From Hermite to stationary subdivision schemes in one and several variables

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From Hermite to stationary subdivision schemes in one and several variables

Jean-Louis Merrien, Tomas Sauer

To cite this version:

Jean-Louis Merrien, Tomas Sauer. From Hermite to stationary subdivision schemes in one and sev- eral variables. Advances in Computational Mathematics, Springer Verlag, 2012, 36 (4), pp.547-579.

�10.1007/s10444-011-9190-7�. �hal-00468245�

From Hermite to Stationary Subdivision Schemes in One and Several Variables

Jean–Louis Merrien^∗ Tomas Sauer^† March 30, 2010

Abstract

Vector and Hermite subdivision schemes both act on vector data, but since the latter one interprets the vectors as function values and consecutive deriva- tives they differ by the “renormalization” of the Hermite scheme in any step. In this paper we give an algebraic factorization method in one and several variables to relate any Hermite subdivision scheme that satisfies the so–called spectral con- dition to a vector subdivision scheme. These factorizations are natural extensions of the “zero atπ” condition known for the masks of refinable functions. More- over, we show how this factorization can be used to investigate different forms of convergence of the Hermite scheme and why the multivariate situation is concep- tionally more intricate than the univariate one. Finally, we give some examples of such factorizations.

Keywords: Subdivision, Hermite, Taylor expansion, Factorization.

AMS Subject classification: 41A60, 65D15, 13P05.

1 Introduction

Subdivision algorithms are iterative methods for producing curves and surfaces with a built-in multiresolution structure. They are now used in curve and surface modelling in computer-aided geometric design, video games, animation and many other applica- tions. Stationary and homogeneous subdivision schemes iterate the same subdivision operator and use rules that, independently of the location, compute new values of a refined discrete sequence only from a certain amount of local and neighboring data.

This data can be scalar or vector even matrix valued, the matrix case been done by doing vector schemes columnwise, and the stationary, i.e. local and neighboring rule is always acting on a vector sequencecas

S_Ac= X

α∈Zs

A(· −2α)c(α),

∗J.-L. Merrien, INSA de Rennes, 20 av. des Buttes de Coesmes, CS 14315, 35043 RENNES CEDEX, France, email: Jean-Louis.Merrien@insa-rennes.fr

†T. Sauer, Lehrstuhl f¨ur Numerische Mathematik, Justus–Liebig–Universit¨at Gießen, Heinrich-Buff- Ring 44, D–35392 Gießen, Germany, email: Tomas.Sauer@math.uni-giessen.de

where the finitely supported sequence of coefficients is referred to as the mask of the subdivision scheme. This operator is iterated, leading to a sequence S_Aⁿc, n ∈ N, of vector valued sequences, which, when related to the finer and finer grid 2⁻ⁿZ^s, converges to a limit vector field with certain properties. Such schemes have been in- vestigated for example in [4, 13, 16, 20, 21] and there exists quite a substantial amount of literature on vector subdivision schemes in one and several variables meanwhile, in- vesting convergence as well as smoothness of the limit functions or the multiresolution structures generated by these functions.

If the vector data to be processed represents the value and consecutive differences of a function in the so–called Hermite subdivision schemes, the situation changes sig- nificantly due to the particular meaning of the vector’s coefficients. Hermite subdivi- sion schemes have also been studied by many authors, for example, [5, 11, 12, 17, 19].

As a consequence, a Hermite subdivision scheme is a so called non stationary scheme with a mask at stepngiven byA_n =D⁻ⁿ⁻¹ADⁿ, see (6), whereAis a usual mask andDis a diagonal matrix with diagonal entries2^−j,j∈ {0, . . . , d}coming from the derivatives ofψ(·) = φ(2·). This matrix creates a new problem since the successive powers ofD⁻¹ have no limit whenn goes to ∞ and almost all the components of Dⁿ goes to0. So it is interesting to transform a Hermite subdivision scheme into a Stationary Subdivision Scheme using a Taylor operator which comes from the usual Taylor expansion.

2 Notation and basic concepts

Vectors inR^r will be labeled by lowercase boldface letters asy = [yj]_j=1,...,r. In particular,e_j,j = 1, . . . , r, stands for the canonical unit vectors for which(e_j)_k = e_jk = δ_jk,k = 1, . . . , r, holds true. Matrices inR^r×r will be written as uppercase boldface letters, likeA= [ajk]j,k=1,...,r.

ByA ∈ ℓ^r×r(Z^s), we also denote a multiindexed sequence of matrices, that is, for allα ∈ Z^sthe sequence element A(α) = [a_jk(α)]j,k=1,...,r ∈ R^r×r is an r×r matrix. Any such sequence will be called a mask provided that it is finitely supported, that is, there existsN ∈Nsuch that

suppA:={α∈Z^s : A(α)6=0} ⊆[−N, N]^s.

