Subdivision Schemes and Irregular Grids

Numerical Algorithms 35: 1–28, 2004. 2004 Kluwer Academic Publishers. Printed in the Netherlands.

Subdivision schemes and irregular grids

Voichita Maxim and Marie-Laurence MazureLaboratoire de Modélisation et Calcul (LMC-IMAG), Université Joseph Fourier, BP 53,

38041 Grenoble Cedex, FranceE-mail: {maxim,mazure}@imag.fr

Received 21 November 2002; accepted 15 September 2003Communicated by C. Brezinski

The present paper deals with subdivision schemes associated with irregular grids. Wefirst give a sufficient condition concerning the difference scheme to obtain convergence. Thiscondition generalizes a necessary and sufficient condition for convergence known in the caseof uniform and stationary schemes associated with a regular grid. Through this sufficientcondition, convergence of a given subdivision scheme can be proved by comparison withanother scheme. Indeed, when two schemes are equivalent in some sense, and when onesatisfies the sufficient condition for convergence, the other also satisfies it and it thereforeconverges too. We also study the smoothness of the limit functions produced by a schemewhich satisfies the sufficient condition. Finally, the results are applied to the study of Lagrangeinterpolating subdivision schemes of any degree, with respect to particular irregular grids.

Keywords: subdivision, equivalent schemes, irregular grids, Lagrange interpolation

AMS subject classification: 41A05, 65D05, 65D17, 65D10, 65Q05

1. Introduction

In computer aided geometric design, subdivision schemes are a classical processto construct curves and surfaces. The topic has been widely explored and our list of ref-erences is extremely limited compared to the abundant literature existing in the domain.This paper will deal only with curves. The principle is simple. A curve in R

d is obtainedas the limit of a sequence of polygonal lines in this space. At each level, each vertex ofthe polygonal line is calculated from a finite number of vertices of the polygonal line ofthe previous level (actually, as a linear combination in the linear case). The limit curveis expected to be at least continuous, and if possible, to satisfy additional smoothnessproperties. As an instance, starting from a bi-infinite sequence of initial vertices, sup-pose that the new vertices are located at one fourth and three fourths respectively on eachsegment of the initial polygonal line. Repeating this process describes the well-knownChaikin algorithm [2], and the limit curve it produces is the quadratic C1 polynomialspline curve associated with regularly spaced knots, the control polygon of which is theinitial polygonal line [13].

2 V. Maxim, M.-L. Mazure / Subdivision schemes and irregular grids

The latter particular example is naturally related to the regular grid of R. The analy-sis of the convergence of any subdivision schemes is generally done by reference to theregular grid too, that is by considering the convergence of the sequence of piecewiseaffine functions parameterizing the polygonal line of level j , with knots at the points2−j k, k ∈ Z. However the limitation to regular grids is far too restrictive. The ne-cessity of allowing non regular ones naturally arises, for instance, through the exampleof spline curves. Any such curve of any degree, associated with non regularly spacedknots, can indeed be viewed as the limit curve through a subdivision scheme, obtainedby inserting arbitrary new knots at each step. Hence the underlying grid is not regular.Another motivation for the consideration of non regular grids comes from wavelets andmultiresolution analysis to which subdivision schemes are closely related. For a longtime wavelets were considered on regular grids. Yet, the necessity of extending themto the nonregular case appeared in recent years with the introduction of second genera-tion wavelets (see, for instance, [14]). For subdivision schemes concerning semi-regulargrids (that is grids obtained as in the regular case by inserting the midpoints at each level,but the original points being nonregularly spaced) we refer to [15], and for more generalirregular grids to [3,4].

The present paper studies the convergence of subdivison schemes defined rela-tive to irregular grids. The main idea is to derive the convergence of a given subdivisionscheme by comparison with another subdivision scheme to which it is equivalent in somesense. The idea of comparing subdivision schemes is not new. In the regular setting, itwas used in [10], where the authors deduce, in particular, the convergence of nonsta-tionary schemes from that of stationary ones. Let us also quote a recent paper [12] inwhich the authors consider subdivision schemes associated which irregular grids whichgeneralize the four-point scheme introduced in [11]. Provided that the grids satisfy con-ditions which are similar to those we shall require in the present paper, convergence ofsuch schemes to continuously differentiable functions is obtained by comparison withthe corresponding regular case.

Our paper is organized as follows. The second section gathers the basic tools aboutsubdivision schemes, along with the notion of convergence which we shall use. In thethird section we establish a sufficient condition for convergence for subdivision schemesin terms of the difference schemes. This condition extends a well-known necessaryand sufficient condition for stationary and uniform schemes associated with the regu-lar grid. We also show that if the nonregular grid is not so far from the regular one(quasi-regularity), the limit functions are more regular than just continuous, exactly asin the regular case. In the fourth section we introduce the notion of equivalence betweensubdivision schemes, and we show that if two schemes are equivalent and one of themsatisfies the sufficient condition established in section 3, the other also satisfies it, andtherefore it converges. Applying the latter result to the derived schemes, we show in thefifth section how to derive smoothness of higher order of the limit functions produced bya given subdivision scheme from properties satisfied by an equivalent one. Finally, thelast section illustrates the previous results through the study of Lagrange interpolatingsubdivision schemes associated with particular quasi-regular grids. The convergence is

V. Maxim, M.-L. Mazure / Subdivision schemes and irregular grids 3

then deduced from that of the corresponding regular Lagrange interpolating schemes.Convergence of Lagrange interpolating subdivision schemes relative to irregular gridswas already addressed in [3], but only for degrees 3 and 5. A weak additional assump-tion on the grids enabled us to obtain interesting results of convergence for any (odd)degrees. A detailed comparison with the results of [3] is given at the end of our paper.

2. Subdivision schemes and grids

This section provides the reader with the vocabulary we shall use throughout thepaper. Even though some notions and results can be considered classical, we shall giveshort proofs, first for the sake of completeness, and also to emphasize the precise rôle ofeach assumption.

Though most schemes we know are defined relative to a grid (it will be so in par-ticular for the Lagrange intepolatory schemes studied in section 6), it is on purpose thatwe shall first introduce subdivision schemes independently of any grid. This choice willenable us to have a better grasp of which properties or assumptions are attached to theschemes themselves, and which involve more deeply the grids themselves.

2.1. Subdivision schemes

We define a subdivision scheme as an infinite sequence S = {Sj , j � 0} of matricesSj := (Sj,k,�)k,�∈Z. When selecting such a subdivision scheme, we shall always assumethat:

(SS1) the scheme is (binarily) local in the sense that there exists an integer nS such thatfor any j � 0, and any k, � ∈ Z,

Sj,k,� �= 0 ⇒ |k − 2�| � nS , (2.1)

(SS2) the scheme is bounded in the sense that there exists a number MS < +∞ suchthat, for all j � 0,

‖Sj‖ := Sup{|Sj,k,�|, k, � ∈ Z

}� MS . (2.2)

By nS we shall actually denote the smallest integer satisfying (2.1) and we shall referto it as the length of the local subdivision scheme S . Condition (SS1) means that, atany level j , the column of index � of the matrix Sj contains at most 2nS + 1 nonzeroelements centered around the row of index 2�. As for the number of nonzero elementsof a given row, it is bounded above by nS + 1.

Let us denote by f0 := (f0,k)k∈Z a given bi-infinite sequence in Rd , for some

d � 1, so that f0 represents a matrix with d columns and a bi-infinite number of rows.Due to our requirement (SS1), the subdivision scheme enables us to calculate recursivelya sequence fj := (fj,k)k∈Z, j � 0, as follows:

fj+1 := Sjfj , j � 0. (2.3)


The second requirement (SS2) guarantees, in particular, that if we start with a bounded f0

(in the sense that ‖f0‖∞ := Supk∈Z‖f0,k‖ < +∞, where ‖ · ‖ denotes a given norm in

Rd), then each fj will also be bounded.

