Extra Regularity of Hermite Subdivision Schemes

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Extra Regularity of Hermite Subdivision Schemes

Jean-Louis Merrien, Tomas Sauer

To cite this version:

Jean-Louis Merrien, Tomas Sauer. Extra Regularity of Hermite Subdivision Schemes. Dolomites Research Notes Approximation, 2021, 14 (2), pp.85-94. �hal-03095865�

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Extra Regularity of Hermite Subdivision Schemes

Jean-Louis Merrien^* Tomas Sauer^† January 4, 2021

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 of a function and its consecutive derivatives. Starting with data on`^r(Z),r=d+1, interpreted as function value andd=r−1 consecutive derivatives, we compute successive itera- tions to define values on`^r(2⁻ⁿZ) and anr-vector valued limit function for whose first componentC^d–smoothness is generally expected.

In this paper, we construct Hermite subdivision schemes such that, beginning with the same data, it is possible to reach a limit function with smoothnessd+pfor anyp>0.

The result is obtained with a generalized Taylor factorization and a smoothness condition for vector subdivision schemes.

keywords: Taylor operator, Hermite subdivision, Vector Subdivision, Smoothness.

1 Introduction

Subdivision schemes create curves or surfaces by applying stationary refinement rules on data defined on the integers. This refinement process extends the data to a discrete function defined on the half integers, quarter integers and so on, until eventually the values become so dense that one could speak of alimit function.Stationary subdivision[1] means that any step of the subdivision process is a stationary process which defines data on next level in a convolution like way as

g_n+1=Sagn:=X

β∈Za(· −2β)gn(β);

in this expression,astands for themask, a finitely supported sequence and the valuesgnon different iteration levels are normalized to be discrete functionsgn:Z→Rwith the under- standing thatgn(α) stands for a value at 2⁻ⁿα,α∈Z. Such subdivision schemes with scalar

*Univ Rennes, INSA de Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, 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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coefficients can be trivially extended to the generation of curves by actingcomponentwise, on vector data, resulting in the iteration

g_n+1=Sag_n:= X

β∈Za(· −2β)g_n(β), g_n:Z→R^r.

Vector subdivisiongoes one step further by applying amatrix valuedmask to the data, allow- ing for interaction between the components of the data vectors:

g_n+1=SAgn:=X

β∈Z

A(· −2β)g_n(β), A:Z→R^r×r,

again with the assumption thatA is finitely supported. Finally, inHermite subdivisionthe components of the vectorf_n(α)∈R^r,r=d+1, are considered as function value anddcon- secutive derivatives of a function at 2⁻ⁿα. Due to the chain rule, the refinement scheme now takes alevel dependentform, that is, the operator depends on the iteration levelnas

f_n+1=D⁻ⁿ⁻¹SADⁿf_n=X

β∈Z

D⁻ⁿ⁻¹A(· −2β)Dⁿf_n(β), D=





 1

1 2 . ..

2^−d





 .

All such types of subdivision schemes are covered extensively in the literature, see e.g. [2, 3, 4, 5, 7, 14], just to name a few specific references on Hermite subdivision schemes. Standard questions to consider are the convergence of the iterative schemes and the regularity of the associated limit functions. This is well-known to be closely related to the way how the subdivision operators act on polynomial sequences, a property that can in turn be conveniently characterized by means of operator factorizations.

In the next section, we will review the basic definitions of vector and Hermite subdivision schemes and the appropriate notions of convergence. We will point out what vector subdivision schemes and Hermite subdivision schemes have in common and where they differ.

Introducing Taylor operators, we will also present the transformation of a Hermite subdivision scheme into vector subdivision schemes via the Taylor factorizations. We illustrate the different schemes with an example where, in particular the limit functions are shown.