To any mask A we associate the stationary subdivision operator SA : ℓ^r(Z^s) → ℓ^r(Z^s), defined as

S_Ac(α) := X

β∈Zs

A(α−2β) c(β), c∈ℓ^r(Z^s). (1) Polynomials and Laurent polynomials play a fundamental role in the study of sub- division schemes, so we recall the definitions ofΠ = R[x] = R[x1, . . . , xs] and Λ =R

x, x⁻¹

, respectively, where, using the monomialsx^α:=x^α₁¹. . . x^α_s^s, f(x) = X

α∈N^s

0

f_αx^α∈Π, f(x) = X

α∈Zs

f_αx^α∈Λ, (2)

but in both cases only finitely many of the coefficientsf_α are different from zero so that we are dealing with finite sums in (2). Though polynomials and Laurent poly- nomials appear to be almost the same – after all, any Laurent polynomial is obtained by multiplying a polynomial of the formx^α, α ∈ Z^s, which is a unit in Λ – they are of surprisingly different algebraic structure, cf. [22]. ByΠ_kwe denote the finite dimensional vector space of all polynomials of total degree at mostk, that is

f(x) = X

|α|≤k

f_αx^α∈Π_k, |α|:=

Xs j=1

α_j, α∈N^s₀,

wheredim Π_k=r_k:= ^k+s_s .

Recall thatsdenotes the number of variables and write, fork∈N,s_k:= ^s+k−1_s−1 for the dimension of the space

Π⁰_k:=span{x^α : |α|=k}

of homogeneous polynomials of degree k. Obviously, the dimension r_k of Π_k = Lk

j=0Π⁰_ksatisfiesr_k=s0+· · ·+s_k.

A convenient way of studying masks, well used in signal processing, is to consider its symbol which is the – in our case matrix valued – Laurent polynomial

A^∗(z) := X

α∈Zs

A(α)z^α, z∈C^s_×, C_×=C\ {0}.

Moreover,A^∗ z⁻¹

is the well–knownz–transform from signal processing, cf. [14, 15, 28]. In particular, forc∈ℓ^r(Z^s)we have the identity

(SAc)^∗(z) =A^∗(z)c^∗ z² .

Ifǫ_j is thej-th vector of the canonical basis inR^s, we will use the forward differ- ence operators∆_j,j = 1, . . . , s, defined by∆_jc =c(·+ǫ_j)−cand the difference operator

∆ : ℓ^r(Z^s)→ℓ^sr(Z^s), ∆c=





∆₁c ...

∆_sc



 (3)

which plays the role of a discrete gradient.

We will first state and prove the Taylor factorization in one variable in Section 3 to explain its basic idea and then turn to the notationally and conceptionally more intricate multivariate situation later (Section 4).

One key argument is the following factorization result: If a ∈ ℓ(Z^s), satisfies, for allα ∈ Z^s, the condition P

β∈Zsa(α−2β) = 0, then there exist b_j ∈ ℓ(Z^s), j= 1, . . . , s, such that

a^∗(z) = Xs j=1

z_j⁻²−1

b^∗_j(z), (4)

see [1, 23] for a proof. This is a generalization of the well known result in one dimen- sion,s= 1, that

X

β∈Z

a(α−2β) = 0, α∈Z, ⇒ a^∗(z) = (z⁻²−1)b^∗(z). (5) Now we define the Hermite subdivision schemeHA. In dimensionswithdderiva- tives, starting withf₀∈ℓ^rd(Z^s), forn= 0, . . ., we definef_n+1 ∈ℓ^rd(Z^s)by

Dⁿ⁺¹f_n+1(α) = X

β∈Zs

A(α−2β) Dⁿf_n(β), c∈ℓ^rd(Z^s), (6)

where

D =





 1

1 2I_s1

. ..

1 2^dI_sd







is the diagonal matrix with diagonal entries2^−j repeated s_j times, respectively, j = 0, . . . , d.

For the iterated application of the Hermite subdivision scheme, we decompose the vector into the block form

f_n(β) =h

f^(j)_n (β)i

j=0,...,d, β∈Z^s, f^(j)_n ∈ℓ^sj(Z^s),

and the first componentf⁽⁰⁾_n (β) can be interpreted as the value of a functionφ_n at β/2ⁿ, while thes1following ones,f⁽¹⁾_n (β), describe the gradient or first total deriva- tiveD¹φn(β/2ⁿ), and so on, up to the lasts_dones,f^(d)_n (β)which areD^dφn(β/2ⁿ), whereD^j :=h

D^α = _∂x^∂jα

i

|α|=j,j= 0, . . . , d.

