Generalized Taylor operators, polynomial chains, and Hermite subdivision schemes

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Generalized Taylor operators, polynomial chains, and Hermite subdivision schemes

Jean-Louis Merrien, Tomas Sauer

To cite this version:

Jean-Louis Merrien, Tomas Sauer. Generalized Taylor operators, polynomial chains, and Her- mite subdivision schemes. Numerische Mathematik, Springer Verlag, 2019, 142 (1), pp.167-203.

�10.1007/s00211-018-0996-9�. �hal-01731590v2�

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Generalized Taylor operators, polynomial chains, and Hermite subdivision schemes

Jean-Louis Merrien^* Tomas Sauer^† August 15, 2018

Abstract

Hermite subdivision schemes act on vector valued data that is not only considered as functions values of a vector valued function fromRtoR^r, but as evaluations ofrconsec- utive derivatives of a function. This intuition leads to a mild form of level dependence of the scheme. Previously, we have proved that a property called spectral condition or sum rule implies a factorization in terms of a generalized difference operator that gives rise to a “difference scheme” whose contractivity governs the convergence of the scheme. But many convergent Hermite schemes, for example, those based on cardinal splines, do not satisfy the spectral condition. In this paper, we generalize the property in a way that pre- serves all the above advantages: the associated factorizations and convergence theory.

Based on these results, we can include the case of cardinal splines in a systematic way and are also able to construct new types of convergent Hermite subdivision schemes.

Keywords: Taylor operator Hermite subdivision spectral condition polynomial chain.

1 Introduction

Subdivision schemes, as established in [1], are efficient tools for building curves and surfaces with applications in design, creation of images and motion control. For vector subdivision schemes, cf. [8, 10, 18], it is not so straightforward to prove more than the Hölder regularity of the limit function, due to the more complex nature of the underlying factorizations. On the other hand, Hermite subdivision schemes [7, 11, 12, 13, 9] produce function vectors that consist of consecutive derivatives of a certain function, so that the notion of convergence automatically includes regularity of the leading component of the limit. Such schemes have even been considered also on manifolds recently [19] and have also been used for wavelet constructions [5]. While vector subdivision schemes are quite well–understood, nevertheless there are still surprisingly many open questions left in Hermite subdivision. In particular, a characterization of convergence in terms of factorization and contractivity is still missing as

*INSA de Rennes, IRMAR - UMR 6625, 20 avenue des Buttes de Coesmes, CS 14315, 35043 RENNES CEDEX, France, email:jmerrien@insa-rennes.fr

†Lehrstuhl für Mathematik mit Schwerpunkt Digitale Bildverarbeitung & FORWISS, Universität Passau, Innstr.

43, D-94032 PASSAU, email:Tomas.Sauer@uni-passau.de

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it is known in the scalar case: a subdivision scheme is convergent if and only if it can be fac- torized by means of difference operators and the resultingdifferencescheme is contractive.

In previous papers [6, 15, 16], we established an equivalence between a so–calledspectral conditionand operator factorizations that transform a Hermite scheme into a vector scheme for which analysis tools are available. Under this transformation, the usual convergence of the vector subdivision scheme implies convergence for the Hermite scheme and thus regularity of the limit function. It was even conjectured for some time that the spectral condition, sometimes also called thesum rules[4, 12] of the Hermite subdivision scheme, might be nec- essary for convergence. Already in [14] this was relaxed to some extent by considering proper similarity transforms of the mask that gave slightly generalized sum rules.

In this paper we show, among others results, that this conjecture does not hold true. We define a new set of significantly more general spectral conditions, calledspectral chains, that widely generalize the classical spectral condition from [6] and show that these spectral conditions are more or less equivalent to the existence of a factorization with respect to respective generalized Taylor operators and allow for a description of convergence by means of contractivity. Indeed, we conjecture that these factorization can be used to eventually characterized the convergence of Hermite subdivision schemes by means of contractive different schemes.

We then define a process that allows us to construct Hermite subdivision schemes of arbi- trary regularity with guaranteed convergence and, in particular, give examples of convergent Hermite subdivision schemes that do not satisfy the spectral condition. In addition, our new method can be applied to an example based on B–splines and their derivatives which was one of the first examples of a convergent Hermite subdivision scheme that does not satisfy the spectral condition, [14].

The paper is organized as follows: after introducing some basic notation and the concept of convergent vector and Hermite subdivision schemes, we introduce the new concept of chains and generalized Taylor operators in Section 4 and use them for the factorization of subdivision operators in Section 4. These results allow us to extend the known results about the convergence of the Hermite subdivision schemes to this more general case in Section 5.

Section 6 is devoted to the construction of a convergent Hermite subdivision scheme emerg- ing from a properly constructed contractive vector subdivision scheme by reversing the factorization process, even in the generality provided by generalized Taylor operators. Finally, we give some examples of the results of such constructions in Section 7, and also provide a new approach for the aforementioned spline case.