We will see that the definitions of the smoothness of the two schemes are significantly different. By construction, thelimit function

φ=





 φ0

... φd





,

of a Hermite scheme satisfiesφj=φ^(j)₀ for j=0, . . . ,d, so thatφ0∈C^dwhenever all components ofφ are continuous. The limit of a Hermite subdivision scheme always has to have a certain amount of regularity in the sense of differentiability. In this paper we investigate the question under which circumstances we can have extra regularity, that is,φ0∈C^d⁺^p for

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some integerp≥0. We will relate this to a combined factorization, one due to the nature of Hermite subdivision schemes and one coming from a smoothness condition for vector subdivision schemes that is due to [13]. A similar approach has been used to characterize overreproductionof polynomials as an algebraic properties of the matrix symbols in [15].

Section 3 will be devoted to the B-spline case. Here the splines are obtained as the limit of, firstly, a scalar subdivision scheme, secondly, a Hermite subdivision scheme. The smoothness of such functions are well known and can be as large as wanted.

In the final Section 4, we will give a generic construction to obtain convergent Hermite subdivision schemes with any order of extra smoothness.

2 Vector and Hermite subdivision schemes

We begin by fixing some notation to describe subdivision schemes. Vectors inR^r,r∈N, will generally be labeled by lowercase boldface letters: y =£

yj¤

j=0,...,r−1 or y =£ y^(j)¤

j=0,...,r−1, where the latter notation is used to highlight the aforementioned fact that in Hermite subdivision the components of the vectors correspond to consecutive derivatives. Moreover, in Hermite subdivision we denote the highest derivative byd, so that throughout the paper we will always have the relationshipr=d+1.

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 mostn will be written as Πn, with the usual conventionΠ−1={0}, whileΠwill 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 by`^r(Z). For a sequencey∈`^r(Z), theforward differenceis defined as

∆y:=y(· +1)−y, and iterated to

∆^jy:=∆³

∆^j−1y´

=∆^j−1y(· +1)−∆^j−1y(·)=

j

X

k=0

(−1)^k−j Ãj

k

!

y(· +k), j≥1.

We use0to indicate zero vectors and matrices. If we want to highlight the dimension of the object, we will use a subscript like0r, but to avoid too cluttered notation, we will often drop them if the size of the object is clear from the context.

For a finitely supported sequence of matricesA∈`^r₀^×r(Z), called themaskof the subdivision scheme, we define the associatedstationary subdivision operator

SA:g7→X

β∈ZA(· −2β)g(β), g∈`^r(Z).

Using potentially different masksA_n∈`^r₀^×^r(Z),n∈N, these operators can be iterated into a subdivision schemethat creates sequencesg_n∈`^r₀(Z),n≥0,

g_n₊₁:=SA_ng_n:=X

β∈Z

An¡

· −2β¢

g_n(β), n≥0, (1) from a giveng₀. An important algebraic tool for stationary subdivision operators is thesym- bolof the mask, the matrix valued Laurent polynomial

A^∗(z) := X

α∈ZA(α)z^α, z∈C\ {0}. (2)

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In avector subdivision schemeas defined in [1], we simply set An=A∈`^r₀^×r(Z) and define convergence as follows.

Definition 1 Thevector subdivision operatorSA:`^r(Z)→`^r(Z)is called C^p–convergent, p≥ 0, if for any datag₀:=g∈`^r(Z)and the refinements from(1)there exists a functionψg:R→R^r with C^p components such that for any compact K⊂Rthere exists a sequenceεn with limit0 that satisfies

α∈Z∩2maxⁿK

°

°g_n(α)−ψg¡ 2⁻ⁿα¢°

°∞≤εn. (3)

For aHermite scheme, in (1), we set

An(α)=D⁻ⁿ⁻¹A(α)Dⁿ, α∈Z, D:=





 1

1 2 . ..