The main goal of this paper is to prove that, under suitable hypotheses, namely the spectral condition, a Hermite subdivision operator can be transformed into an equiva- lent stationary subdivision operator which we will call the Taylor subdivision operator associated toHA(Section 3 for the dimensions= 1and Section 4 fors >1).

Then, in Section 5, fors= 1, we will prove that if the Taylor subdivision operator is convergent as a stationary vector subdivision scheme, thenHAis convergent in the sense of Hermite subdivision schemes. Fors >1, when adding further hypothesis, we will give a generalization of the previous results.

In the last Section, we give examples of interpolatory and non interpolatory schemes and their transformation into stationary subdivision schemes.

3 The Taylor factorization in one variable, s = 1

We will use a function and itsdderivatives, but since s = 1, we are in the special situation that, for anyk∈N,s_k= 1andr=r_d=d+ 1.

In this section, we will show that any subdivision operator that satisfies the spec- tral condition will admit a factorization in terms of operators. In fact, this result was

already proved in [10]. We give a new proof using different operators and a more algebraic approach.

To introduce the definition of the spectral condition, we associate to any function f ∈C^d(R)the vector sequencev_f ∈ℓ^d+1(Z)with

v_f(α) :=





 f(α) f^′(α)

... f^(d)(α)





, α∈Z. (7)

Definition 1 A maskAor its associated subdivision operatorSAsatisfies the spectral condition of orderd^′ ≤dif there exist polynomialsp_j ∈Π_j,degp_j =j,j = 0, . . . , d^′, such that

SAv_j = 1

2^jv_j, v_j :=v_pj =





 p_j p^′_j ... p^(d)_j





, (8)

where we will always assume thatp_jis normalized such thatp_j(x) = _j!¹x^j +· · ·. It was proved in [10] that the spectral condition is also equivalent to the sum rule introduced by Bin Han et al [16, 17].

Definition 2 The Taylor operatorT_dof orderd, acting onℓ^d+1(Z), is defined as

T_d:=











∆ −1 . . . −_(d−1)!¹ −_d!¹

∆ . .. ... ... . .. −1 ...

∆ −1

1











. (9)

The name “Taylor operator” is easily explained: Iff ∈C^d(R)then, for anyα∈Z, (T_dv_f)_j(α) =f^(j)(α+ 1)−

Xd−j k=0

1

k!f^(j+k)(α), j= 0, . . . , d−1, (10) are precisely the terms appearing in the Taylor expansion off, as well as(T_dv_f)_d(α) = f^(d)(α).

Remark 3 An alternative version of the Taylor operator, namely

T_d^′ :=











∆ −1 . . . −_(d−1)!¹ 0

∆ . .. ...

. .. −1 ...

∆ 0

1









 was introduced and investigated in [10].

To transform the Hermite subdivision scheme into a stationary subdivision scheme, we will also need a slight modification of the Taylor operator from (9), namely the complete Taylor operator acting onℓ^(d+1)(Z)

Te_d:=











∆ −1 . . . −_(d−1)!¹ −_d!¹

∆ . .. ... ... . .. −1 ...

∆ −1

∆











We notice thatTe₀^∗(z) =z⁻¹−1and fork= 1, . . . , d,

Te_k+1^∗ (z) =

Te_k^∗(z) −w_k 0 z⁻¹−1

, w_k:=







1 (k+1)!

1 k!... 1





. (11)

To simplify notation, we will writeIand0for identity and zero matrices of appropriate size whenever their dimensions are clear from the context.

Theorem 4 If the maskA∈ℓ(d+1)×(d+1)(Z)satisfies the spectral condition of order d, then there exist two finitely supported masksB,Be ∈ℓ(d+1)×(d+1)(Z)such that

T_dSA= 2^−dSBT_d and Te_dSA= 2^−dS_fBTe_d. (12) We will split the proof of Theorem 4 into several pieces, beginning with a simple observation onT_d.

Lemma 5 Forp∈Π_d, we have thatT_dv_p =e_dp^(d)(0)andTe_dv_p =0.

Proof: Making use of (10), we see that for anyα∈Zthere existξ₀, . . . , ξ_d−1∈(0,1) such that

(T_dv_p)_j(α) = 1

(d−j+ 1)!p^(d+1)(α+ξj) = 0, j= 0, . . . , d−1, and that(T_dv_p)_d(α) =p^(d)(α) =p^(d)(0),sincep∈Π_d.

The proof ofTe_dv_p=0is similar.

Expanding, at first, any p ∈ Π_d−1 with respect to the basis{p0, . . . , p_d−1}, and taking into account thatp^(d)_d = 1, the following result is an immediate consequence of Lemma 5.

Corollary 6 If the maskAsatisfies the spectral condition of orderd, then

T_dSAv_p= 0, p∈Π_d−1, T_dSAv_d= 2^−de_d (13) andTe_dSAv_p = 0for anyp∈Π_d.