Whatever the initial f0, it is usual to write fj as

fj =∑i∈Z

ϕ〈i〉j f0,i, j � 0, (2.4)

where, for any i ∈ Z, we denote by ϕ〈i〉j , j � 0, the sequence in R

Z defined by

ϕ〈i〉j+1 := Sjϕ〈i〉

j , ϕ〈i〉0 := (

ϕ〈i〉0,k

)k∈Z

:= (δi,k)k∈Z. (2.5)

Relation (2.4) comes from the obvious equality f0 = ∑i∈Zϕ

〈i〉0 f0,i , and it is made valid

by condition (2.1) implying that, at any level j � 0, only a finite number of ϕ〈i〉j,k are non-

zero. We can more precisely state:

ϕ〈i〉j,k �= 0 ⇒ 2j i − (

2j − 1)nS � k � 2j i + (

2j − 1)nS . (2.6)

Let us consider the matrix � := (�k,�)k,�∈Z defined by

�k,� := δk+1,� − δk,�.Given any sequence w = (wk)k∈Z in R

d , the product �w is well-defined and its rowsare the differences of order 1 of the components of w, namely, (�w)k = wk+1 − wk forall k ∈ Z. With a subdivision scheme it is classical to associate its difference schemeaccording to the definition below.

Proposition and definition 2.1. Let S be a local and bounded subdivision scheme.Then, the subdivision scheme D = {Dj, j � 0} defined by

Dj,k,� :=∑i��+1

(Sj,k+1,i − Sj,k,i), j � 0, k, � ∈ Z, (2.7)

is called the difference scheme associated with S . It is bounded and it satisfies the so-called commutation formula:

�Sj = Dj�, j � 0. (2.8)

Proof. Due to S being local, the sum in (2.7) is actually a finite one, with at most nS+2nonzero elements. Hence the scheme D is well-defined and it is also bounded since Sitself is bounded.

Each side of (2.8) is well-defined, and equality (2.8) results from the simple obser-vation that

Sj,k+1,� − Sj,k,� =∑i��(Sj,k+1,i − Sj,k,i)−

∑i��+1

(Sj,k+1,i − Sj,k,i)

=Dj,k,�−1 −Dj,k,�. �


Without first making sure that each row of each Dj has only a finite number ofnonzero elements, (2.8) makes the equalities

�(Sjv) = Dj(�v), j � 0, (2.9)

valid only for all sequences v = (vk)k∈Z in Rd such that all vk are equal to 0 except

for a finite number of indices k. Hence the importance of ensuring for instance that thedifference scheme is local. From (2.1) we can deduce that, if 2� > k + nS − 1, thenDj,k,� = 0 for all j � 0. However, without any additional assumption on S , we cannotassert that the difference scheme D is local.

Definition 2.2. We say that a local and bounded subdivision scheme S reproduces con-stants if, whenever f0,k = a ∈ R

d for all k ∈ Z, then fj,k = a for all j � 0 and allk ∈ Z.

In other words, the subdivision scheme S reproduces constants iff, at any levelj � 0, we have ∑

i∈Z

ϕ〈i〉j = 1, (2.10)

where 1 denotes the element of RZ with all components equal to one. Clearly, the repro-

duction of constants is satisfied iff∑�∈Z

Sj,k,� = 1, j � 0, k ∈ Z. (2.11)

This is thus a natural assumption, which means that each point fj+1,k ∈ Rd is obtained

as an affine combination of nS + 1 or nS points fj,� of consecutive indices centeredat k/2 (depending on whether k has the same parity as nS or not).

Proposition 2.3 (see [4]). Assume that the local and bounded subdivision scheme S re-produces constants. Then the difference scheme D defined by (2.7) is not only bounded,but also local, with length less than or equal to that of S . More precisely, for any j � 0,and any k, � ∈ Z:

Dj,k,� �= 0 ⇒ −nS � 2�− k � nS − 1. (2.12)

Moreover, the difference scheme enables us to calculate recursively the differences oforder 1 obtained from fj , that is, �fj = (fj,k+1 − fj,k)k∈Z,

�fj+1 = Dj�fj , j � 0. (2.13)


Proof. We have already observed that Dj,k,� = 0 as soon as 2� > k+nS −1. Considernow two integers k, � ∈ Z such that 2� < k − nS . Due to (2.1), for any i � �, we haveSj,k+1,i = Sj,k,i = 0. Therefore, we can write

Dj,k,� =∑i∈Z

Sj,k+1,i −∑i∈Z

Sj,k,i, 2� < k − nS .

In case S reproduces constants, this is equal to 0 on account of (2.11). Hence, (2.12) isproved and D is local. Equality (2.9) is now valid for any v, and (2.13) is obtained whenapplying it to v = fj . �

2.2. Grids and convergence

In geometric design, subdivision schemes are classical ways of producing curvesin R

d , as “limits" of the polygonal lines defined by the fj , that is the polygonal lineswith vertices the points fj,k, k ∈ Z. This requires the choice of a notion of convergencefor such sequences of polygonal lines. For this purpose, we shall introduce parameteri-zations by choosing a grid according to the definition below.

Definition 2.4. A (binary) grid is a set X = {xj,k, j � 0, k ∈ Z} of real numbers xj,k ,meeting the following requirements:

(G1) at the coarsest level, we have limk→−∞ x0,k = −∞, limk→+∞ x0,k = +∞,

(G2) at any level j � 0, the sequence xj,k , k ∈ Z, is strictly increasing,

(G3) the set of points of a given level j � 0 is contained in the set of points of the nextlevel j + 1, in the sense that xj+1,2k = xj,k for any j � 0, k ∈ Z.

Such a grid is said to be regular when, at any level j , the difference xj,k+1 − xj,kbetween two consecutive points does not depend on k. Without any loss of generality, inthe regular case we shall always suppose that xj,k = 2−j k for any j � 0 and any k ∈ Z,which it is so anyway up to a translation of indices and to a multiplication by a positiveconstant. In this paper we are concerned with irregular grids.

The three requirements (Gi), i = 1, 2, 3, ensure that

xj,k < xj+1,2k+1 < xj,k+1, j � 0, k ∈ Z, (2.14)

and that for any level j � 0,

limk→−∞

xj,k = −∞, limk→+∞

xj,k = +∞. (2.15)

Given a local and bounded subdivision scheme S , and given a sequence f0 := (f0,k)k∈Z

in Rd , we consider the sequence fj , j � 0, defined by (2.3). Due to (2.15), for any

nonnegative integer j , the function Fj : R → Rd such that

Fj is affine on [xj,k, xj,k+1], Fj (xj,k) = fj,k, k ∈ Z, (2.16)


is well-defined. It is a particular parameterization of the polygonal line defined by fj .Moreover, since S is bounded, in case f0 is bounded, each function Fj is bounded on R.

Definition 2.5. We say that the subdivision scheme S converges with respect to a grid Xif, for any bounded f0 ∈ (Rd)Z, the corresponding sequence Fj , j � 0, convergesuniformly on R. We say that subdivision scheme S converges if there exists a grid Xwith respect to which S converges.

Remarks 2.6. 1. When the subdivision scheme S converges, it may converge with re-spect to different grids. Nevertheless, the limit curve C := {F(t), t ∈ R} is unique. Thisis due to the uniform convergence. Indeed, suppose that S converges with respect to twodifferent grids X and X . Then, at a given level j , the corresponding two parameteriza-tions Fj and Fj of the polygonal line defined by fj are linked by:

Fj (t) = Fj(ψj(t)

), t ∈ R, (2.17)

where ψj : R → R is the function which satisfies ψj(xj,k) = xj,k for all k ∈ Z and whichis affine on each [xj,k, xj,k+1]. Given P = F(t0) = limj→+∞ Fj(t0) ∈ C, let k0 denotethe unique integer such that t0 ∈ [x0,k0 , x0,k0+1[. Due to the axioms on grids, for anyj � 0, the function ψj is increasing on Z, and ψj(t0) belongs to [x0,k0, x0,k0+1[. Hencewe can find a convergent subsequence ψjr (t0), r ∈ N. Setting t0 := limr→+∞ ψjr (t0),the uniform convergence of the sequence Fj , j � 0, to a function F , guarantees that

F (t0) = limr→+∞ Fjr

(ψjr (t0)

),

i.e., according to (2.17), P = F (t0). Hence, in that case, the two limit functions F andF are two different parameterizations of the same curve C.