1 2^d







, (4)

so thatr=d+1 and fork=0, . . . ,dthe k-th component ofc_n(α) corresponds to an approximation of the k-th derivative of some functionϕnatα2⁻ⁿ. Starting from an initial sequence

f₀∈`^r(Z), a Hermite scheme

f_n+1:=HAnf_n:=D⁻ⁿ⁻¹SADⁿf_n, n≥0, can be rewritten as

g_n₊₁:=Dⁿ⁺¹f_n₊₁=SADⁿf_n=SAg_n, n≥0, (5) based on the relation

g_n=Dⁿf_n, n≥0. (6)

To capture the intuition of vectors with consecutive derivatives, the convergence of Hermite schemes is a little bit more intricate and defined as follows.

Definition 2 The Hermite subdivision scheme with respect to the maskA∈`^r×r(Z)as defined by(5)is calledconvergentif for any dataf₀∈`^r(Z)there exists a functionΦ∈C(R,R^r)such that for any compact K ⊂Rthere exists a sequenceεnwith limit0which satisfies

0≤maxj≤d max

α∈Z∩2ⁿK

¯

¯f_n^(j)(α)−φi¡ 2⁻ⁿα¢¯

¯

¯≤εn. (7)

Moreover, the scheme HAnis said to be C^p–convergentwith p≥d if in additionφ0∈C^p(R,R) and

φ⁽₀^j)=φj, 0≤j≤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.

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Remark 4 The two concepts of convergence based on SAand HAnare significantly different as can be seen immediately from(6). Indeed, if the Hermite subdivision scheme is convergent it follows that

g_n=





 f_n⁽⁰⁾ 2⁻ⁿf_n⁽¹⁾

... 2^−ndf_n^(d)







→





 φ0

0 ... 0





 ,

henceΨg =φ0e₀. In particular, the components ofg_n have to converge to zero even with a prescribed rate. Therefore, in general it cannot be ensured thatD⁻ⁿg_nconverges or is bounded, even ifg_nconverges to a multiple ofe0. This is the reason why the factorization properties and the convergence analysis for Hermite subdivision cannot be obtained in a straightforward way from that of the vector subdivision operator, even if they are based on the same mask, see Fig. 1 for a particular example.

As a consequence of Remark 4 we observe that whenever a mask Adefines a convergent Hermite subdivision scheme, the associated vector subdivision scheme based onSAis a so- calledrank-1subdivision schemeas defined in [12, 13]. In concrete terms, this means that the mask as to satisfy

Q₀e0=Q₁e0=e0, Q_²:= X

α∈Z

A2α+1, ²∈{0, 1},

and that the two matricesQ₀andQ₁have no further common eigenvector with respect to the eigenvalue 1.

To give convergence criteria for vector and Hermite subdivision schemes, we need three different types of difference operators from [9, 10].

Definition 5 Thesimple difference operator(of rank-1type) is defined as

D_d:=∆e_de^T_d=





 1

. ..

1

∆







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Ageneralized incomplete Taylor operatoris an operator of the form

Td:=







∆ −1 ∗ . . . ∗ . .. ... ... ...

. .. ... ∗

∆ −1 1







=

·∆I 1

¸ +£

tj k¤

j,k=0,...,d, (9)

where

t_j,j+1= −1 and tj k=0, k≤j.

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In the same way, thegeneralized complete Taylor operatoris of the form

Te_d:=D_dT_d=







∆ −1 ∗ . . . ∗ . .. ... ... ...

. .. ... ∗

∆ −1

∆







=∆I+£ t_{j k}¤

j,k=0,...,d. (10)

The nameTaylor operatorstems from the fact that, motivated by observations from [6], the (incomplete) operator had originally be introduced in [9] as

Td:=







∆ −1 −¹₂ . . . −_d!¹ . .. ... ... ...

. .. ... −¹₂

∆ −1 1







as a comparison between the difference of function values and the derivative terms of the Taylor expansion. Since for anyφ∈C^d⁺¹(R) one has that

T˜d





 φ φ⁰ ... φ^(d)





 (x)=







φ^(d⁺¹⁾(ξ0) φ^(d+1)(ξ1)

... φ^(d⁺¹⁾(ξd)







, ξj∈(x,x+1), j=0, . . . ,d,

the operator clearly annihilates all polynomials of degree at mostd. Moreover, it enables us to give a sufficient criterion for the convergence of Hermite subdivision schemes by means offactorization.