Proposition 7 If S_Cv_p = 0 for all p ∈ Π_k, k ≤ d, then there exists a (finitely supported) maskB_k ∈ℓ(d+1)×(d+1)(Z)such that

SC =SB_k

Te_k 0 0 I

. (14)

Proof: WritingC^∗(z) = [c^∗₀(z)· · ·c^∗_d(z)] in terms of its column vectors, we shall prove thatS_Cv_p= 0,p∈Π_k, implies the existence ofb₀, . . . ,b_ksuch that

C^∗(z) =

b^∗₀(z)· · ·b^∗_k(z)c^∗_k+1(z)· · ·c^∗_d(z) Te_k^∗ z² 0

0 I

, (15)

which defines

B^∗_k(z) :=

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

. (16)

This will be done by induction onk.

In the casek= 0the assumption implies thatpis a constant polynomial, hence 0= (SCe₀)(α) =X

β∈Z

c₀(α−2β),

so that, by (5), there existsb^∗₀(z)such thatc^∗₀(z) = z⁻²−1

b^∗₀(z). Consequently, C^∗(z) =

z⁻²−1

b^∗₀(z)c^∗₁(z)· · ·c^∗_d(z)

=B^∗₀(z)

z⁻²−1 0

0 I

= B^∗₀(z)

Te₀^∗ z² 0

0 I

, which shows the validity of (15) fork= 0.

Now, let us assume that (14) is satisfied for somek≥0and thatSCv_p= 0for all p∈Π_k+1. A simple computation based on (10), like in the proof of Lemma 5, shows that forp=p_k+1we get forv^′_p =h

(v_p)_ji

j=0,...,k ∈ℓ^k+1(Z)that Te_kv^′_p=w_kp^(k+1)(0) =w_k,

wherew_kis defined in (11). By the induction hypothesis and our assumption, 0 = S_Cv_p =S_Bk





Te_k 0 0 0 1 0 0 0 I







 v^′_p

1 0



=S_Bk



 w_k

1 0





That is, X

β∈Z

Xk j=0

1

(k+ 1−j)!b_j(α−2β) +c_k+1(α−2β) = 0, α∈Z, and the same argument as before shows that

Xk j=0

1

(k+ 1−j)!b^∗_j(z) +c^∗_k+1(z) =b^∗_k+1(z) z⁻²−1 ,

in other words,

c^∗_k+1(z) =

b^∗₀(z)· · ·b^∗_k+1(z)

−w_k z⁻²−1

, or

B^∗_k(z) =B^∗_k+1(z)





I −w_k 0 0 z⁻²−1 0

0 0 I



.

Substituting this into (15) we thus get C^∗(z) = B^∗_k(z)

Te_k^∗ z² 0

0 I

= B^∗_k+1(z)





I −w_k 0 0 z⁻²−1 0

0 0 I









Te_k^∗ z² 0 0

0 1 0

0 0 I





= B^∗_k+1(z)





Te_k^∗ z²

−w_k 0 0 z⁻²−1 0

0 0 I



=B^∗_k+1(z)

Te_k+1^∗ z² 0

0 I

,

which completes the proof.

Corollary 8 For a maskC ∈ ℓ(d+1)×(d+1)(Z) the following statements are equiva- lent:

1. S_Cv_p =0for allp∈Π_d−1,

2. there exists as maskB^′such thatSC =S_B^′T_d^′, 3. there exists as maskBsuch thatS_C =S_BT_d. Proof: By Proposition 7,

C^∗(z) =B^∗_d−1(z)

Te_d−1 z² 0

0 1

=B^∗_d−1(z)T_d^′ z² ,

hence,B^′ =B_d−1satisfies 2). To prove 3), we expand the above identity into C^∗(z) = B^∗_d−1(z)

I w 0 1

I −w 0 1

T_d^′ z²

= B^∗_d−1(z)

I w 0 1

T_d z²

=:B^∗(z)T_d z² .

Conversely, since forp ∈Π_d−1 we have thatT_d^′v_p =T_dv_p = 0, it also follows that SAv_p =S_B^′T_d^′v_p=SBT_dv_p =0forp∈Π_d−1.

Corollary 9 A maskC ∈ℓ(d+1)×(d+1)(Z)satisfiesSCv_p = 0for allp∈ Π_dif and only if there exists a maskBe such thatS_C =S_fBTe_d.

With these observations in hand, the proof of Theorem 4 is now easily completed.

Proof of Theorem 4: DefiningC by C^∗(z) = 2^dT^∗_d(z)A^∗(z) and noting that for p=p_k∈Π_k,k= 0, . . . , d−1, we get that

SCv_k=T_dSAv_k= 1

2^kT_dv_k =0, v_k=v_pk.

