+1−j−1)P1(n)γ1 · · ·Pj(n)γj δ′|γ|

=Dn(µj+1−1)

2CDj

∑

γ∈Γj

P1(n)γ1 · · ·Pj(n)γj δ′|γ|

.

Since µj+1 − 1 < 0 and P1, · · · , Pj are polynomials, it follows that

∞∑n=1

|Fn+1D (Dj(n+1)∆jSn+1x)−Fn+1

D (Dj(n+1)∆jSSnx)|∞

6∞∑

n=1

Dn(µj+1−1)

2CDj

∑

γ∈Γj

P1(n)γ1 · · ·Pj(n)γj δ′|γ|

< ∞.

Using Lemma 2.2, we have

|Fn+1D (Dj(n+1)∆jSSnx)−Fn

D(Djn∆jSnx)|∞=Djn|Fn+1

D (Dj∆jSSnx)−FnD(∆jSnx)|∞

=Djn|Fn+1D (Sj∆jSnx)−Fn

D(∆jSnx)|∞6CDjn|∆j+1Snx|∞6CDjnD(−j−1+µj+1)nPj+1(n)|∆x|∞6CDn(µj+1−1)Pj+1(n)δ′.

Since µj+1 − 1 < 0 and Pj+1 is a polynomial, it follows that

∞∑n=1

|Fn+1D (Dj(n+1)∆jSSnx)−Fn

D(Djn∆jSnx)|∞ 6∞∑

n=1

CDn(µj+1−1)Pj+1(n)δ′ < ∞.

Combining all the above, we obtain

∞∑n=1

|Fn+1D (Dj(n+1)∆jSn+1x)−Fn

D(Djn∆jSnx)|∞ < ∞.

Hence FnD(Djn∆jSnx) is a Cauchy sequence with respect to | · |∞. Therefore Fn

D(Djn∆jSnx) convergesuniformly as n →∞.

21

Lemma A.3. Let S be a subdivision operator defined on XM,δ and S be a linear subdivision operator suchthat its derived subdivision operators S1, · · · , Sk all exist. Suppose S and S have the same dilation factor. Ifthere exists C1 > 0 such that for any x ∈ XM,δ and j = 1, · · · , k

|∆j−1Sx−∆j−1Sx|∞ 6 C1Ωj(x).

Then for any n ∈ N, there exists Cn > 0 such that for any x ∈ XM,δ and j = 1, · · · , k

|∆j−1Snx−∆j−1Snx|∞ 6 CnΩj(x).

Proof: Suppose the dilation factor of S and S is D. Then Sj∆j = Dj∆jS for j = 1, · · · , k. Hence

∆jSn = D−njSnj ∆j , ∀n ∈ N, j = 1, · · · , k.

We use induction to prove the result. When n = 1, the result is true by assumption. Now assume theresult is true for n− 1. Then for j = 1, · · · , k

|∆j−1Snx−∆j−1Snx|∞ 6 |∆j−1Snx−∆j−1SSn−1x|∞ + |∆j−1SSn−1x−∆j−1Snx|∞6 Cn−1Ωj(Sn−1x) + D1−j |Sj−1∆j−1Sn−1x− Sj−1∆j−1Sn−1x|∞6 Cn−1Ωj(Sn−1x) + D1−j |Sj−1|∞Cn−1Ωj(x).

Since for j = 1, · · · , k,

|∆jSn−1x|∞ 6 |∆jSn−1x|∞ + |∆jSn−1x−∆jSn−1x|∞6 D−(n−1)j |Sn−1

j ∆jx|∞ + Cn−1Ωj(x)

6 D−(n−1)j |Sn−1j |∞|∆jx|∞ + Cn−1Ωj(x).

It follows that

Ωj(Sn−1x) =∑

γ∈Γj

|∆Sn−1x|γ1∞ · · · |∆jSn−1x|γj∞

6∑

γ∈Γj

(D−(n−1)|Sn−1

1 |∞|∆x|∞ + Cn−1Ω1(x))γ1 · · ·

(D−(n−1)j |Sn−1

j |∞|∆jx|∞ + Cn−1Ωj(x))γj

.