2 Notation and fundamental concepts

Vectors inR^r,r ∈N, will generally be labeled by lowercase boldface letters: y=£ yj¤

j=0,...,r−1

ory=£ y^(j)¤

j=0,...,r−1, where the latter notation is used to highlight the fact that in Hermite subdivision the components of the vectors correspond to derivatives. Matrices inR^r×rwill be written as uppercase boldface letters, such asA=£

a_{j k}¤

j,k=0,...,r−1. The space of polynomials in one variable of degree at mostnwill be written asΠn, with the usual conventionΠ−1={0}, whileΠ will denote the space of all polynomials. Vector sequences will be considered as functions fromZtoR^r and the vector space of all such functions will be denoted byℓ(Z,R^r)

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orℓ^r(Z). Fory(·)∈ℓ(Z,R^r), theforward differenceis defined as∆y(α) :=y(α+1)−y(α),α∈Z, and iterated to∆ⁱ⁺¹y:=∆¡

∆ⁱy¢

=∆ⁱy(· +1)−∆ⁱy(·),i≥0.

We use0to indicate zero vectors and matrices. If we want to highlight the dimension of the object, we will use subscript0d, but to avoid too cluttered notation, we will often drop them if the size of the object is clear from the context. Moreover, we will use the convenient Matlab notationAj:j^′,k:k^′andaj:j^′to denote submatrices and subvectors.

Given a finitely supported sequence of matricesA=(A(α))α∈Z∈ℓ^r×r(Z), called themask of the subdivision scheme, we define the associatedstationary subdivision operator

SA:c7→ X

β∈Z

A(· −2β)c(β), c∈ℓ^r(Z).

The iteration of subdivision operatorsSAn,n∈N, is called asubdivision schemeand consists of the successive applications of level-dependent subdivision operators, acting on vector valued data,SA_n:ℓ^r(Z)→ℓ^r(Z), defined as

cn+1(α)=SAncn(α) :=X

β∈Z

An¡ α−2β¢

cn(β), α∈Z, c∈ℓ^r(Z) . (1) An important algebraic tool for stationary subdivision operators is thesymbolof the mask, which is the matrix valued Laurent polynomial

A^∗(z) := X

α∈Z

A(α)z^α, z∈C\ {0}. (2)

We will focus our interest on two kinds of such schemes, the first one being “traditional” vector subdivision schemes in the sense of [1], whereA_n is independent ofn, i.e.,A_n(α)=A(α) for anyα∈Zand anyn≥0. In the following, such schemes for which an elaborate theory of convergence exists, will simply be called avector scheme. Their convergence is defined in the following way.

Definition 1 Let SA :ℓ^r(Z) →ℓ^r(Z) be a vector subdivision operator. The operator is C^p– convergent, p≥0, if for any datag ∈ℓ^r(Z)and corresponding sequence of refinementsg_n= Sⁿ_Ag, g₀:=g, there exists a functionψg ∈C^p(R,R^r)such that for any compact K ⊂Rthere exists a sequenceεnwith limit0that satisfies

α∈maxZ∩2ⁿK

°°g_n(α)−ψ_g¡ 2⁻ⁿα¢°°

∞≤εn. (3)

As the second type of, now even level–dependent, schemes we consider theHermite scheme

whereAn(α)=D⁻ⁿ⁻¹A(α)Dⁿforα∈Zandn≥0 with the diagonal matrixD:=





 1

1 2 . ..

1 2^d





. In this caser=d+1 and fork=0, . . . ,dthe k-th component ofcn(α) corresponds to an ap- proximation of the k-th derivative of some functionϕn atα2⁻ⁿ. Starting from an initial se- quencec₀, a Hermite scheme can be rewritten

Dⁿ⁺¹cn+1(α)=Dⁿ⁺¹SADⁿcn(α)= X

β∈Z

A¡ α−2β¢

Dⁿcn(β), α∈Z, n≥0. (4)

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Convergence of Hermite schemes is a little bit more intricate and defined as follows.

Definition 2 Let A∈ℓ(d+1)×(d+1)(Z) be a mask and HA the associated Hermite subdivision scheme onℓ^d+1(Z)as defined in(4). The scheme isconvergentif for any data f₀∈ℓ^d+1(Z) and the corresponding sequence of refinements f_n =[f_n⁽⁰⁾, . . . ,f_n^(d)]^T, there exists a function Φ=[φi]0≤i≤d ∈C¡

R,R^d+1¢

such that for any compact K ⊂Rthere exists a sequenceεn with limit0which satisfies

0≤i≤dmax max

α∈Z∩2ⁿK

¯¯

¯f_n⁽ⁱ⁾(α)−φi

¡2⁻ⁿα¢¯¯¯≤εn. (5) The scheme H_Ais said to be C^p–convergentwith p≥d if moreoverφ₀∈C^p(R,R)and

φ⁽ⁱ⁾₀ =φi, 0≤i≤d.