Since{p₀, . . . , p_d−1}form a basis ofΠ_d−1, we have thatSCv_p = 0,p ∈Π_d−1, and Corollary 8 tells us that there existsBsuchS_C =S_BT_d, henceT_dS_A = 2^−dS_BT_d as claimed.

The equalityTe_dSA= 2^−dS_fBTe_dis obtained withC^∗(z) = 2^dTe^∗_d(z)A^∗(z),k=d

andBe =B_din Proposition 7.

4 The multivariate case

In the multivariate case, the definition, theorems and proofs follow the same lines as in the univariate situation, but are conceptionally slightly more intricate.

We still consider polynomials of total degree at most dand to any function f ∈ C^d(R^s)we now associate the vector valued sequence defined as

v_f(γ) :=





 f(γ) D¹f(γ)

... D^df(γ)





, γ ∈Z^s,

where we recall thatD^j :=h

D^α= _∂x^∂jα

i

|α|=j stands for the total differential of order j. The partial differences∆j,j = 1, . . . , d, are now defined as∆jf(γ) =f(γ+ǫj)−

f(γ)forγ ∈Z^s. Again, we use a partial univariate Taylor expansion in the direction ofǫ_j, to observe that for|α| ≤d

D^αf(γ+ǫ_j) =D^αf(γ) +

d−|α|−1

X

k=1

1

k!D^α+kǫjf(γ) +Rf, γ ∈Z^s, with Rf = 0 if f ∈ Π_d−1, so that the j–th partial Taylor operator operating on ℓ^rd(Z^s)takes the form

T_d^j :=











∆_jI₀ T^j_0,1 · · · T^j_0,d−1 T^j_0,d

∆jI₁ . .. ... ... . .. T^j_d−2,d−1 ...

∆jI_d−1 T^j_d−1,d I_d









 ,

whereI_k ∈R^sk×sk,k= 0, . . . , d, are identity matrices of dimensions_k×s_kand the matricesT^j_k,ℓ∈R^sk×sℓ,ℓ=k+ 1, . . . , d, have zero entries except

T^j_k,ℓ

α,α+(ℓ−k)ǫj

=− 1

(ℓ−k)!, |α|=k, ℓ=k+ 1, . . . , d.

The total Taylor operator is then defined as T_d:=



 T_d¹

... T_d^s



 (17)

For example, fors= 2andd= 2, we have

f =









 f^(0,0) f^(1,0) f^(0,1) f^(2,0) f^(1,1) f^(0,2)











, T2f =























f^(0,0)(·+ǫ₁)−f^(0,0)(·)−f^(1,0)(·)−¹₂f^(2,0)(·) f^(1,0)(·+ǫ₁)−f^(1,0)(·)−f^(2,0)(·) f^(0,1)(·+ǫ₁)−f^(0,1)(·)−f^(1,1)(·)

f^(2,0)(·) f^(1,1)(·) f^(0,2)(·)

f^(0,0)(·+ǫ₂)−f^(0,0)(·)−f^(0,1)(·)−¹₂f^(0,2)(·) f^(1,0)(·+ǫ₂)−f^(1,0)(·)−f^(1,1)(·) f^(0,1)(·+ǫ₂)−f^(0,1)(·)−f^(0,2)(·)

f^(2,0)(·) f^(1,1)(·) f^(0,2)(·)























∈R¹².

As in one variable we define the complete (partial and total) Taylor operators as

Te_d^j :=











∆jI₀ T^j_0,1 · · · T^j_0,d−1 T^j_0,d

∆_jI₁ . .. ... ... . .. T^j_d−2,d−1 ...

∆_jI_d−1 T^j_d−1,d

∆jI_d











, Te_d:=



 Te_d¹

... Te_d^s



.

Again for the examples= 2andd= 2, we obtain

Te₂f =























f^(0,0)(·+ǫ₁)−f^(0,0)(·)−f^(1,0)(·)−¹₂f^(2,0)(·) f^(1,0)(·+ǫ₁)−f^(1,0)(·)−f^(2,0)(·) f^(0,1)(·+ǫ₁)−f^(0,1)(·)−f^(1,1)(·)

f^(2,0)(·+ǫ₁)−f^(2,0)(·) f^(1,1)(·+ǫ₁)−f^(1,1)(·) f^(0,2)(·+ǫ₁)−f^(0,2)(·)

f^(0,0)(·+ǫ₂)−f^(0,0)(·)−f^(0,1)(·)−¹₂f^(0,2)(·) f^(1,0)(·+ǫ₂)−f^(1,0)(·)−f^(1,1)(·) f^(0,1)(·+ǫ₂)−f^(0,1)(·)−f^(0,2)(·)

f^(2,0)(·+ǫ₂)−f^(2,0)(·) f^(1,1)(·+ǫ₂)−f^(1,1)(·) f^(0,2)(·+ǫ₂)−f^(0,2)(·)























∈R¹².