Definition 6 The masksB,Be∈`^r×r(Z)are called aTaylor factorizationand acomplete Taylor factorizationofA, respectively, if they satisfy

TdSA=2^−dSBTd and TedSA=2^−dS_B_eTed, (11) respectively.

In a certain sense, the factorization always exists. We simply have to write 6 as

T^∗(z)A^∗(z)=2⁻^dB^∗(z)T^∗(z²), T^∗(z) :=







z⁻¹−1 −1 ∗ . . . ∗ . .. ... . .. ...

. .. . .. ∗ z⁻¹−1 −1

1







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with an analogous identity for the complete factorization, to obtain that

B^∗(z)=2^dT^∗(z)A^∗(z)T^∗(z²)⁻¹ and A^∗(z)=2⁻^dT^∗(z)⁻¹B^∗(z)T^∗(z²),

respectively. Given a finitely supported maskA, the resultingB^∗(z) is usually a nonpolyno- mial rational function, hence the factorB is an infinitely supported mask. Unfortunately, the same also holds true forA^∗which, for givenB can only be guaranteed to be a rational function, even if

detA^∗(z) = 2⁻^d¡

detT^∗(z)¢₋₁

detB^∗(z) detT^∗(z²)=2⁻^d(z⁻²−1)^d+1

(z⁻¹−1)^d⁺¹ detB^∗(z)

= 2

µz⁻¹+1 2

¶^d+1

detB^∗(z)

is a Laurent polynomial inz. This is in contrast to scalar subdivision schemes where raising the order of the zero at−1 is the standard way to increase the smoothness of the limit function. Nevertheless, the existence of a factorization is the key to the construction of convergent Hermite subdivision schemes.

Theorem 7 ([10], Corollary 4) If a given maskAhas a complete Taylor factorizationTe_dS_A= 2^−dS_B_eTedwhere

1. Be∈`^r^×^r(Z)isfinitely supported, 2. S_B_e is a contraction on`^r_∞(Z), 3. (B^∗(1))11=1,

then HA_nis C^d-convergent.

Hence, in order to construct aC^d-convergent subdivision scheme, we can start with a finitely supportedBesuch that the associated subdivision satisfies the contractivity condition 2) and the normalization condition 3) at the same time. Note that the latter prohibits a simple rescal- ing ofB, i.e., a multiplication with a small constant.

This, however, is not enough as one also has to ensure thatT^∗(z)⁻¹B^∗(z)T^∗(z²) is a matrix valued Laurent polynomial which leads to additional conditions onB^∗. A generic construction for such aB has been given, foranygeneralized Taylor operator, in [10] which shows that for any generalized Taylor operator and anydthere exists aC^dconvergent Hermite subdivision scheme that if factorizable with respect to this generalized Taylor operator. We will later extend this construction by means of asupercompleteTaylor factorization, but first we illustrate the concept by looking at a special case that actually motivated the development of generalized Taylor factorizations.

In Figure 1, with a given mask, {A(·)}, we plot the ”limit” functions for the two vector schemes,SA,SB (after Taylor incomplete factorization), and the Hermite schemeHAn. We notice that the first functions forSAandHAnare identical, corresponding toφ0in Definition 2 and similarly the last ones ofSBandHA_n corresponding toφ^(d)₀ in the same definition.

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Figure 1: The different schemes:SA,SBandHA_n

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3 The B–spline case

In this section, we rewrite the well known cardinal splines, [16] in term of a scalar subdivision scheme and extend it into Hermite schemes of different orders. From the properties of cardinal splines, we have convergence of the schemes and regularity of the limit.

Our presentation, already proposed in [8], is based on a construction detailed by Michelli in [11] and in summarized in the following.