2. In order to prove the convergence of a subdivision scheme, we can limit our-selves to the case d = 1, which we shall always do from now on. We shall use thefollowing notations: for any local and bounded scheme S ,

‖Sj‖ � ‖Sj‖∞ := Supk∈Z

∑�∈Z

|Sj,k,�| � (nS + 1)‖Sj‖,

and, for any real valued function f continuous and bounded on R, ‖f ‖∞ :=Supx∈R

|f (x)|.3. Let S be a given local and bounded subdivision scheme. Choosing a grid X , for

any i ∈ Z, denote by �〈i〉j the piecewise affine function interpolating the data ϕ〈i〉

j,k (see(2.5)) at the points xj,k . Due to (2.6), it satisfies

�〈i〉j (x) = 0 if x /∈ ]x0,i−nS , x0,i+nS [.

In case the scheme converges with respect to X , the support of the corresponding limitfunction �〈i〉 := limj→+∞�

〈i〉j thus satisfies

Supp(�〈i〉) ⊂ [x0,i−nS , x0,i+nS ]. (2.18)


If in addition S reproduces constants, the equality (2.10) implies that∑i∈Z�

〈i〉j (x) = 1

for all x. Hence we have ∑i∈Z

�〈i〉(x) = 1, x ∈ R. (2.19)

3. How to obtain convergence?

In this section, we are concerned with finding conditions to ensure the convergenceof a subdivision scheme, according to the definition we gave in the previous one. Inthe classical case of uniform and stationary schemes associated with the regular grid,necessary and sufficient conditions are known. Inspired by this case, we will actuallyobtain a condition on the difference scheme sufficient for convergence.

3.1. A sufficient condition for convergence

It is usual to say that a subdivision scheme S is stationary if Sj = S for all non-negative j , and that it is uniform if Sj,k,� = Sj,k+2,�+1 for all indices j � 0, k, � ∈ Z. Incase S is both uniform and stationary, then the associated difference scheme D is uni-form and stationary, and we denote by D = Dj for all j � 0 the corresponding matrix.Let us recall the following result.

Theorem 3.1 [7,8]. Let S be a local, uniform, and stationary subdivision scheme whichreproduces constants. Then, the following two properties are equivalent:

(i) S is convergent (with respect to the regular grid),

(ii) there exits an integer K > 0 such that

µ := ∥∥DK∥∥∞ < 1, (3.1)

where D is the matrix of the (uniform and stationary) difference subdivision schemeassociated with S .

The next result generalizes the previous one to the irregular case. However we shallonly be able to obtain a sufficient condition for convergence. The proof is modeled onthat of theorem 3.1.

Theorem 3.2. Assume that the local and bounded subdivision scheme S reproducesconstants and that its difference scheme D satisfies the following property:

there exist two integers J,K � 0, there exists a number µ ∈ ]0, 1[ such that

‖Dj+K · · ·Dj+1Dj‖∞ � µ for all j � J. (∗)

Then the scheme S converges with respect to any grid.

The proof of this theorem will make use of the following two lemmas.


Lemma 3.3. Given two local and bounded subdivision schemes T = {Tj , j � 0} andU = {Uj , j � 0}, there exists a bounded subdivision scheme A = {Aj , j � 0} suchthat

Tj − Uj = Aj�, j � 0. (3.2)

Furthermore, if both T and U reproduce constants, then the scheme A is also local.

Proof. For a given j � 0, we are actually looking for a matrix Aj satisfying

Tj,k,� − Uj,k,� = Aj,k,�−1 − Aj,k,�. (3.3)

The two schemes T and U being local, the following quantities are well defined:

Aj,k,� :=∑i��(Uj,k,i − Tj,k,i), k, � ∈ Z.

The corresponding matrix Aj does satisfy (3.3) and the boundedness of T and U impliesthat of the subdivision scheme A so defined. Since Tj,k,� = 0 for |k − 2�| > nT andUj,k,� = 0 for |k−2�| > nU , for 2� < k−max(nT , nU), we have Aj,k,� = 0. Moreover,for any j � 0, we also have

Aj,k,� =∑i∈Z

Uj,k,i −∑i∈Z

Tj,k,i, 2� > k + max(nT , nU ).

This is equal to 0 in case both T and U reproduce constants. Hence, if so, the scheme Ais local, with nA � max(nT , nU). �

Lemma 3.4. If the difference scheme of the subdivision scheme S satisfies (∗), thenthere exists a constant H such that, for any bounded initial vector f0,

‖�fj‖∞ � Hµj‖�fJ‖∞, j � J, (3.4)

where µ := µ1/(K+1).

Proof. For any j � 0, iterating (2.13) yields

�fj+K+1 = (Dj+K · · ·Dj)�fj .Hence, for j � J , (∗) leads to

‖�fj+K+1‖∞ � ‖Dj+K · · ·Dj‖∞ ‖�fj‖∞ � µ‖�fj‖∞.

Iteration of the latter inequality gives, for any q � 0,

‖�fj+q(K+1)‖∞ � µq‖�fj‖∞, j � J. (3.5)

Any integer j � J can be written as j = J + q(K + 1) + r, with 0 � r � K. Sinceµ ∈ ]0, 1[ and q � (j − J −K)/(K + 1), by application of (3.5) we obtain

‖�fj‖∞ � µq‖�fJ+r‖∞ � µj µ−(J+K)‖�fJ+r‖∞, j � J.


Using the commutation formula again, one can write �fJ+r = (DJ+r−1 · · ·DJ)�fJ ,whence the announced equality (3.4) with

H := µ−(J+K) max(1, ‖DJ‖∞, . . . , ‖DJ+K−1 · · ·DJ‖∞

). (3.6)

�

Proof of theorem 3.2. Assume the local and bounded subdivision scheme S to repro-duce constants and its difference scheme to satisfy (∗). Given a fixed grid X = {xj,k,j � 0, k ∈ Z}, we shall prove the convergence of S with respect to X . Let f0 = (f0,k)k∈Z

denote a fixed bounded element of RZ. For any nonnegative integer j , fj = (fj,k)k∈Z is

obtained through (2.3) and Fj denotes the corresponding piecewise affine interpolatingfunction as introduced in (2.16). This function can be written as follows:

Fj =∑�∈Z

fj,� (j,�, (3.7)

where, for any � ∈ Z, (j,� : R → R denotes the function which is affine on each[xj,k, xj,k+1] and which satisfies the interpolation conditions:

(j,�(xj,k) = δk,�, k, � ∈ Z. (3.8)

In order to prove the convergence of the sequence Fj , j � 0, we shall first evaluate thedifference

Fj+1 − Fj =∑k∈Z

fj+1,k(j+1,k −∑�∈Z

fj,�(j,�. (3.9)

Let us introduce the following notation:

dj,k := xj,k+1 − xj,k, j � 0, k ∈ Z. (3.10)

For any integer �, we can write

(j,� =∑k∈Z

Uj,k,�(j+1,k, (3.11)

with

Uj,k,� := (j,�(xj+1,k) =

dj+1,2�−2/dj,�−1 if k = 2� − 1,1 if k = 2�,dj+1,2�+1/dj,� if k = 2� + 1,0 otherwise.

(3.12)

Using both the relation fj+1 = Sjfj , and (3.11), equality (3.9) becomes

Fj+1 − Fj =∑k∈Z

[∑�∈Z

(Sj,k,� − Uj,k,�)fj,�](j+1,k. (3.13)

The function |Fj+1 − Fj | is bounded and piecewise affine. On any interval [xj+1,k,

xj+1,k+1] it reaches its maximum at either xj+1,k or xj+1,k+1. Hence,


‖Fj+1 − Fj‖∞ = Supk∈Z

∣∣∣∣∑�∈Z

(Sj,k,� − Uj,k,�)fj,�∣∣∣∣ = ∥∥(Sj − Uj)fj

∥∥∞, (3.14)

where Uj := (Uj,k,�)k,�∈Z. The corresponding subdivision scheme U is local (withnU = 1), bounded (withMU = 1), and it reproduces constants, as well as S . Therefore,lemma 3.3 ensures the existence of a local and bounded subdivision scheme A such that

Sj − Uj = Aj�, j � 0.

On this account, (3.14) leads to

‖Fj+1 − Fj‖∞ � ‖Aj‖∞‖�fj‖∞.