Remark 3 Since the intuition of Hermite subdivision schemes is to iterate on function values and derivatives, it usually only makes sense to consider C^p–convergence for p≥d . Note, how- ever, that the case p>d leads to additional requirements.

The (classical)spectral conditionof a subdivision operator has been introduced in [6]. It re- quests that there exist polynomialspj∈Πj,j=0, . . . ,d, such that

SA





 pj

p^′_j ... p^(d)_j







=2^−j





 pj

p^′_j ... p^(d)_j





, j=0, . . . ,d. (6)

This spectral condition is a special case of aspectral chainthat will be defined in Definition 21.

3 Generalized Taylor operators and chains

In this section, we introduce the concept of generalized Taylor operators and show that they form the basis of symbol factorizations. The first definition concerns vectors of almost monic polynomials of increasing degree.

Definition 4 ByV_d we denote the set of all vectorsv of polynomials inΠdwith the property that

v=



 vd

... v0



, vj= 1

j!(·)^j+uj∈Πj, uj∈Πj−1. (7) A vector inV_d thus consists of polynomials vj of degreeexactly j whose leading coefficient is normalized to ¹_j!, and the remaining part of the polynomial vjof lower degree is denoted by uj. Note that in (7) we always havev0=1 andu0=0. Also keep in mind that the vectorsv are indexed in a reversed order, but referring directly to the degree of the object, this notion is more comprehensible.

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We will use the convenient notation ofPochhammer symbols(·)j ∈Πj, j ≥0, in the following way:

(·)0:=1, (·)j:=

j−1Y

k=0

(· −k), j≥1, and [·]j:= 1

j!(·)j, j≥0. (8) These polynomials satisfy

∆(·)j=j(·)j−1, ∆[·]j=[·]j−1. (9) Both©

(·)0, . . . , (·)j

ªand©

[·]0, . . . , [·]j

ªare bases ofΠjand allow us to write the Newton interpo- lation formula of degreedat 0, . . . ,din the form

x^j= Xj k=0

1 k!

³

∆^k(·)^j´

(0) (x)_k= Xj k=0

³

∆^k(·)^j´

(0) [x]_k; then, since∆^j(·)^j=j!, we have that

1

j!(·)^j=[·]j+

j−1X

k=0

¡∆^k(·)^j¢ (0) j! [x]k

which implies that

v∈V_d ⇔ vj=[·]j+uj, uj∈Πj−1 j=0, . . . ,d. (10) We will use this form in the future to write eachv∈V_das

v=



 [·]d

... [·]0



+u. (11)

Generalizing the Taylor operators operating on vector functionsR→R^d+1introduced in [6, 15], we define the following concept.

Definition 5 Ageneralized incomplete Taylor operatoris an operator of the form

T_d:=







∆ −1 ∗ . . . ∗ . .. ... ... ...

. .. ... ∗

∆ −1 1







=

·∆I 1

¸ +£

t_{j k}¤

j,k=0,...,d, (12)

where tj,j+1= −1and t_{j k} =0for k ≤j . In the same way, thegeneralized complete Taylor operatoris of the form

Te_d:=







∆ −1 ∗ . . . ∗ . .. ... ... ...

. .. ... ∗

∆ −1

∆







=∆I+£ t_{j k}¤

j,k=0,...,d. (13)

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Remark 6 The Taylor operator becomes generalized for d≥2, otherwise we simply recover the classical case, see Example 16.

Lemma 7 Letv:=[vd, . . . ,v0]^T be a vector of polynomials inΠ^d+1with v0=1. Thenv∈V_dif and only if there exists a generalized complete Taylor operatorTe_dsuch thatTe_dv=0.

Proof: For “⇐” suppose thatTedv=0 and let us prove by induction on j=0, . . . ,d thatvj= [·]j+uj for some appropriateuj ∈Πj−1. The assumption v0=1 ensures that for j =0 by simply settingu₀=0. Now, for 0≤j<d, we assume thatv_j+1is of degreem≥0 and write it in the basis {[·]0, . . . , [·]m} as

vj+1= Xm k=0

c_k[·]_k= Xm k=j+2

c_k[·]_k+cj+1[·]j+1+q,

withq∈Πj, hence∆q∈Πj−1. By induction hypothesis, we have thatvj=[·]j+uj,uj∈Πj−1

andvk∈Πkfork=0, . . . ,j−1. ThenTedv=0 implies at rowd−j−1 that 0 = ∆v_j+1−v_j+

j−1X

k=0

t_{d−j−1,d−k}v_k

= Xm k=j+2

ck[·]k−1+cj+1[·]j+∆q−[·]j−uj+

j−1X

k=0

td−j−1,d−kvk

=

m−1X

k=j+1

c_k+1[·]_k+¡

cj+1−1¢

[·]j+u, u∈Πj−1,

and comparison of coefficients yieldscj+2= · · · =cm =0 as well ascj+1=1, hencevj+1= [·]_j+1+u_j+1withu_j+1∈Πj, which advances the induction hypothesis.