Obviously, in the general case,T_dandTe_dmap anr_d–vector valued sequence to a s·r_d–vector valued sequence.

We will denote the unit vectors inR^rdbye_α,|α| ≤d, and the unit vectors inR^srd bye_j,α,j= 1, . . . , s,|α| ≤d.

Definition 10 The maskA ∈ ℓ^rd×rd(Z^s)is said to satisfy the spectral condition of orderd^′ ≤dif there exists a basis ofΠ_d^′consisting of polynomialsp_α ∈Π_|α|,|α| ≤d^′, such that

S_Av_α= 2^−|α|v_α, v_α =v_pα (18) Remark 11 Since{p_β}_|β|≤kis a basis ofΠ_k, for anyαsuch that|α|=k, the mono- mial _α!¹x^αcan be written as

1

α!x^α = X

|β|=k

λ_α,βp_β+ X

|β|<k

λ_α,βp_β.

Letq_α(x) = P

|β|<kλ_α,βp_β. By (18), the polynomial m_α(x) = P

|β|=kλ_α,βp_β =

1

α!x^α−q_α(x)satisfiesS_Av_mα = 2^−|α|v_mα. Replacingp_αbym_αif necessary, we can thus always assume that the polynomialsp_αof Definition 10 are normalized in such a way that their leading term is _α!¹ x^α.

The counterpart for Lemma 5 is as follows.

Lemma 12 Forp∈Π_dwe have that T_dv_p =

Xs j=1

X

|α|=d

e_j,αD^αp(0). (19)

Proof: For|α|< d, anyγ ∈Z^sandj = 1, . . . , sthere exists someξ=ξ_α,γ,j∈(0,1) such that

e^T_j,αT_dv_p(γ) = T_d^jv_p

α(γ) = 1

(d− |α|+ 1)!Dα+(d−|α|+1)p(γ+ξǫ_j) = 0, (20) while for|α|=dandj= 1, . . . , s

e^T_j,αT_dv_p(γ) =α!pα(γ) =D^αp(0), which can be combined into

T_dv_p = Xs j=1

X

|α|≤d

e_j,αe^T_j,αT_dv= Xs j=1

X

|α|=d

e_j,αD^αp(0),

that is (19).

Corollary 13 Forp∈Π_dwe have thatTe_dv_p =0.

Also the extension of Proposition 7 follows essentially the same lines but now makes use of the associated quotient ideals, cf. [23, 26].

Proposition 14 For some positive integer N, let C ∈ ℓ^N×rd(Z^s). If for k ≤ d, SCv_p = 0for all p ∈ Π_k, then there exists a finitely supported maskB_k ∈ ℓ^N×srd such that

SC =SB_kTe_k.

Proof: Again let us writeC^∗ in terms of its column blocks,

C^∗(z) = [C^∗₀(z)· · ·C^∗_d(z)], C^∗_k(z) = [c^∗_α(z)]_|α|=k, k= 0, . . . , d.

The proof consists of showing that wheneverSCv_p = 0for allp ∈ Π_k, then there exist finitely supported masksB_j,k ∈ℓ^N×sj(Z^s),j = 1, . . . , sandk= 0, . . . , d, thus B_k∈ℓ^N×rd(Z^s),k= 0, . . . , d, such that

C^∗(z) =

B^∗₁₀(z)· · ·B^∗_1k(z)C^∗_k+1(z)· · ·C^∗_d(z)| · · ·

· · · |B^∗_s0(z)· · ·B^∗_sk(z)C^∗_k+1(z)· · ·C^∗_d(z)









 Te_k¹∗

z² 0

0 I

... ... Te_k^s∗

z² 0

0 I









 (21)

=: B^∗_k(z) Te_k^∗ z²

, I ∈R^{(rd−rk)×(rd−rk)}. Once more, induction is used to buildB_korB^∗_k(z).

For k = 0 we only have to consider p = 1, that is v_p = e₀, which leads to 0=S_Ce₀ such that, by (4), we have the vector identity

c^∗₀(z) = Xs j=1

z⁻²_j −1 b^∗_0j(z),

and withB^∗_0j =b^∗_0j we thus get that

C^∗(z) = [B^∗₀₁(z)C^∗₁(z)· · ·C^∗_d(z)| · · · |B^∗_0s(z)C^∗₁(z)· · ·C^∗_d(z)]









z₁⁻²−1 0

0 I

... ... z_s⁻²−1 0

0 I









= B^∗₀(z)









 Te₀¹∗

z² 0

0 I

... ... Te₀^s∗

z² 0

0 I











=B^∗₀(z)Te₀^∗ z² ,

which is (21) fork= 0.