According to lemma 3.4, the assumption (∗) implies that ‖�fj‖∞ � H µj‖�fJ‖∞, forany j � J , where µ := µ1/(K+1) ∈ ]0, 1[. This eventually proves the existence of aconstant ) (independent of f0) such that

‖Fj+1 − Fj‖∞ � )µj‖�fJ‖∞, j � J, ) := H MA (nA + 1). (3.15)

By a standard argument it follows that

‖Fj+q − Fj‖∞ � C µj‖�fJ‖∞, j � J, q � 0, (3.16)

with C := )/(1 − µ). The uniform convergence on R of the sequence Fj , j � 0, isproved. �

Note that in case the subdivision scheme S is uniform and stationary, our con-dition (∗) is nothing but condition (3.1). Therefore, according to theorem 3.1, in thisparticular case, condition (∗) is automatically ensured by the convergence of the schemewith respect to the regular grid. As a corollary of theorem 3.2, we can thus state:

Corollary 3.5. Let S be a local, stationary, and uniform subdivision scheme which re-produces constants. Then, the scheme S converges with respect to the uniform grid iff itconverges with respect to any grid.

Just by making the integer q go to +∞ in (3.16) we can state the following result.

Proposition 3.6. The assumptions are the same as in theorem 3.2. Then, there exist apositive C (independent of f0) such that the limit function F satisfies

‖F − Fj‖∞ � Cµj‖�fJ‖∞, j � J. (3.17)

3.2. Quasi-regular grids

The limits obtained in the previous subsection are continuous functions as uni-form limits of continuous functions. However, without any additional assumption on thegrid, we cannot say more on them, while in the regular case, it is known that the limitsgenerated by uniform and stationary schemes satisfy a Lipschitz condition. Under our


condition (∗), the same result is actually valid in the irregular case as soon as we usegrids which are not so far from being regular, according to the definition below.

Definition 3.7. A grid X is said to be quasi-regular if there exist positive numbers a, bsuch that

a2−j � dj,k � b2−j , j � 0, k ∈ Z, (3.18)

with the notation dj,k introduced in (3.10).

As a matter of fact, in the present subsection, only the left part of (3.18) is needed.The right part will be useful to obtain differentiability results in section 5.

Quasi-regularity implies in particular density of the grid in R. The regular grid is ofcourse quasi-regular. Apart from this trivial case, it is easy to obtain examples of quasi-regular grids. For instance, choose any increasing sequence x0,k, k ∈ Z, which satisfiesboth (G1) and a quasi-regularity condition (3.18) at level j = 0, then compute the nextlevels recursively by xj+1,2k+1 := (xj,k + xj,k+1)/2. This provides a quasi-regular grid.This case is often referred to as the semi-regular case.

As in the regular, uniform, and stationary setting, using quasi-regular grids ensuresmore regularity of the limit functions, as stated in the following theorem.

Theorem 3.8. The assumptions are the same as in theorem 3.2. Let X be a quasi-regulargrid satisfying (3.18). Then, for any bounded f0, the limit function F generated by Sis Lipschitz of index ν := − log2 µ (where µ := µ1/(K+1)), i.e., there exists a positivenumber M such that ∣∣F(x)− F(y)∣∣ � M |x − y|ν , x, y ∈ R. (3.19)

Proof. For any bounded f0, the limit function F is continuous and bounded. Hence,in order to prove the inequality (3.19), it is sufficient to consider real numbers x, y suchthat

0 <|x − y|a

� 2−J .

The interval ]0, 2−J ] being the disjoint union of all the intervals ]2−j−1, 2−j ], j � J ,there exists a unique j0 � J such that

2−j0−1 <|x − y|a

� 2−j0 . (3.20)

Using this particular integer j0, we can write∣∣F(x)− F(y)∣∣ �∣∣F(x)− Fj0(x)∣∣ + ∣∣F(y)− Fj0(y)∣∣+ ∣∣Fj0(x)− Fj0(y)∣∣. (3.21)

The right inequality in (3.20) implies the existence of an integer k0 such that both x andy are located in the interval [xj0,k0 , xj0,k0+2]. This allows us to write∣∣Fj0(x)− Fj0(y)∣∣ � 2‖�fj0‖∞ � 2Hµj0‖�fJ‖,


the left inequality resulting from the piecewise affinity of the function Fj0 , and the rightone from lemma 3.4.

On the other hand, according to (3.17), each of the two quantities |F(x)− Fj0(x)|and |F(y)− Fj0(y)| is bounded above by Cµj0‖�fJ‖∞. Consequently, (3.21) yields∣∣F(x)− F(y)∣∣ � Lµj0‖�fJ‖∞, (3.22)

with L := 2C + 2H .Finally, the left part in (3.20) gives

µj0 = (2−j0)− log2 µ <

(2

a|x − y|

)− log2 µ

.

Hence, (3.22) gives∣∣F(x) − F(y)∣∣ � )‖�fJ‖∞ |x − y|− log2 µ, for all x, y such that |x − y| � a 2−J ,(3.23)

where ) := L(2/a

)− log2 µ is independent of f0. �

Remark 3.9. If a local and bounded subdivision scheme S reproduces constants andfulfils the condition (∗), then the number µ satisfies:

1

2K+1� µ < 1. (3.24)

Indeed suppose that µ < 1/2K+1. Then, according to theorem 3.8, any limit function Fwith respect to a quasi-regular grid X , would be Lipschitz of index ν > 1. This wouldimply that F is constant on R. As a particular case, for any i ∈ Z, the function �〈i〉introduced in 2.6, 3, being constant and of support contained in [x0,i−nS , x0,i+nS ] wouldbe equal to 0. This would contradict equality (2.19). Hence µ � 1/2K+1.

4. Equivalent schemes and convergence

The problem we are concerned with in this section is how to make sure that thedifference scheme D of a given subdivision scheme S satisfies condition (∗). This willbe done by comparing the subdivision scheme S to a reference scheme the differencescheme of which does satisfy (∗). With this aim in mind, let us introduce the followingdefinition.

Definition 4.1. We say that two subdivision schemes S and S are equivalent if thereexist two positive numbers α, β such that∥∥Sj − Sj

∥∥ � α 2−βj , j � 0. (4.1)

Theorem 4.2. Let S and S be two local and bounded subdivision schemes which re-produce constants and which are equivalent. If S satisfies property (∗), then S in turnsatisfies property (∗).


The proof relies on the following preliminary result.

Lemma 4.3. Let S and S be two local and bounded subdivision schemes which repro-duce constants. If S and S satisfy (4.1), then for each r � 0, there exits a positivenumber Cr such that∥∥Dj+r · · ·Dj − Dj+r · · · Dj

∥∥� Cr 2−βj , j � 0. (4.2)

Proof. We shall prove (4.2) by induction on r. For r = 0, we just have to prove that∥∥Dj − Dj∥∥ � C0 2−βj , j � 0, (4.3)

for some positive C0. Now, according to (2.7), for any k, � ∈ Z,

Dj,k,� − Dj,k,� =∑i��+1

(Sj,k+1,i − Sj,k,i)−∑i��+1

(Sj,k+1,i − Sj,k,i

)=∑i��+1

(Sj,k+1,i − Sj,k+1,i

)−∑i��+1

(Sj,k,i − Sj,k,i

). (4.4)

Each sum in (4.4) contains at most max(nS , nS) + 1 nonzero elements, the absolutevalues of which are bounded above by ‖Sj − Sj‖. Hence, (4.4) clearly leads to (4.3)with

C0 := 2α[max(nS , nS)+ 1

].