Let

ϕ0(x)=χ[0,1]=

½ 1 ifx∈[0, 1], 0 ifx∉[0, 1].

Form=1, 2, . . ., we build ϕm by means of autoconvolution asϕm =ϕ0∗ϕm−1orϕm(x)= Rx

x−1ϕm−1(t)d t.

Let us recall that ϕm is aC^m⁻¹ piecewise polynomial of degreem with finite support [0,m+1].

Moreover,ϕm(x)=₂¹^mP

α∈Z¡_m+1

α

¢ϕm(2x−α) where¡_i

j

¢= ( _i!

j!(i−j)! if 0≤j≤i,

0 otherwise.

Consideringv(x)=P

α∈Zf₀⁽⁰⁾(α)ϕm(x−α), which is a finite sum for anyx∈Rsinceϕm

has finite support, we deduce forn∈N0thatv(x)=P

α∈Zf_n⁽⁰⁾(α)ϕm(2ⁿx−α) where f_n+1⁽⁰⁾ (α)= 1

2^m X

β∈Z

µm+1 α−2β

¶

f_n⁽⁰⁾(β)=:X

β∈Z

am(α−2β)f_n⁽⁰⁾(β), α∈Z, (12) that is,

am(α)= 1 2^m

Ãm+1 α

!

, α∈Z. (13)

This is a scalar subdivision scheme.

Then, the well–known derivative formula for cardinal B–spline yields dⁱv

d xⁱ(x)=X

α∈Z

2ⁿⁱ∆ⁱf_n⁽⁰⁾(α−i)ϕm−i¡

2ⁿx−α¢

, i=0, . . . ,m−1. (14) We have a particular case wheni =m−1. Since the function ϕ1 is piecewise linear with ϕ1(α)=δ1α, we obtain

d^m⁻¹v

d x^m−1(β/2ⁿ)=2^n(m−1)∆^m−1f_n⁽⁰⁾(β−m+1).

With this formula, we define a Hermite subdivision scheme of orderd<mwith mask {A(α)}

and support [0,m+d+1] by applying differences to the maskam, yielding

A(α)=







am(α) 0 . . . 0

∆am(α−1) 0 . . . 0 ...

∆^da_m(α−d) 0 . . . 0







, (15)

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thus

A^∗(z)=(1+z)^m⁺¹ 2m







1 0 . . . 0 (1−z) 0 . . . 0

...

(1−z)^d 0 . . . 0







. (16)

Beginning withf₀∈`^r, defined by (5) we notice that forn≥1 andi=1, . . . ,d:

f_n⁽ⁱ⁾(α)=2^{i n}∆ⁱf_n⁽⁰⁾(α−i). (17) Now with (12) and (14), forn>0,

dⁱv

d xⁱ(x)= X

α∈Z

f_n⁽ⁱ⁾(α)ϕm−i(2ⁿx−α), i=0, . . . ,d.

In [8], we had proved that the generalized Taylor operators are given by

T_d:=







∆ −1 . . . −1 . .. ... ...

∆ −1 1







andTe_d:=







∆ −1 . . . −1 . .. ... ...

∆ −1

∆







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Thus the corresponding vector scheme in the factorization is given by

Be^∗(z)=z(1+z)^m⁻^d 2^m−d





 1 1 ... 1







£(1−z²)^d z²((1−z²)^d⁻¹ . . . z²(1−z²) z²¤

. (19)

It is of rank 1. Let us also notice that

Te^∗(z)A^∗(z)=2⁻^dBe^∗(z)T^∗(z²)=2⁻^mz⁻¹(1+z)^m⁺¹(1−z)^d⁺¹







1 0 . . . 0 1 0 . . . 0 ... ... ... 1 0 . . . 0





 .

We did not plot the graphs of the B-splines which are well known and probably everyone has seen one already.

4 Hermite schemes with extra regularity: a generic construction

In this section we will show that for any generalized Taylor operator of any orderdthere exists aC^d-convergent subdivision scheme with an a extra regularity ofpfor any givenp≥0.