For the converse “⇒”, we note that for anyv∈V_dwe have that for j≥1

∆vj−vj−1=[·]j−1+∆uj−[·]j−1−uj−1=∆uj−uj−1∈Πj−2

and since©

v0, . . . ,vj−2

ªis a basis ofΠj−2, the polynomial∆vj−vj−1can be uniquely written as

c₀v₀+ · · · +c_j−2v_j−2= − Xd ℓ=d−j+2

t_d−j,ℓv_d−ℓ

which defines the remaining entries of rowd−jofTe_din a unique way such thatTe_dv=0.

The last observation in the above proof can be formalized as follows.

Corollary 8 For each v ∈V_d there exists a uniquegeneralized complete Taylor operatorTe_d such thatTe_dv=0.

Definition 9 The generalized Taylor operator of Corollary 8, uniquely defined by

Te(v)v=0, (14)

is called theannihilatorofv∈V_dand written asTe(v). We can skip the subscript “d ” because it is directly given by the dimension ofv.

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Definition 10 Achainof length d+1is a finite sequenceV :=[v0, . . . ,vd]of vectors

vj=



 v_j,_j

... vj,0



=



 [·]_j

... [·]0



+uj∈V_j, j=0, . . . ,d,

that satisfies thecompatibility condition

wj+1:=



 w_j+1,1

... wj+1,j+1



:=Te(vj)





v_j+1,j+1 ... vj+1,1



∈R^j+1, j=0, . . . ,d−1. (15)

Remark 11 Compatibility is a strong requirement on the interaction betweenvjandvj+1. In general,Te(vj)





vj+1,j+1

... vj+1,1



can only be expected to be a vector of polynomials inΠj, . . . ,Π0, while compatibility requires all these polynomials to be constants.

Due to and by means of the compatibility condition, chains uniquely define a generalized Taylor operator.

Lemma 12 IfV is a chain of length d+1, then wj j=1, j=1, . . . ,d .

Proof: Sincevj+1,1=[·]1+cfor some constantcdue tovj∈V_j, it follows immediately from the definition (15) that

w_j+1,j+1=∆v_j+1,1=1,

as claimed.

We introduce the convenient abbreviation ˆ

vj:=

· vj

0d−j

¸

∈R^d+1, j=0, . . . ,d, (16)

where the dimensiondis clear from the context.

Proposition 13 ForV =[v0, . . . ,vd],vj ∈V_j, j=0, . . . ,d , of length d+1the following state- ments are equivalent:

1. V is a chain of length d+1.

2. For j=1, . . . ,d , we have

Te(vj)=· eT(vj−1) −wj

∆

¸

=







∆ −w1,1 . . . −wj,1

∆ . .. ... . .. −wj,j

∆







, wj∈R^j. (17)

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3.

Te(v_d) ˆvj=0, j=0, . . . ,d. (18) Proof: To show that 1)⇒2), we note that again (15) yields that

0 = Te(vj)





vj+1,j+1

... v_j_+1,1



−wj+1=£

Te(vj)| −wj+1

¤







vj+1,j+1

... vj+1,1

1







= £

Te(vj)| −wj+1

¤vj+1. SinceTe(v_j+1) is unique, we deduce that

Te(vj+1)=· eT(vj) −wj+1

∆

¸

, j=0, . . . ,d−1, (19) which directly yields (17).

For 2)⇒3) we simply notice that Te(v_d) ˆvj=· eT(vj) ∗

0 ∗

¸· vj

0_d−j

¸

=· eT(vj)vj

0

¸

=0,

while for 3)⇒1) we first observe for j<dthat

0=Te(vd)vj=· eT(vd)0:j,0:jvj

0

¸

and the uniqueness of the annihilators from Corollary 8 yields thatTe(vd)0:j,0:j=Te(vj). This, in turn, implies together with (18) that

0=Te(vd) ˆvj+1=





Te(v_j) −w_j+1 ∗

∆ ∗

∗











vj+1,j+1

... vj+1,1

1 0







=





 Te(vj)





vj+1,j+1

... vj+1,1



−wj+1

0 0





 ,

which is the compatibility identity (15), henceV is a chain.

The above proof shows thatTe(vj)=Te(v_d)0:j,0:j, j=0, . . . ,d, hence all generalized Taylor operators associated to a chain depend only onvd. This justifies the following definition.

Definition 14 The unique generalized Taylor operatorTe(vd)for a chainV will be written as Te(V).

Remark 15 Since complete and incomplete Taylor operators differ only on the∆or1in lower right corner, there is an obvious extension of the definition to T(V)and the two operators are equivalent.