Now suppose that the validity of (21) has been proved for some k and assume thatS_Cv_p = 0for all p ∈ Π_k+1. By the induction hypothesis, (21) holds, so that C^∗(z) =B^∗_k(z)Te_k^∗ z²

. Fixα∈N^s

0 with|α|=k+ 1. For the monic polynomialp_α withp_α(x)−_α!¹x^α∈Π_|α|−1, we again consider the truncated vector sequence

v^′_p =h (v_p)_βi

|β|≤k∈ℓ^rk(Z^s)

for which (20) yields that there exist some ξ ∈ R^d such that for j = 1, . . . , s and

|β| ≤k

e^T_j,βTe_kv^′_p = T_d^jv_p

β = 1

(k− |β|+ 1)!Dβ+(k−|β|+1)ǫjp(ξ)

= 1

(k− |β|+ 1)!δβ+(k−|β|+1)ǫj,α, hence,

Te_k^jv^′_p= Xk ℓ=0

1

(k−ℓ+ 1)!e_{α−(k−ℓ+1)ǫj} = Xk+1 ℓ=1

1

ℓ!e_α−ℓǫj =:w_j ∈R^rk, (22) with the usual convention thate_β =0wheneverβ ∈Z^s\N^s

0, that is, wheneverβhas a negative component. The definition ofw_j in (22) is also valid inR^rd, and we will denote this natural embedding bywe_j. Now,

0 = S_Cv_p =S_BkTe_k v^′_p+e_α

=S_Bk



 e w₁+e_α

... e w_s+e_α





= S_Bk Xs j=1

e_j,α+ Xk+1 ℓ=1

1

ℓ!e_{j,α−ℓǫj}

! ,

which can be summarized by stating that 0=SB_kw_α, w_α:=

Xs j=1

e_j,α+

k+1X

ℓ=1

1

ℓ!e_{j,α−ℓǫj}

!

∈R^srd, |α|=k+ 1.

(23) Recalling that(B_k)^∗_j,α=c^∗_αforj= 1, . . . , sand|α|=k+ 1, (23) means in terms of the columns ofB_kthat forj= 1, . . . , sand|α|=k+ 1

0 =c^∗_α(z) +

k+1X

ℓ=1

1

ℓ!b^∗_{j,α−ℓǫj}(z), z∈ {±1}^s, so that there existb^∗_m,j,α,m= 1, . . . , s,|α|=k+ 1, such that

c^∗_α(z) +

k+1X

ℓ=1

1

ℓ!b^∗_{j,α−ℓǫj}(z) = 1 s

Xs m=1

z⁻²_m −1

b^∗_m,j,α(z). (24) Averaging this identity overj, settingb^∗_m,α(z) := ¹_sPs

j=1b^∗_m,j,α(z)and replacingm

byj, we obtain

c^∗_α(z) = 1 s

Xs j=1

−

k+1X

ℓ=1

1

ℓ!b^∗_{j,α−ℓǫj}(z) +

z⁻²_j −1 b^∗_j,α(z)

!

= 1 s

Xs j=1

B^∗_j,0(z)· · ·B^∗_j,k+1(z)"

−w_j z_j⁻²−1

e_α

#

= 1 s

B^∗_j,0(z). . .B^∗_j,k+1(z)

j=1,...,s









−w1

z₁⁻²−1 e_α ...

−w_s z_s⁻²−1

e_α







 ,

which can be combined into

C^∗_k+1(z) = 1 s

B^∗_j,0(z)· · ·B^∗_j,k+1(z)

j=1,...,s









−w₁1^T

sk+1

z₁⁻²−1 I_sk+1 ...

−w_s1^T

sk+1

z_s⁻²−1 I_sk+1







 .

If we then substitute this expression for the columnC^∗_k+1of B^∗_k(z) =

B^∗_j,0(z)· · ·B^∗_j,k(z)C^∗_k+1(z)· · ·C^∗_d(z)

j=1,...,s, we obtain the recursive relationshipB^∗_k(z) =B^∗_k+1(z)Y (z)where

Y (z) =



















I_rk −w₁ s 1^T

sk+1 . . . −w₁

s 1^T

sk+1

z₁⁻²−1

s I_sk+1 z₁⁻²−1

s I_sk+1

0 I_rd−rk+1 0

... . .. ...

−w_s s 1^T

s_k+1 I_rk −w_s

s 1^T

s_k+1

z_s⁻²−1

s I_sk+1 z₁⁻²−1

s I_sk+1

0 . . . 0 I_rd−rk+1

















 .

(25) Substituting (25) into (21), we thus get that

C^∗(z) = B^∗_k(z)T_k^∗ z²

= 1

sB^∗_k+1(z)Y (z)T_k^∗ z² .