For simplicity we shall use the notation Bj,r := Dj+r · · ·Dj and a similar one forthe scheme S . Let us fix an integer r � 0. Then,

Bj,r+1 − Bj,r+1 =Dj+r+1Bj,r − Dj+r+1Bj,r

= (Dj+r+1 − Dj+r+1

)Bj,r + Dj+r+1

(Bj,r − Bj,r

). (4.5)

The scheme Br := {Bj,r , j � 0} is clearly local and bounded, and so is the scheme Br .Using the fact that nD � nS and nD � nS (proposition 2.3), we can derive from (4.5):∥∥Bj,r+1 − Bj,r+1

∥∥�[max(nS, nS)+ 1

]∥∥Dj+r+1 − Dj+r+1

∥∥MBr + (nS + 1)MD∥∥Bj,r − Bj,r

∥∥.According to (4.3) we know that ‖Dj+r+1 − Dj+r+1‖ � C02−β(j+r+1) for all j � 0. Ifwe assume that (4.2) has been proved for r, then ‖Bj,r − Bj,r‖ � Cr2−βj for all j � 0.The inequality (4.2) is thus proved for r + 1 with

Cr+1 := [max(nS, nS)+ 1

]MBr C02−β(r+1) + (nS + 1)MDCr. �

Proof of theorem 4.2. Assume the existence of a number µ ∈ ]0, 1[, and of two non-negative integers J , K such that∥∥Dj+K · · · Dj+1Dj

∥∥∞ � µ for all j � J . (4.6)


Choose any number µ in ]µ, 1[ and define K as K := K . With the notations introducedin the proof of lemma 4.3, we want to justify the existence of an integer J such that‖Bj,K‖∞ � µ for any j � J . Now,∥∥Bj,K − Bj,K

∥∥∞ �[max(nBK , nBK )+ 1

]∥∥Bj,K − Bj,K∥∥

�[max(nBK , nBK )+ 1

]CK 2−βj , j � 0,

the latter inequality resulting from (4.2). Hence we can find an integer J1 so that∥∥Bj,K − Bj,K∥∥∞ � µ− µ, j � J1.

For any j � J := max(J1, J ), we then have ‖Bj,K‖∞ � µ. �

As a particular case, suppose that the reference scheme S is local, stationary, anduniform, and that it is known to converge (with respect to the regular grid). Then, ac-cording to theorem 3.1, it satisfies (3.1) which is nothing but condition (∗). Therefore,in this case, theorem 3.2 gives:

Corollary 4.4. Let S denote a local, stationary, and uniform subdivision scheme whichreproduces constants and is convergent (with respect to the regular grid). Then, any localand bounded subdivision scheme S , which reproduces constants and is equivalent to S,converges (with respect to any grid).

Remark 4.5. Let S be a local and bounded subdivision scheme reproducing constantsand satisfying (4.6). Choose any other subdivision scheme S equivalent to S . Accordingto theorem 4.2 and to its proof, we can apply theorem 3.8 to S with K := K , and anyµ ∈ ]µ, 1[. Therefore, using any quasi-regular grid X , the limit functions produced by Sare then Lipschitz of index ν for any positive ν < −1/(K + 1) log2 µ. The regularityobtained in theorem 3.8 for the limit functions thus depends neither on the scheme S ,nor on the chosen quasi-regular grid X .

5. Differentiability of the limit functions

In the previous sections we have dealt with the difference scheme of a given sub-division scheme S . We will now consider the derived schemes of S , which, unlike thedifference scheme, do depend on the chosen grid. The derived schemes help, in particu-lar, to study the differentiability of the limit functions.

Throughout the section, X denotes a fixed grid.

5.1. Order of a subdivision scheme

The notions and notations introduced in this subsection are those of [3] to whichwe refer. Nevertheless, we shall give full details on them in order to emphasize whereexactly the quasi-regularity of the grid is important as well as the exact indices involvedin the derived schemes.


Let S denote a local and bounded subdivision scheme. We assume that S repro-duces constants. From now on, we shall as well refer to the latter property by saying thatS is of order greater than or equal to 1. Then, as stated in proposition 2.3, the differencesof order 1,�fj , j � 0, are given by the local and bounded difference scheme D. Insteadof dealing with the differences of order 1, at a given level j � 0, we shall now considerthe divided differences of order 1 of the function Fj based on consecutive points of thegrid X at level j , namely,

f[1]j,k := [xj,k, xj,k+1]Fj = fj,k+1 − fj,k

xj,k+1 − xj,k = (�fj)k

dj,k, k ∈ Z. (5.1)

From the equality �fj+1 = Dj�fj , it follows that the divided differences of order 1 canbe calculated by means of a subdivision scheme S [1] = {S[1]

j , j � 0}:

f[1]j+1 = S[1]

j f[1]j , j � 0, with f [1]

j := (f

[1]j,k

)k∈Z, (5.2)

and where

S[1]j,k,� := dj,�

dj+1,kDj,k,� = dj,�

dj+1,k

∑i��+1

(Sj,k+1,i − Sj,k,i

), k, � ∈ Z. (5.3)

The difference scheme D being local, so is the derived scheme S [1]. From (2.12) weknow more precisely that

S[1]j,k,� �= 0 ⇒ −nS � 2�− k � nS − 1. (5.4)

Due to the presence of the ratio dj,�/dj+1,k in (5.3), a priori the derived scheme S [1] isnot bounded. However, in case the grid satisfies the quasi-regularity condition (3.18),then we have

0 <2a

b� dj,�

dj+1,k� 2b

a, j � 0, k, � ∈ Z. (5.5)

We can therefore state the following obvious result.

Proposition 5.1. Given a local and bounded subdivision scheme S of order greater thanor equal to 1, its derived scheme S [1] with respect to any quasi-regular grid is not onlylocal but also bounded.

From now on we shall assume that the grid X is quasi-regular, the quasi-regularitybeing given by (3.18). Then, generalizing the notation introduced in (3.10), at eachlevel j , we shall set

d[p]j,k := xj,k+p − xj,k, k ∈ Z, p � 1,


so that, in particular, d [1]j,k = dj,k. Note that condition (3.18) then implies:

0 <2a

b�

d[p]j,�

d[p]j+1,k

� 2b

a, j � 0, k, � ∈ Z. (5.6)

From now on, we shall also set S [0] := S and D[1] := D, so that D[1] is the differ-ence scheme of the subdivision scheme S [0]. With the local and bounded subdivisionscheme S [1], we can associate its difference scheme D[2], defined as follows:

D[2]j,k,� :=

∑i��+1

(S

[1]j,k+1,i − S[1]

j,k,i

), j � 0, k, � ∈ Z. (5.7)

This scheme is bounded. Suppose now that S [1] is of order greater than or equal to 1(i.e., that it reproduces constants. We shall then say that S is of order greater thanor equal to 2). Then, according to proposition 2.3, the scheme D[2] is local too, withnD[2] � nS [1] = nD[1] � nS [0] . Actually, taking (5.4) and (5.7) into account, one cancheck that, more precisely:

D[2]j,k,� �= 0 ⇒ −nS � 2�− k � nS − 2. (5.8)

On the other hand, proposition 2.3 says that the scheme D[2] satisfies

�f[1]j+1 = D[2]

j �f[1]j , j � 0. (5.9)

Consider the divided differences of order two based on consecutive points of the grid atlevel j � 0:

f[2]j,k := [xj,k, xj,k+1, xj,k+2]Fj = f

[1]j,k+1 − f [1]

j,k

xj,k+2 − xj,k = (�f[1]j )k

d[2]j,k

, k ∈ Z. (5.10)

Setting f [2]j := (f [2]

j,k )k∈Z, we have

f[2]j+1 = S[2]

j f[2]j , S

[2]j,k,� := d

[2]j,�

d[2]j+1,k

D[2]j,k,�, j � 0, k, � ∈ Z. (5.11)

The derived scheme S [2] is local, and it is also bounded due to (5.6). When possiblewe shall continue the same way. We say that S is of order greater than or equal to pif we have been able to define a local and bounded subdivision scheme S [p−1] enablingus to calculate the divided differences of order (p − 1), and if S [p−1] is of order greaterthan or equal to 1, that is, if it reproduces constants. By proposition 2.3, the differencescheme D[p] of S [p−1] is then both local and bounded with

D[p]j,k,� �= 0 ⇒ −nS [p−1] � 2�− k � nS [p−1] − 1. (5.12)


At any level j � 0, it satisfies�f [p−1]j+1 = D[p]

j ·�f [p−1]j . The derived scheme of order p,

can be calculated recursively as follows:

S[p]j,k,� = d

[p]j,�

d[p]j+1,k

D[p]j,k,� = d

[p]j,�

d[p]j+1,k

∑i��+1

(S

[p−1]j,k+1,i − S[p−1]

j,k,i

), j � 0, k, � ∈ Z. (5.13)

It is the subdivision scheme which allow the calculation of the divided differences oforder p at level j + 1 from those at level j :

f[p]j+1 = S[p]

j f[p]j , f

[p]j := (

f[p]j,k

)k∈Z. (5.14)

This scheme is local as D[p], and, due to (5.6) it is bounded too, but it may or notreproduce constants. Using (5.12) and (5.13), a simple recursive argument shows that:

S[p]j,k,� �= 0 ⇒ −nS � 2�− k � nS − p. (5.15)

Note that being of order greater than or equal to p is related to the grid X withrespect to which the derived schemes are constructed. It will not necessarily be so whenconsidering the derived schemes with respect to another quasi-regular grid. For instance,the Chaikin subdivision scheme is of order greater than or equal to 2 with respect to agrid X iff X is regular, in which case it is of exact order 4. In any case, due to (5.15),the order of a subdivision scheme S cannot exceed 2nS .