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Example 16 Let pj=[·]j+qj, qj∈Πj−1, j=0, . . . ,d , be given. Then vj=

h

pj,p^′_j, . . . ,p^(j)_j iT

is a chain for the classical complete Taylor operator

TeC,d:=







∆ −1 −1/2! −1/3! . . . −1/d!

∆ −1 −1/2! . . . −1/(d−1)!

∆ −1 ...

. .. . .. ...

∆ −1

∆







. (20)

This is exactly the relationship for the classical spectral condition from [6, 15]. Similarly, vj=£

pj,∆pj, . . . ,∆^jpj

¤T

is a chain for the operator

Te_∆,d:=







∆ −1 0 . .. ...

. .. −1

∆







. (21)

Another interesting generalized Taylor operator is

Te_S,d:=







∆ −1 . . . −1 . .. ... ...

. .. −1

∆







, (22)

whose chains, connected to B–splines, we will consider in Example 46 later.

Lemma 17 For any generalized complete Taylor operatorTed there exists a chainV of length d+1such thatTe_d=Te_d(V).

Proof: The construction of the chainV is carried out inductively. To that end, we recall that ifp∈Πis of the form∆p=[·]kfor somek∈N, thenp=[·]k+1+cwith somec∈R.

Next, letvj∈V_j,j=0, . . . ,d, be any solution of 0=Tedvˆj=· eTj ∗

0 ∗

¸· vj

0d−j

¸ ,

or, equivalently, ofTe_jv_j=0. Such a solution can be found by settingv_j0=1 and then solving, recursively fork=1, . . . ,j, the equation given by rowj−kof the Taylor operator,

0=∆vj,k−vj,k−1+

k−2X

ℓ=0

tj−k,j−ℓvj,ℓ. (23)

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Equivalently, this can be written with respect to the basis {[·]0, . . . , [·]k−1} and usingvj,k−1= [·]_k−1+u_j,k−1,u_j,k−1∈Πk−2, as

0=∆vj,k−[·]k−1+

k−2X

ℓ=0

sj−k,ℓ[·]ℓ, sj−k,ℓ∈R, yielding

v_{j k}=[·]_k+

k−1X

ℓ=1

s_{j−k,ℓ−1}[·]_ℓ+c_k0, k=0, . . . ,j,

where the constants ck0∈Rcan be chosen freely. This process yields polynomial vectors vj∈V_jsuch thatTejvj=0,j=0, . . . ,d.

Thus, it follows from the uniqueness of the annihilating Taylor operator from Corollary 8 thatTej=Te(vj), and decomposing the identity

0=Te(vj+1)vj+1=Tej+1vj+1=· eT(vj) −w

0 ∆

¸

vj+1, w∈R^j+1, yields

Te(vj)





vj+1,j+1

... v_j+1,1



=w=:wj+1, (24)

which is exactly the compatibility condition (15) needed forV to be a chain.

Corollary 18 In the chainV from Lemma 17 the constant coefficients of the polynomials v_{j k}, j=1, . . . ,d , k=1, . . . ,j , can be chosen arbitrarily.

Remark 19 The chain associated to a generalized Taylor operator is not at all unique, see also Example 16.

The next result shows that any polynomial vector inV_d can be reached by a chain of length d+1.

Proposition 20 For anyv∈V_dthere exists a chainV =[v0, . . . ,vd]of length d+1withvd=v, i.e.,Te(V)=Te(v).

Proof: Again we prove the claim by induction ond. The cased =0 is trivial as the only chain of length 0 consists ofv=1. For the induction step, we choosev∈V_d,d>0 and the associated generalized Taylor operatorTe(v) as in Definition 9. Then we know from Lemma 17 that there exists a chainV =[v0, . . . ,vd] of lengthd+1 such thatTe(v)V =0. Suppose that v_d6=vand, in particular, thatv_d,1(0)=v1(0)−1, which is possible according to Corollary 18.

With

v=



 [·]_d

... [·]0



+u, v_d=



 [·]_d

... [·]0



+u_d, u0=u_d,0=0,

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we find that

0=Te(v) (v−vd)=Te(v)







ud−ud,d

... u1−u_d,1

0







=:Te(v)

· v^′ 0

¸

whereu₁−u_d,1=v₁(0)−v_d,1(0)=1. In addition, Lemma 7 yields thatv^′∈V_d−1and therefore the decomposition

Te(v)=· eT(v^′) −w

0 ∆

¸

, w∈R^d, and

0=Te(v)v=· eT(v^′) −w

0 ∆

¸





 vd

... v1

1







=





 Te(v^′)





 vd

... v1

1







−w

∆







compatibility betweenv^′andv. By the induction hypothesis, there exists a chainV^′of length dwithvd−1=v^′and sincev^′is compatible withv, this chain can be extended to lengthd+1

withv^′_d=v.

4 Chains and factorizations

We now relate the existence of a spectral chain to factorizations of the subdivision operators, thus extending the results first given in [15] for the classical Taylor operator.