If we partitionT_k^∗according to the structure ofY as

T_k^∗ =











 T_k¹∗

I_sk+1

I_rd−rk+1 ... ... ... (T_k^s)^∗

I_sk+1

I_rd−rk+1











 ,

then a simple matrix multiplication yields that

Y^∗(z) T_k^∗ z²

=











 T_k¹∗

(z₂) −w₁1^T 0 0 z₁⁻²−1

I_sk+1 0

0 0 I

... ... ...

(T_k^s)^∗ z²

−w_s1^T 0 0 z_s⁻²−1

I_sk+1 0

0 0 I













=T_k+1^∗ z² ,

which shows that indeedC^∗(z) =B^∗_k+1(z)T_k+1^∗ z²

, thus advancing the induction

hypothesis and completing the proof.

With an appropriate renormalization like in the proof of Theorem 4, we thus obtain its counterpart for dimensions.

Theorem 15 If the maskA ∈ ℓ^rd×rd(Z^s)satisfies the spectral condition of orderd then there exists two (finitely supported) masksB,Be ∈ℓ^srd×srd(Z^s)such that

TdSA= 2^−dSBTd, and TedSA= 2^−dS_fBTed. (26) Remark 16 It is worthwhile to note that the proof of Theorem 15 gives a constructive method to compute the symbols of the Taylor schemesBandB. The crucial step is thee computation of the decomposition in (24), a task that is already implemented in some Computer Algebra Systems, for example in CoCoA, [2]. Recall, however, that due to the existence of syzygies this decomposition is usually not unique, cf. [3] and so also BandBe are not unique.

Definition 17 With the hypotheses of the previous Theorem, the subdivision schemes SBis called the Taylor subdivision scheme associated withHAand the schemeS_fB is the complete Taylor subdivision scheme.

Remark 18 For initial dataf₀ ∈ ℓ^rd(Z^s), we consider the sequence of iterations f_n,n ∈Nbuilt from the schemeHAdefined in (6). If we now define the associated sequence

g_n:= 2^ndTdDⁿf_n∈ℓ^srd(Z^s), n∈N, (27)

and ifAsatisfies the spectral condition, then we deduce from (6) that forn ∈ Nwe have

g_n = 2^ndT_dDⁿf_n= 2^ndT_dS_ADⁿ⁻¹f_n−1 = 2^(n−1)dS_BT_dDⁿ⁻¹f_n−1

= S_Bg_n−1= (S_B)ⁿg₀,

henceg_nis generated by the stationary subdivision schemeSB. Similarly ifh_n:= 2^ndTe_dDⁿf_nthen,h_n= S_fBn

h₀.

5 Convergence of Subdivision Operators

Definition 19 LetB ∈ ℓ^r×r(Z^s) be a mask andSB : ℓ^r(Z^s) → ℓ^r(Z^s) the asso- ciated stationary subdivision operator defined in (1). The operator isC⁰–convergent if for any datag₀ ∈ ℓ^r(Z^s)and corresponding sequence of refinementsg_n = S_Bⁿg₀ there exists a functionΨg ∈ C(R^s,R^r) such that for any compact K ⊂ R^s there exists a sequenceε_nwith limit0that satisfies

α∈maxZs∩2ⁿK

g_n(α)−Ψg 2⁻ⁿα

∞≤ε_n. (28)

Definition 20 LetA∈ℓ^rd×rd(Z^s)be a mask andHAthe associated Hermite subdi- vision scheme onℓ^rd(Z^s)as defined in (6). The scheme is convergent if for any data f₀ ∈ℓ^rd(Z^s)and the corresponding sequence of refinementsf_n, there exists a func- tionΦ = [φ_α]_|α|≤d ∈ C(R^s,R^rd)such that for any compactK ⊂ R^sthere exists a sequenceε_nwith limit0which satisfies

max|α|≤d max

β∈Zs∩2ⁿK

f_n^(α)(β)−φ_α 2⁻ⁿβ≤ε_n. (29) The schemeHAis said to beC^d–convergent if moreoverφ0 ∈C^d(R^s,R)and

∂^αφ₀

∂x^α =φ_α, |α| ≤d.

Theorem 21 Given a maskA ∈ℓ^rd(Z^s)which satisfies the spectral condition. Sup- pose that for any dataf₀ ∈ ℓ^rd(Z^s) and associated refinement sequencef_n of the Hermite schemeHA,

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

2. the associated Taylor subdivision schemeSBisC⁰–convergent and for any ini- tial datag₀ =T_df₀, the limit functionΨ = Ψg ∈C(R^s,R^srd)satisfies

Ψ =1_s⊗ 0_r

d−s_d

Ψ_d

=







 0 Ψ_d

... 0 Ψ_d









, Ψ_d= [ψ_α]_|α|=d. (30)

ThenH_Ais convergent.