Theorem 8 Given p≥0and a Taylor operator Td of order d , there exists a finitely mask A∈

`^r×rsuch that the subdivision scheme is C^d-convergent with a limit functionφ∈C^d+p(R)and SAadmits a Taylor factorization with respect to T_d.

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The idea behind the supercomplete construction is simple and to some extent even follows the same concept as usual scalar subdivision: starting with the Taylor factorBsuch that T_dS_A=2⁻^dS_BT_d, we create an additional the order of smoothness of the limit function ofS_B by constructing a scheme whose symbol has extra (matrix) factors. This means thatSBfrom the (incomplete) Taylor factorization should be further factorizable into

· Id

∆

¸^p

SB=2⁻^pS_B_e

· Id

∆

¸^p

, (20)

whereBe is also a finitely supported mask. From [12, 13] we recall the following result on smoothing limit functions.

Theorem 9 The vector subdivision scheme SBhas C^plimit function if

· I_d

∆

¸p+1

SB= 1 2^pS_B_e

· I_d

∆

¸p+1

(21) and S_B_e is contractive. The converse does not hold.

For the construction of an appropriateBe, we partition it as

Be^∗(z)=

·

Be^∗₁₁(z) Be^∗₁₂(z) Be^∗₂₁(z) Be^∗₂₂(z)

¸

, Be11∈`^d×d(R),Be12,Be^T₂₁∈`^1×d(R),Be11∈`^1×1(R) Since (21) can be rewritten as

B^∗(z) = 1 2^p

·I_d

z⁻¹−1

¸−p−1·

Be^∗₁₁(z) Be^∗₁₂(z) Be^∗₂₁(z) Be^∗₂₂(z)

¸ ·I_d

z⁻²−1

¸p+1

= 1 2^p

·

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

¸

, (22)

we can record the following immediate consequence of Theorem 9.

Corollary 10 SBconverges to a C^plimit function of the form f_c=fcedif 1. SBe is contractive,

2. Be^∗₂₁has a zero of order p+1at1, 3. Beis normalized asBe^∗₂₂(1)=1.

The corollary tells us that contractive schemes are at the heart of the construction of a convergent Hermite subdivision scheme. Note that contractivity of a schemeC means that the spectral radius

ρ(SC) :=lim sup

n→∞ kS_Cⁿk^1/n, kSCk:= sup

kck∞=1kSCck∞,

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based on the operator norm of thesubdivision operatoris less than one, where kck∞=sup

α∈Z max

0≤k<r|c_k(α)|.

The following simple sufficient condition for contractivity of a vector subdivision scheme is most likely known in the folklore, but state it and give a quick proof for the sake of complete- ness and the reader’s convenience.

Lemma 11 IfC is a lower triangular mask, i.e., all components ofC(α)are lower triangular matrices and the diagonal elements c00, . . . ,cr−1,r−1∈`(Z)are scalar contractive schemes, then C defines a contractive vector subdivision scheme.

Proof: WriteC=D+N whereD∈`^r^×^r(Z) is a diagonal scheme andN∈`^r^×^r(Z) is strictly lower diagonal one, thenS_Cⁿ=Sⁿ_D+SNnfor some strictly lower diagonalNn∈`^r×r(Z),n∈N. Since the diagonal elements are contractions, there exist somen∈Nsuch that°

°Sⁿ_D°

°<1. With

E_ε:=





 1

ε . ..

ε^r⁻¹







we have that

E_εS_CⁿE⁻_ε¹=Sⁿ_D+S_E_ε_Nⁿ_E−1

ε , E_εNⁿE⁻_ε¹=





 0 ε∗ 0

... . .. ...