5.2. Comparison of subdivision schemes of higher orders

Assume that, for some p � 1, the derived scheme S [p] is defined. For a givenbounded initial f0, we can calculate the divided differences at level 0, and then at anylevel j by means of (5.14). We shall denote by F [p]

j : R → R the function defined by

F[p]j is affine on [xj,k, xj,k+1], F

[p]j (xj,k) = f

[p]j,k , k ∈ Z. (5.16)

Our assumption that the grid X is quasi-regular enables us to make use of the followingresult, for which we refer to [3, lemma 5].

Theorem 5.2. Let S [0] be a local and bounded subdivision scheme, which is of ordergreater than or equal to P � 1 with respect to a quasi-regular grid X . Suppose that S [0]as well as each of its derived schemes (with respect to X ) S [p], 1 � p � P−1, converges(with respect to X ). Then, for any given bounded f0 ∈ R

Z, the limit function F producedby the scheme S is (P − 1)-times differentiable on R. Setting F [p] := limj→+∞ F

[p]j ,

1 � p � P − 1, the derivatives of F are given by

F (p) = p!F [p], 1 � p � P − 1. (5.17)

In the rest of the paper, a function G : R → R will be said to be Hölder of indexα > 0 if it fulfils the following requirements:


(H)1 G is n times differentiable, where n denotes the nonnegative integer such thatε := α − n ∈ ]0, 1],

(H)2 each among the functions G,G′, . . . ,G(n) is bounded on R,

(H)3 G(n) is Lipschitz of index ε.

With this definition, combining theorem 5.2 with the results of section 3, we canstate:

Theorem 5.3. Let S [0] be a local and bounded subdivision scheme, which is of ordergreater than or equal to P � 1 with respect to a quasi-regular grid X . Suppose that,for 1 � p � P , the difference scheme D[p] of the derived scheme S [p−1] (with respectto X ) satisfies property (∗), namely, for 1 � p � P :

there exist two integers Jp,Kp � 0, there exists a number µp ∈ ]0, 1[ such that∥∥D[p]j+Kp · · ·D[p]

j+1D[p]j

∥∥∞ � µp for all j � Jp.(∗)p

Then, for any bounded f0 ∈ RZ, the subdivision scheme S [0] provides a limit function

F which is Hölder of index P − 1 + νP , where νP := −1/(KP + 1) log2 µP � 1.

Proof. By application of theorem 3.2, we obtain the convergence of each local andbounded subdivision scheme S [p], 0 � p � P −1. Given a bounded f0, the correspond-ing limit function F [p] are bounded, and, applying theorem 3.8 to S [P−1] enables us toconclude that F [P−1] is Lipschitz of index νP . Since S [P−1] reproduces constants andD[P ] satisfies (∗)P , remark 3.9 allows us to state that νP � 1. Now, according to theo-rem 5.2, F = F [0] is (P − 1)-times differentiable on R, its derivatives being obtainedby (5.17). They are thus bounded, and F (P−1) is Lipschitz of index νP . It follows that Fis Hölder of index P − 1 + νP . �

Suppose that S is of order greater than or equal to P . In order to prove that eachof the schemes D[1], . . . ,D[P ], satisfies property (∗), we shall again compare S witha reference subdivision scheme. For this purpose, we need to introduce the notion ofequivalent grids.

Definition 5.4. We say that two grids X and X are equivalent if, for any given positiveinteger N , there exist two positive numbers γ , η such that, for any k, � ∈ Z, and anyp � 1

−N � 2� − k � N − p ⇒∣∣∣∣∣ d

[p]j,�

d[p]j+1,k

− d[p]j,�

d[p]j+1,k

∣∣∣∣∣ � γ 2−ηj , j � 0. (5.18)


Example 5.5. Assume that X is the regular grid. Then, one can check that another gridX is equivalent to X if and only if, for any positive Q, there exist positive γ , η such that

−Q � 2�− k � Q− 1 ⇒∣∣∣∣ dj,�dj+1,k

− 2

∣∣∣∣ � γ 2−ηj , j � 0, (5.19)

or if an only if

Max

(∣∣∣∣ dj,�dj,�+1− 1

∣∣∣∣, ∣∣∣∣dj,�+1

dj,�− 1

∣∣∣∣) � γ 2−ηj , � ∈ Z, (5.20)

for some positive γ , η. Using (5.20) it is easy to check that a semi-regular grid (i.e., a gridsatisfying at any level j , xj+1,2k+1 := (xj,k + xj,k+1)/2 for any k ∈ Z) is equivalent tothe regular grid iff it is regular. Examples of quasi-regular grids equivalent to the regularone will be presented in the next section. Note that (5.20) is exactly the condition onthe grids used in [12] to prove the convergence of a generalized version of the four-pointscheme.

The equivalence relation between quasi-regular grids is exactly what is needed toobtain the following result.

Proposition 5.6. Let S and S be two local bounded subdivision schemes of ordergreater than or equal to P � 1 with respect to two quasi-regular grids X and X , re-spectively. Suppose that S and S are equivalent and that the two grids are equivalenttoo. Then, for 1 � p � P − 1, the derived subdivision schemes S [p] (with respect to X )and S [p] (with respect to X ) are equivalent.

Proof. We shall prove the equivalence between S [p] and S [p] by induction on p, whichholds for p = 0 by assumption. Assume this equivalence to hold for a given integer p,0 � p < P − 1, and let us prove it for p + 1. According to (5.13), we have, for anyj � 0, and any integers k, � ∈ Z,

S[p+1]j,k,� − S[p+1]

j,k,� = d[p+1]j,�

d[p+1]j+1,k

D[p+1]j,k,� − d

[p+1]j,�

d[p+1]j+1,k

D[p+1]j,k,� ,

=(d

[p+1]j,�

d[p+1]j+1,k

− d[p+1]j,�

d[p+1]j+1,k

)D

[p+1]j,k,� + d

[p+1]j,�

d[p+1]j+1,k

(D

[p+1]j,k,� − D[p+1]

j,k,�

). (5.21)

Denote by a, b the positive numbers provided by the quasi-regularity condition of thegrid X . Then, on account of (5.6), equality (5.21) gives

∣∣S[p+1]j,k,� − S[p+1]

j,k,�

∣∣ �∣∣∣∣∣d

[p+1]j,�

d[p+1]j+1,k

− d[p+1]j,�

d[p+1]j+1,k

∣∣∣∣∣∥∥D[p+1]j

∥∥+ 2b

a

∥∥D[p+1]j − D[p+1]

j

∥∥. (5.22)


Since S [p] and S [p] are equivalent, there exist positive numbers αp, βp such that∥∥S[p]j − S[p]

j

∥∥ � αp 2−βpj , j � 0.

The scheme D[p+1] is the difference scheme of S [p] and D[p+1] is the difference schemeof S [p]. Therefore, by applying lemma 4.3 to the equivalent schemes S [p] and S [p], wecan derive the existence of a positive number )p such that∥∥D[p+1]

j − D[p+1]j

∥∥ � )p2−βpj , j � 0. (5.23)

Note that, in order to find an upper bound for ‖S [p+1]j −S [p+1]

j ‖, we only have to considerthe quantities |S [p+1]

j,k,� − S [p+1]j,k,� |, with −Q � 2�−k � Q−p, whereQ := max(nS, nS).

Hence, using the corresponding property (5.18), we can derive from (5.22) and (5.23):∥∥S [p+1]j − S [p+1]

j

∥∥ � γ 2−ηj MD[p+1] + 2b

a)p 2−βpj , j � 0.