Definition 21 A chainV of length d+1is calledspectral chainfor a vector subdivision scheme with maskA∈ℓ(d+1)×(d+1)(Z)if

SAvˆj=2⁻^jvˆj, j=0, . . . ,d. (25) withvˆ_jfrom(16).

Remark 22 The spectral chain is an extension of the classical spectral condition which, in turn, corresponds to the special choicevj=h

pj,p^′_j, . . . ,p^(d)_j iT

, see also Example 16.

We will prove in Theorem 25 that the existence of spectral chains is equivalent to the existence of generalized Taylor factorizations. The main tool for this proof is the following result.

Proposition 23 IfC∈ℓ(d+1)×(d+1)(Z)is a finitely supported mask for which there exists a chain V such that SCvˆj=0, j=0, . . . ,d , then there exists a finitely supported maskB∈ℓ(d+1)×(d+1)(Z) such that SC=SBTe(V).

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Proof: We follow the idea from [15] and prove by induction onkthat the symbolC^∗(z) satisfies

C^∗(z)=B^∗_k(z)· eT(v_k)^∗(z²) 0

0 I

¸

, k=0, . . . ,d. (26)

with the columnwise written matrix B^∗_k(z)=£

b^∗₀(z)· · ·b^∗_k(z)c^∗_k₊₁(z)· · ·c^∗_d(z)¤

. (27)

The construction makes repeated use of the well known factorization for a scalar subdivision

schemeS_a: X

α∈Z

a(α−2β)=0 ⇒ a^∗(z)=(z⁻²−1)b^∗(z), (28) see, for example, [1] for a proof.

For casek=0, the annihilation of the vector ˆv0=e0=[1, 0, . . . , 0]^T immediately gives the decompositionc^∗₀(z)=¡

z⁻²−1¢

b^∗₀(z) and therefore C^∗(z) = £

b^∗₀(z)c^∗₁(z)· · ·c^∗_d(z)¤· z⁻²−1

I

¸

= £

b^∗₀(z)c^∗₁(z)· · ·c^∗_d(z)¤· eT(v0)^∗(z²) I

¸ .

Now suppose that (26) holds for somek≥0. Then the fact thatV is a chain yields, by means of the compatibility condition

w_k+1=Te(v_k)





v_k_+1,k+1 ... vk+1,1





that

0=SCvˆk+1=SBk



 Te(vk)

1 I





·v_k+1 0

¸

=SBk



 w_k+1

1 0



, or, applying (28) to each row of the preceding equation,

£b^∗₀(z)· · ·b^∗_k(z)¤T

wk+1+c^∗_k+1(z)=¡

z⁻²−1¢

b^∗_k+1(z), which is

c^∗_k+1(z)=£

b^∗₀(z)· · ·b^∗_k+1(z)¤T·

−wk+1

z⁻²−1

¸ , or

C^∗(z)=£

b^∗₀(z)· · ·b^∗_k₊₁(z)c^∗_k+2(z)· · ·c^∗_d(z)¤





Te(v_k)^∗(z²) −w_k+1 z⁻²−1

I



. (29) Since

Te(vk+1)^∗(z)=· eT(v_k)^∗(z) −w_k+1 z⁻¹−1

¸ ,

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(29) yields (26) withkreplaced byk+1 and advances the induction hypothesis.

Remark 24 Proposition 23 shows that, in the terminology of [2], the generalized Taylor oper- ator is aminimal annihilatorfor the chainV since it annihilates the chain and factors any subdivision operator that does so, too.

Now we can show that the existence of a spectral chain results in the existence of a factorization by means of generalized Taylor operators. Since the Taylor operator corresponds to computing differences, the schemeSB from (30) is often called thedifference schemeofSA

with respect to the generalized Taylor operatorTe(V).

Theorem 25 If S_A possesses a spectral chainV of length d+1then there exists a finite mask B∈ℓ(d+1)×(d+1)(Z)such that

Te(V)SA=SBTe(V). (30)

Proof: SinceS_C:=Te(V)S_Ahas the property that

S_Cvˆ_k=Te(V)S_Av_k=2^−kTe(V)v_k=0,

an application of Proposition 23 proves the claim.

Remark 26 For the validity of Theorem 25, which is of a purely algebraic nature, the concrete eigenvalues of the spectral set are irrelevant. Their normalization will play a role, however, as soon as convergence is concerned.

Next, we generalize a result from [16] that serves as a converse of Theorem 25. The proof is a modification of the former.

Theorem 27 Suppose that for a finitely supported maskA∈ℓ(d+1)×(d+1)there exists a finitely supportedB and a generalized incomplete Taylor operator T_dsuch that T_dS_A=2^−dS_BT_dand SBed=ed. If a chainV =[v0, . . . ,vd]withTed=Te(V)satisfies

SAvˆj∈span©

vˆ0, . . . , ˆvj

ª, j=0, . . . ,d, (31)

then there exists a spectral chainV^′for S_A.