ε^r−1∗ . . . ε∗ 0







and hence there existsε>0 such thatρ:=°

°E_εSⁿ_CE⁻¹_ε °

°<1. Hence, for anym∈N, kS^mn_C k ≤ kE_εkkE⁻_ε¹k°

°E_εS_C^mnE_ε⁻¹°

°=ε¹⁻^r°

°

¡E_εS_CⁿE⁻_ε¹¢m°

°

°≤ ρ^r e^r⁻¹

which becomes<1 formsufficiently large. Sinceρ(SC)≤ kS^mn_C k^1/(mn)for any choice ofm,n∈

N, this completes the proof thatSC is a contraction.

Next, note that the second condition onBein Corollary 10 ensures thatB^∗in (22) is a Laurent polynomial while the normalization yields

B^∗(1)=

·∗ 0 0 2

¸

, B^∗(−1)=

·∗ 0

∗ 0

¸ , hence

Ã X

α∈Z

B(2α)

! e_d=

Ã X

α∈Z

B(2α+1)

!

e_d=e_d,

which is the necessary condition for the limit function to be of the formfced. Combining the two factorizations into one, we arrive at the following definition.

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Definition 12 The (generalized)supercomplete Taylor operatorof order d and extra regular- ity p is of the form

T_d,p:=







∆ −1 ∗ . . . ∗ . .. ... ... ...

. .. ... ∗

∆ −1

∆^p⁺¹







=

· Id

∆

¸^p Te_d=

· Id

∆

¸^p+1

T_d. (23)

The special cases are T_d=T_d,−1andTe_d=T_d,0.

A factorization with respect to a supercomplete operator,T_d,pSA=2^−p−dS_B_bT_d,pis equivalent to

A^∗(z) = 1 2^p+d

³T^∗_d,p(z)´₋1

Bb^∗(z)Tb^∗_d,p(z²)

= 1

2^p⁺^d

¡

Te^∗(z)¢−1· Id

(z⁻¹−1)^−p

¸ Bb^∗(z)

·Id

(z⁻²−1)^p

¸ Te^∗(z²)

= 1

2^p+d

¡

Te^∗(z)¢−1·I_d

z⁻¹−1

¸−p−1

Bb^∗(z)

·I_d

z⁻²−1

¸p+1

Te^∗(z²)

=: 1 2^d

¡

Te^∗(z)¢−1

C^∗(z)Te^∗(z²).

Thus, if we can find maskCassociated to a contractive scheme and normalized as (C^∗(1))_{d d}= 1, such thatH_Ais aC^d-convergent subdivision scheme, then we can computeBb^∗(z) as

Bb^∗(z) = 2^p

·Id

z⁻¹−1

¸^p+1

C^∗(z)

·Id

z⁻¹−1

¸−p−1

= 2^p

"

Ce^∗₁₁(z) ¡ ₁

z⁻²−1

¢p+1

Ce^∗₁₂(z) (z⁻¹−1)^p⁺¹Ce^∗₂₁(z) ¡ ₁

z⁻¹+1

¢p+1

Ce^∗₂₂(z)

# .

The construction ofC has been pointed out in [10]. More precisely, given any symbolh^∗_{d d} of a maskhsuch that the univariate scalar stationary subdivision schemeS_h is contractive, then there exists a recursive scheme [10, eq. (65) in the proof of Theorem 5] to compute h^∗_d,d−1, . . . ,hd,0such that for anyh^∗_{j k},k=0, . . . ,j−1,j=1, . . . ,d−1, the upper triangular symbol

C^∗(z)=







z⁻¹−1 2

(z⁻¹−1)²h₁₀^∗(z) ^(z⁻¹₄⁻¹⁾²

... . .. . ..