It is easy to verify that for any γ ∈ Γj , there exists Cγ,n−1 > 0 such that(D−(n−1)|Sn−1

1 |∞|∆x|∞ + Cn−1Ω1(x))γ1 · · ·

(D−(n−1)j |Sn−1

j |∞|∆jx|∞ + Cn−1Ωj(x))γj

6 Cγ,n−1Ωj(x),

where Cγ,n−1 depends on D, γ, Cn−1 and |Sn−11 |∞, · · · , |Sn−1

j |∞. Therefore

Ωj(Sn−1x) 6

∑

γ∈Γj

Cγ,n−1

Ωj(x).

Hence

|∆j−1Snx−∆j−1Snx|∞ 6 Cn−1

∑

γ∈Γj

Cγ,n−1

Ωj(x) + D1−j |Sj−1|∞Cn−1Ωj(x)

6 CnΩj(x),

where

Cn = Cn−1 max16j6k

∑

γ∈Γj

Cγ,n−1 + D1−j |Sj−1|∞

.

22

References

[1] M. P. do Carmo. Differential Geometry of Curves and Surfaces. Prentice-Hall, 1976.

[2] N. Dyn. Subdivision Schemes in Computer-Aided Geometric Design, pages 36–104. Advances in NumericalAnalysis II, Wavelets Subdivision Algorithms and Radial Basis Functions. Clarendon Press, Oxford, 1992.

[3] N. Dyn, J. Gregory, and D. Levin. Analysis of uniform binary subdivision schemes for curve design. ConstructiveApproximation, 7:127–147, 1991.

[4] N. Dyn and D. Levin. Subdivision schemes in geometric modelling. Acta Numerica, 11:73–144, 2002.

[5] P. Grohs. Higher order smoothness of interpolatory multivariate subdivision in Lie groups. April 2007.

[6] P. Grohs. Smoothness equivalence properties of univariate subdivision schemes and their projection analogues.Manuscript, June 2007.

[7] P. Grohs and J. Wallner. Log-exponential analogues of univariate subdivision schemes in lie groups and theirsmoothness properties. 2007.

[8] S. Lang. Real and Functional Analysis. Springer-Verlag, 3rd edition, 1993.

[9] J. Munkres. Topology. Prentice Hall, Inc., 1975.

[10] I. Ur Rahman, I. Drori, V. C. Stodden, D. L. Donoho, and P. Schroder. Multiscale representations for manifold-valued data. Multiscale Modeling and Simulation, 4(4):1201–1232, 2005.

[11] O. Rioul. Simple regularity criteria for subdivision schemes. SIAM J. Math. Anal., 23(6):1544–1576, November1992.

[12] J. Wallner. Smoothness analysis of subdivision schemes by proximity. Constructive Approximation, 24(3):289–318, 2006.

[13] J. Wallner and N. Dyn. Convergence and C1 analysis of subdivision schemes on manifolds by proximity. ComputerAided Geometric Design, 22(7):593–622, 2005.

[14] J. Wallner, E. Nava Yazdani, and P. Grohs. Smoothness properties of Lie group subdivision schemes. MultiscaleModeling and Simulation, 6(2):493–505, 2007.

[15] G. Xie. Analysis of Subdivision Schemes in Nonlinear and Geometric Settings. PhD thesis, Department ofMathematical Sciences, Rensselaer Polytechnic Institute, May 2006.

[16] G. Xie and T. P.-Y. Yu. Smoothness equivalence properties of interpolatory Lie group subdivision schemes. Toappear in IMA Journal on Numerical Analysis, 2007.

[17] G. Xie and T. P.-Y. Yu. Smoothness equivalence properties of manifold-valued data subdivision schemes basedon the projection approach. SIAM Journal on Numerical Analysis, 45(3):1200–1225, 2007.

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200,000

Fifth Derivative

6th

Figure 3: Divided difference plots of level j = 8 subdivision of Donoho’s original log-exp scheme and ourmodified log-exp scheme on S2-valued data.

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