Proof: Relying on Lemma 17, we choose a chainV such thatTed=Te(V), which particularly yields thatT_dv_d=e_d. Then

T_dv_d=e_d=SBe_d=SBT_dv_d=2^dT_dSAv_d implies thatT_d¡

2^−dv_d−SAv_d¢

=0, hence

SAvd=2^−dvd+v,˜ 0=Tdv˜=· eT_d−1 ∗ 1

¸ v˜,

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so that ˜v0=0 and thereforeTed−1v0:d−1=0. Since ˆv0, . . . , ˆvd−1form a basis for the kernel of Te_d with last component equal to zero, it follows thatv∈span{ ˆv0, . . . , ˆv_d−1}. Making use of the two–slantedness ofSA, one can literally repeat the arguments of the proof of [16, Theo- rem 2.11] to conclude that

v₀, . . . , ˆv_j−1ª ,

henceSA[ ˆv0, . . . , ˆvd]=[ ˆv0, . . . , ˆvd]U, whereU∈R(d+1)×(d+1)is an upper triangular matrix with diagonal entries 1, . . . , 2^−d. Using the upper triangularSsuch thatS⁻¹U Sis diagonal, we can then defineV^′by£

vˆ^′₀, . . . , ˆv^′_d¤

=[ ˆv0, . . . , ˆvd]S, which is a chain since Te(v_d)

Ã _j X

k=0

c_kvˆ_k

!

=0, j=0, . . . ,d,

due to Proposition 13.

5 Convergence

From [15, 16] we know that the Hermite subdivision schemeHAconverges to aC^d function according to Definition 2 if

1. there exists a schemeSBsuch thatT_C,dSA=2^−dSBT_C,d andSB is a convergentvector subdivision schemewith limit functionψ_g=e_df_g, for given input datag, wheree_d= [0, . . . , 0, 1]^T andfg is ascalar valuedfunction,

2. there exists a schemeS_B_e such thatTe_C,dS_A=2^−dS_B_eTe_C,dandS_B_eiscontractive.

Note that the normalization with the factor 2^−d now becomes relevant since it affects the normalization and contractivity property ofS_BandS_B_e, respectively.

Before we give the results about the convergence replacingTC,d andTeC,d byT andTe, respectively, we will now consider conditions to guarantee thatBe is the result of such a factorization. To that end, we recall the factorization identity

·I_d

∆

¸

S_B=S_B_e

·I_d

∆

¸

(32) from vector subdivision [18]. This relationship does not depend on the form of the Taylor operator. In terms of symbols, (32) becomes

·Id

z⁻¹−1

¸·B^∗₁₁(z) B^∗₁₂(z) B^∗₂₁(z) B^∗₂₂(z)

¸

=·eB^∗₁₁(z) Be^∗₁₂(z) Be^∗₂₁(z) Be^∗₂₂(z)

¸·Id

z⁻²−1

¸

, (33)

hence

B^∗(z) =

·Id

z⁻¹−1

¸⁻¹·eB^∗₁₁(z) Be^∗₁₂(z) Be^∗₂₁(z) Be^∗₂₂(z)

¸·Id

z⁻²−1

¸

=

· Be^∗₁₁(z) (z⁻²−1)Be^∗₁₂(z) (z⁻¹−1)⁻¹Be^∗₂₁(z) (z⁻¹+1)Be^∗₂₂(z)

¸

. (34)

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Lemma 28 SB converges to a continuous limit function of the formψg =fged if and only if S_B_e is contractive,Be21(1)=0andBe22(1)=1.

Proof: That convergence of the above type is equivalent to factorization and contractivity has been shown in [18], which already gives “⇒”. For “⇐”, however, we also must ensure thatB^∗ as defined in (34) is a Laurent polynomial. To that end, we must haveBe^∗₂₁(1)=0, otherwise (z⁻¹−1)⁻¹Be^∗₂₁(z) has a pole at 1. Second, the conditionSBed=ed is equivalent toB^∗(−1)ed =0 and B^∗(1)ed =2ed. The first one of these requirements is automatically satisfied according to (34), the second one becomes 2B^∗₂₂(1)=2.

Remark 29 Not thatBe^∗₂₂from(33)is just the scalar valued Laurent polynomialbe_{d d}^∗ .

Now we study the convergence of the Hermite scheme whenever we have one of the factorizations:T Se A=2^−dS_B_eTeorT SA=2^−dSBT. To that end, we first recall the one dimensional case of [15, Lemma 3].