(z⁻¹−1)^dh^∗_d

−1,0(z) . . . (z⁻¹−1)^dh^∗_d

−1,d−2(z) ^(z⁻¹₂⁻d¹⁾^d

c_d0^∗ (z) . . . c_d,d^∗

−2(z) c_d,d^∗

−1(z) c^∗_{d d}(z)







with

c_{d j}^∗ (z)=¡

z⁻¹−1¢d−j

h^∗_{d j}(z⁻¹), j=0, . . . ,d,

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defines a Taylor factor with a contractive associated subdivision scheme by Lemma 11. Note, in particular, thatC^∗₁₂=0 and thatc_{d d}^∗ =h_{d d}^∗ . If, in addition, we choose

h^∗_{d d}(z)=(z+1)^p⁺¹

2^p+1 a(z), a(1)=2,

in a B-spline fashion, thenBb^∗(z) is a matrix Laurent polynomial andS_A, defined by A^∗(z) = 1

2^d

¡

Te^∗(z)¢−1

C^∗(z)Te^∗(z²)= 1 2^p⁺^d

³T^∗_d,p(z)´−1

Bb^∗(z)Tb^∗_d,p(z²)

defines aC^dconvergent Hermite subdivision scheme by Theorem 7 and has ap-supercomplete Taylor factorization with factorBb. The symbolBb^∗is a lower triangular matrix with the same diagonal structure asC^∗and thus also defines a contraction. Hence, by Theorem 9 and Corol- lary 10, the last component of the limit£

φ,φ⁰, . . . ,φ^(d)¤

ofH_Abelongs toC^p(R) which eventually verifies thatφ∈C^d+p(R).

This also concludes the proof of Theorem 9.

We finish with revisiting one example from [10] where already a mask with a supercomplete Taylor factorization was constructed.

Example 13 In the case n=5with the functions from [10, Example 5], we can get a factoriza- tion with p=4and obtain

Bb^∗(z) =







−^{8 (z}_z⁻¹⁾

16 (z−1)² z²

32 (z−1)⁶(5−4a−3z+4a z) z⁷

0

4 (z−1)² z²

−^{8 (z}⁻¹⁾

5(¹⁵−20a+16a²−18z+32a z−32a²z+7z²−12a z²+16a²z²)

z⁷

0 0

1+z 2z





as well as A^∗(z)

=







−⁽¹⁺^z)(−20a+16a²−23z+48a z−32a²z+15z²−28a z²+16a²z²)

2z³

−^(z⁻^{1) (1}⁺^{z) (11}_z⁻3^8a⁻⁷^z⁺⁸^{a z)}

2 (z−1)²(1+z) (5−4a−3z+4a z) z⁴

30a−40a²+32a³+31z−46a z+56a²z−32a³z+24z²−78a z²+72a²z²−32a³z²−45z³+94a z³−88a²z³+32a³z³ 4z³

(z−1)(−31+40a−32a²−24z+16a z+43z²−56a z²+32a²z²)

4z³

−^(z−1)²(−15+20a−16a²−12z+8a z+19z²−28a z²+16a²z²)

2z⁴

a+z+7a z+8z²+22a z²+30z³+42a z³+72z⁴−183a z⁴−119z⁵+63a z⁵ 32z⁵

(z−1)(−1−8z−30z²−72z³+119z⁴+32a z⁴)

32z⁵

−^(z⁻¹⁾

3(¹+9z+39z²+111z³)

32z⁶





.

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Figure 2: The function and its first and second derivatives

In this expression, a is the free parameter of the associated generalized Taylor operator with complete form

Te_d=





∆ −1 a

∆ −1

∆



.

In Fig. 2, we have plotted the ”limit” function and its first and second derivatives. Since the process does not compute the next derivatives, the approximations for higher derivatives in Fig. 3 have been determined using the successive finite differences off⁽²⁾.

5 Conclusions

Convergent Hermite subdivision schemes have a limit function that belongs toC^d, at least if the scheme converges in the sense proper for Hermite subdivision. We have shown that this regularity can be raised to an arbitrary order and provided an explicit recipe to deter- mine such schemes. The approach uses factorizations and contractivity, but in contrast to the scalar univariate case the factor maskBe must satisfy additional, nontrivial conditions, and the main task in the construction is to satisfy these conditions.

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Figure 3: Approximation of the successive derivatives by finite differences

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