Whence the existence of a positive number αp+1 such that∥∥S [p+1]j − S [p+1]

j

∥∥ � αp+1 2− min(βp,η)j , j � 0. �

Corollary 5.7. Let S and S be two local bounded subdivision schemes of order greaterthan or equal to P � 1 with respect to quasi-regular grids X and X , respectively. Sup-pose that:

1. S and S are equivalent,

2. the two grids X and X are equivalent,

3. each of the subdivision schemes D[p] (with respect to X ), 1 � p � P , satisfiesproperty (∗), with some Jp, some Kp, and some µp ∈ ]0, 1[.

Then, the subdivision scheme S converges, and the limit functions it produces areHölder of index (P − 1 + ν) for any ν ∈ ]0,−1/(KP + 1) log2 µP [.

Proof. Fix an integer p, 0 � p � P − 1. According to proposition 5.6, the first tworequirements imply the equivalence between the two schemes S [p] and S [p]. Since thedifference scheme D[p+1] of S [p] satisfies property (∗), according to theorem 4.2, thedifference scheme D[p+1] of S [p] satisfies property (∗). Hence each of the schemes D[p],1 � p � P satisfies property (∗). We can thus apply theorem 5.3, and the fact that thelimit functions are Hölder as announced follows from remark 4.5. �

As an instance, let us see what the previous corollary becomes in the case of auniform and stationary reference subdivision scheme S when the reference grid is theregular one.

Corollary 5.8. Let S and S be two local bounded subdivision schemes of order greaterthan or equal to P � 1, with S uniform and stationary. Suppose that:


1. S and S are equivalent,

2a. the quasi-regular grid X is equivalent to the regular grid X ,

3a. the difference scheme D[P ] of the derived scheme S [P−1] (with respect to X ) satisfiesproperty (∗).

Then, the conclusions of corollary 5.7 are valid.

Proof. Due to the subdivision scheme S being uniform and stationary, and due to thereference grid X being regular, it is known [8, theorem 4.2] that the fact that D[P ] sat-isfies property (∗) implies the convergence of each derived subdivision scheme S [p],0 � p � P − 1, with CP−p−1 limit functions. According to theorem 3.1, the con-vergence of S [p] is equivalent to D[p+1] satisfying property (∗). Hence we can applycorollary 5.7. �

Remark 5.9. A subdivision scheme S is said to be interpolatory if satisfies:

fj+1,2k = fj,k, j � 0, k ∈ Z,

whatever the initial f0 may be, or equivalently, if Sj,2k,� = δk,� for all j � 0 and allk, � ∈ Z. If, in addition to the assumptions used in corollary 5.8, our reference schemeS is assumed to be interpolatory, then, one can replace the condition 3a by:

3b. the subdivision scheme S is convergent and it produces (P −1)-times differentiablelimit functions (with respect to the regular grid).

Indeed, because the scheme is interpolatory, the later condition implies that each derivedscheme S [p], 0 � p � P − 1, is convergent and produces CP−p−1 limit functions [9].We thus obtain the same conclusion as in corollary 5.8.

6. Lagrange interpolating subdivision schemes associated with irregular grids

Given an integer N � 0, we shall now apply the results of the previous sections tothe study of the convergence of Lagrange interpolating subdivision schemes of degree2N + 1, associated with particular irregular grids.

Let us first briefly recall how such a Lagrange interpolating subdivision scheme isdefined. Given a grid X , suppose that we know fj = (fj,k)k∈Z at some level j � 0.Then, for a given k ∈ Z, we denote by Pj,k the polynomial of degree less than or equalto 2N + 1 which interpolates the 2N + 2 data fj,�, k − N � � � k + N + 1, at thepoints xj,�, k −N � � � k +N + 1, namely,

Pj,k(xj,�) = fj,�, k −N � � � k +N + 1. (6.1)

The next level fj+1 is then given by

fj+1,2k := fj,k, fj+1,2k+1 := Pj,k(xj+1,2k+1), k ∈ Z. (6.2)


For a given k ∈ Z, denote by Lj,k,� the Lagrange polynomials of degree 2N + 1 basedon the points xj,k−N, . . . , xj,k+N+1, i.e.,

Lj,k,�(t) :=k+N+1∏i=k−Ni �=�

t − xj,ixj,� − xj,i , k −N � � � k +N + 1. (6.3)

Using the relations

Pj,k =k+N+1∑�=k−N

fj,�Lj,k,�, k ∈ Z,

we can translate (6.2) as fj+1 = Sjfj , with

Sj,2k,� = δk,�, Sj,2k+1,� :={Lj,k,�(xj+1,2k+1) if k −N � � � k +N + 1,0 otherwise.

(6.4)This, obviously, is a local scheme, with length nS = 2N + 1. From (6.3) and (6.4), weobtain the following inequality:

|Sj,2k+1,�| <(

max(d [N+1]j,k , d

[N+1]j,k−N)

min|k−s|�N d[1]j,s

)2N+1

.

In case the grid satisfies the quasi-regularity condition (3.18), this provides us with therough bound:

‖Sj‖ �(b (N + 1)

a

)2N+1

, j � 0.

Hence, in this case the scheme S is bounded withMS := (b(N + 1)/a

)2N+1.

Clearly any such Lagrange interpolating subdivision scheme S of degree 2N + 1reproduces the polynomials of degree less than or equal to 2N + 1 (with respect to thegrid X ), in the sense that, if, for all k ∈ Z, f0,k = P(x0,k), where P is any polynomialof degree at most 2N + 1, then we will have fj,k = P(xj,k) for all k ∈ Z at any otherlevel j . It follows that S is of order greater than or equal to 2N+2 (see [3, section 4.4]).

Let us now specify what kind of a grid we shall use. Our reference grid X will nowbe the regular grid, so that, at any level j � 0, xj,k := k2−j for all k ∈ Z. The grid Xwill be built from the reference grid as follows:

xj,k := G−1(xj,k), j � 0, k ∈ Z, (6.5)

where the function G : R → R is assumed to meet the following two requirements:

(i) G is Hölder of index α > 1, according to the definition given in section 5.2,

(ii) there exist two positive numbers m,M such that its first derivative g := G′ satisfies:

0 < m � g(x) � M, x ∈ R. (6.6)


It is easy to construct such functions G, e.g., G(x) = x + ; sin ηx provided that thepositive numbers ;, η satisfy ;η < 1.

The following preliminary result is an elementary consequence of our two require-ments (i) and (ii).

Lemma 6.1. If the function G satisfy the conditions (i) and (ii) above, there exists apositive ( such that ∣∣∣∣g(u)g(v)

− 1

∣∣∣∣ � (|u− v|α1, u, v ∈ R, (6.7)

with α1 := min(α − 1, 1) ∈ ]0, 1].

Proof. We just have to prove that∣∣g(u)− g(v)∣∣ � λ |u− v|α1, u, v ∈ R, (6.8)

for some positive λ. Relation (6.7) will then follow, with ( := λ/m, by dividing (6.8)by g(v) � m > 0.

Let n be the positive integer such that ε := α − n ∈ ]0, 1]. Let us first consider thecase n = 1. Then we have: α − 1 = ε = min(α − 1, 1) > 0. In this case, (6.8) resultsfrom (H)3, which says that g = G′ is Lipschitz of index ε. Suppose now that n � 2.Then, α1 = 1, g is differentiable, and g′ = G′′ is bounded according to (H)2. Therefore,in this case, (6.8) is simply obtained by application of the mean-value theorem to g. �

Proposition 6.2. The grid X defined in (6.5), where the function G satisfies the twoproperties (i) and (ii) above, is quasi-regular and it is equivalent to the regular grid X .

Proof. For given j � 0, k ∈ Z,

dj,k = xj,k+1 − xj,k = G−1(xj,k+1)−G−1(xj,k).