Lemma 30 Given a sequence of refinementshn=

"

h⁽⁰⁾_n h⁽¹⁾_n

#

∈ℓ(Z,R²)such that 1. there exists a constant c inRsuch thatlimn→+∞h_n⁽⁰⁾(0)=c,

2. there exists a functionξ∈C(R,R)such that for any compact subset K ofRthere exists a sequenceµnwith limit0and

α∈2maxⁿK∩Z

¯¯h_n⁽¹⁾(α)−ξ¡ 2⁻ⁿα¢¯¯

∞ ≤ µn, (35)

α∈2maxⁿK∩Z

¯¯2ⁿ∆h⁽⁰⁾_n (α)−h⁽¹⁾_n (α)¯¯_∞ ≤ µn. (36)

Then there exists for any compact K a sequenceθnwith limit0such that the function ϕ(x)=c+

Z1

0

xξ(t x)d t, x∈R, (37)

satisfies

α∈2maxⁿK∩Z

°°h_n⁽⁰⁾(α)−ϕ¡

2⁻ⁿα¢°°≤θ_n, n∈N. (38)

Theorem 31 Let A,B ∈ℓ(d+1)×(d+1)(Z)be two masks related by the the factorization TdSA= 2^−dSBTdfor some generalized incomplete Taylor operator Td.

Suppose that for any initial data f₀∈ℓ^d+1(Z)and associated refinement sequence f_n of the Hermite scheme HA,

1. the sequencef_n(0)converges to a limity∈R^d+1,

2. the subdivision scheme S_B is C^p−d–convergent for some p≥d , and that for any initial datag₀=Tdf₀, the limit functionΨ=Ψg∈C^p−d¡

R,R^d+1¢

satisfies Ψ=

·0 ψ

¸

, ψ∈C^p−d(R,R) . (39)

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Then HAis C^p–convergent.

Proof: The proof is adapted from the proofs in [6, 14]. Givenf₀∈ℓ^d+1(Z), letg₀=T_df₀. We define the following two sequences: f_n+1=D⁻ⁿ⁻¹SA(D f_n) andg_n+1=SBg_n,n∈N. Since TdSA=2^−dSBTd, we can directly deduce thatf_n+1=2^ndTdDⁿf_n.

With the convergence of f_n(0), letyi :=limn→+∞f_n⁽ⁱ⁾(0), i=0, . . . ,d. Then we defineΦ recursively beginning withφd=ψand setting

φi(x)=yi+ Z₁

0

xφi+1(t x)d t i=d−1, . . . , 0. (40) ThenΦ=[φi]_i=0,...dis continuous withφ^(d−i)_i =ψ.

Fixing a compactK ⊂R, we will prove by a backward finite recursion that fork=d,d− 1, . . . , 0, there exists a sequenceεnwith limit 0 such that

¯¯

¯f_n^(k)(γ)−φ_k¡

2⁻ⁿγ¢¯¯¯≤εn, γ∈Z∩2ⁿK. (41) The casek=d is an immediate consequence of the convergence of the last row ofg_n and g_n^(d)=f_n^(d), which yields for anyγ∈Z∩2ⁿKthat

¯¯

¯f_n^(d)(γ)−ψ(2⁻ⁿγ)

¯¯

¯≤εn, (42)

while, fork<d, the convergence of the appropriate component ofg_nto zero implies that 2^n(d−k)

¯¯

¯∆fn^(k)(γ)− 1

2ⁿf_n^(k+1)(γ)+

d−kX

ℓ=2

tk,k+ℓ

1

2^nℓf_n^(k+ℓ)(γ)

¯¯

¯≤εn, (43) for a sequenceεnthat tends to zero forn→ ∞.

To prove (41) fork =d−1, we define the sequenceshn=[f_n^(d−1),f_n^(d)]^T. Then (43) becomes

¯¯

¯2ⁿ∆fn^(d−1)(·)−f_n^(d)(·)

¯¯

¯≤εn. Because of (42), we can apply Lemma 30 and obtain that

¯¯

¯f_n^(d−1)(γ)−φd−1

¡2⁻ⁿγ¢¯¯¯≤θn, γ∈2ⁿK∩Z, which is (41) fork=d−1.

To prove the recursive stepk+1→k, 0≤k<d−2, we get from (43) that, forγ∈Z∩2ⁿK,

¯¯

¯2ⁿ∆f_n^(k)(γ)−f_n^(k+1)(γ)¯¯¯≤ εn

2^n(d−k)+

d−kX

ℓ=2

|tk,k+ℓ| 2^nℓ

¯¯

¯f_n^(k+ℓ)(γ)¯¯¯ (44) Since (41) holds forj>k, it follows that

n→∞lim

¯¯

¯f_n^(j)(γ)−φj

¡2⁻ⁿγ¢¯¯¯=0

uniformly forγ∈Z∩2ⁿK and sinceφ_j is bounded onK, so is the sequence¯¯¯f_n^(j)(γ)¯¯¯onZ∩ 2ⁿK. Thus the right hand side of (44) converges to zero so that it immediately implies (41)

using again Lemma 30.

As a consequence of Theorem 31 and Lemma 28 we also have the following results.