The mean-value theorem thus guarantees the existence of a real number ζj,k such that

dj,k = 2−j

g(ζj,k), ζj,k ∈ ]xj,k, xj,k+1[. (6.9)

Using (6.6), this leads to the quasi-regularity condition:

1

M2−j � dj,k � 1

m2−j , j � 0, k ∈ Z. (6.10)

Let us now show that the grid X is equivalent to the regular grid. At any level j � 0,and for any � ∈ Z, relations (6.9) and (6.7) provide the following inequality:∣∣∣∣ dj,�dj,�+1

− 1

∣∣∣∣ =∣∣∣∣g(ζj,�+1)

g(ζj,�)− 1

∣∣∣∣ � ( |ζj,�+1 − ζj,�|α1 . (6.11)


Since ζj,�+1 belongs to ]xj,�+1, xj,�+2[ and ζj,� belongs to ]xj,�, xj,�+1[, we have

|ζj,�+1 − ζj,�| � |xj,�+2 − xj,�|.From (6.11) and the quasi-regularity condition (6.10), we can thus derive:∣∣∣∣ dj,�dj,�+1

− 1

∣∣∣∣ � γ 2−α1j , j � 0,

where γ := ((2/m)α1 . The same argument is also valid to bound |dj,�+1/dj,� − 1|. Onaccount of (5.20), the equivalence with the regular grid is proved. �

Our reference scheme S will be the regular Lagrange interpolating subdivisionscheme (that is, the Lagrange interpolating scheme related to the regular grid X ) of thesame degree 2N + 1. It is thus defined by

Sj,2k,� = δk,�, Sj,2k+1,� :={Lj,k,�(xj+1,2k+1) if k −N � � � k +N + 1,0 otherwise,

(6.12)where by Lj,k,� we denote the Lagrange polynomials of degree 2N + 1 based on thepoints xj,k−N, . . . , xj,k+N+1, i.e.,

Lj,k,�(t) :=k+N+1∏i=k−Ni �=�

t − xj,ixj,� − xj,i , k −N � � � k +N + 1. (6.13)

This reference subdivision scheme is uniform and stationary and one can check that‖S‖ = 1, where S = Sj for all j � 0.

Proposition 6.3. Assume that the grid X is given by (6.5), where the function G sat-isfies the two properties (i) and (ii) above. Then the irregular Lagrange interpolatingsubdivision scheme S defined in (6.4) is equivalent to the regular Lagrange interpolatingsubdivision scheme S of the same degree.

The proof will be a simple consequence of the following lemma.

Lemma 6.4. For any j � 0, any k ∈ Z, and any �, k − N � � � k + N + 1, we canwrite

Lj,k,�(G−1(t)

) = cj,k,�(t) Lj,k,�(t), t ∈ R, (6.14)

where the function cj,k,� satisfies∣∣cj,k,�(t)− 1∣∣ � C 2−α1j for any t ∈ [xj,k−N, xj,k+N+1], (6.15)

the positive number C being independent of j, k, �.


Proof. Let us consider a fixed level j and a fixed k ∈ Z. From (6.3) and (6.5), we have,for any given t ∈ R,

Lj,k,�(G−1(t)

) :=k+N+1∏i=k−Ni �=�

G−1(t)−G−1(xj,i )

G−1(xj,�)−G−1(xj,i ), k −N � � � k + N + 1. (6.16)

Let i, � be any two integers such that k − N � i, � � k + N + 1, and i �= �. Applyingthe mean value theorem to the function G−1 guarantees the existence of ξj,i (t) locatedbetween G−1(t) and xj,i , and the existence of ρj,i,� between xj,i and xj,� such that

G−1(t)−G−1(xj,i )

G−1(xj,�)−G−1(xj,i )= g(ρj,i,�)

g(ξj,i(t))

t − xj,ixj,� − xj,i .

Accordingly, (6.16) provides us with the equality (6.14), where

cj,k,�(t) :=k+N+1∏i=k−Ni �=�

g(ρj,i,�)

g(ξj,i (t)). (6.17)

From (6.7), we know that∣∣∣∣ g(ρj,i,�)g(ξj,i(t))− 1

∣∣∣∣ � (∣∣ρj,i,� − ξj,i(t)

∣∣α1, k − N � i, � � k +N + 1, i �= �.

Suppose now that t belongs to [xj,k−N, xj,k+N+1]. Then, the real numbers ρj,i,� andξj,i (t) all belong to the interval [xj,k−N, xj,k+N+1]. The quasi-regularity condition (6.10)thus gives ∣∣ρj,i,� − ξj,i (t)

∣∣ � xj,k+N+1 − xj,k−N � 2N + 1

m2−j ,

which finally yields:∣∣∣∣ g(ρj,i,�)g(ξj,i (t))− 1

∣∣∣∣ � A 2−α1j , k − N � i, � � k +N + 1, i �= �, (6.18)

where the positive number A := (((2N + 1)/m)α1 depends neither on i, �, nor on j, k.Taking all equalities (6.18) into account easily leads to the existence of C > 0, indepen-dent of j, k, �, such that the function cj,k,� defined in (6.17) satisfies (6.15). �

Proof of proposition 6.3. Using (6.4) and (6.12), we have, for k−N � � � k+N +1:∣∣Sj,2k+1,� − Sj,2k+1,�

∣∣ = ∣∣Lj,k,�(G−1(xj+1,2k+1))− Lj,k,�(xj+1,2k+1)

∣∣,that is, on account of (6.14),∣∣Sj,2k+1,� − Sj,2k+1,�

∣∣ = ∣∣cj,k,�(xj+1,2k+1)− 1∣∣∣∣Lj,k,�(xj+1,2k+1)

∣∣.From (6.15) and from the fact that ‖S‖ = 1, we thus obtain the inequality∣∣Sj,2k+1,� − Sj,2k+1,�

∣∣ � C 2−α1j ,


which enables us to state that ‖Sj − Sj‖ = ‖Sj − S‖ � C 2−α1j for all j � 0. Whencethe equivalence between the two schemes S and S . �

Since any Lagrange interpolating subdivision scheme of degree 2N + 1 associatedto a given grid X is of order at least 2N + 2 with respect to that grid, by application ofremark 5.9, the comparison with the regular case gives the following result.

Corollary 6.5. Consider an integer P � 2N + 2 such that the regular Lagrange inter-polating subdivision scheme S of degree 2N + 1 is convergent and produces CP−1 limitfunctions. Then, as soon as a grid X is defined as in (6.5), the Lagrange interpolatingsubdivision scheme S of degree 2N + 1 with respect to X is also convergent and itproduces CP−1 limit functions.

For instance, for N = 1, we can take P = 2, and P = 3 for N = 2. Note that, inthe semi-regular case, the convergence to C1 (respectively C2) functions can be foundin [15].

Actually, according to proposition 5.7, we even know that under the assumptionsof corollary 6.5, the limit functions are Hölder of index P − 1 + ν for any ν ∈ ]0, ν[,where ν is given by the condition (∗) satisfied by the scheme D[P ] of the regular case.

Let us now compare our results with the results obtained in [3]. First of all, ourresult is valid for any N while in [3] only the cases N = 1 and N = 2 are examined. Letus first recall the assumptions on the grid which the authors of [3] use for N = 1. Withany grid X , we can associate the following two numbers

γ := Supj,k

Max(dj,k+1, dj,k−1)

dj,k∈ [1,+∞],

β := Infj,k

Min(dj+1,2k, dj+1,2k+1)

dj,k∈[

0,1

2

].

Note that the quasi-regularity condition (3.18) implies γ � b/a and β � a/(2b). Un-der the weak assumption on the grid γ < +∞ (which automatically implies β > 0),the corresponding Lagrange interpolating subdivision scheme of degree 3 is proved toconverge and to produce C1 functions. Actually, the authors of [3] even show that, for1 � γ � γ0, where γ0 is approximately equal to 2.4, the limit functions are Hölder ofindex 2−ε for any positive ε. Hence, they achieve the same smoothness as in the regularcase (see [6]). Otherwise, when γ0 < γ < +∞, the limit functions are proved to beHölder of index 1 − ε + log(1 − β)/ log β for any positive ε. This index depends on β,that is, on the grid. For β = 1/2 (regular grid), they thus obtain the optimal Hölder index2 − ε. On the other hand, when β is close to 0, the result is not so good. To the con-trary, limiting ourselves to grids defined as in (6.5) provides Hölder indices 1 + ν whichare independent of the grid, since they are derived from the regular case (ν > 1.65).However, we cannot expect the best possible index, since in the regular case, the Hölderindex provided by the condition (∗) is not optimal. For N = 2, the authors of [3] prove


that the scheme is convergent and with C2 limit functions provided that γ is sufficientlyclosed to 1.

Acknowledgements

The original motivation of this work is due to Anestis Antoniadis, without whomthis paper would not exist. We thus want to express warm thanks to him. We are alsoindebted to Albert Cohen for giving us important references.

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Subdivision Schemes and Irregular Grids

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