Convergence of level-dependent Hermite subdivision schemes

Applied Numerical Mathematics 116 (2017) 119–128

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Applied Numerical Mathematics www.elsevier.com/locate/apnum

Convergence of level-dependent Hermite subdivision schemes Costanza Conti a , Mariantonia Cotronei b,∗ , Tomas Sauer c a b c

DIEF, Università di Firenze, Viale Morgagni 40-44, 50134 Firenze, Italy DIIES, Università Mediterranea di Reggio Calabria, Via Graziella, 89122 Reggio Calabria, Italy Lehrstuhl für Mathematik mit Schwerpunkt Digitale Bildverarbeitung and FORWISS, Universität Passau, Innstr. 43, 94032 Passau, Germany

a r t i c l e

i n f o

Article history: Available online 28 February 2017 Keywords: Subdivision schemes Hermite schemes Factorization Annihilators

a b s t r a c t Subdivision schemes are known to be useful tools for approximation and interpolation of discrete data. In this paper, we study conditions for the convergence of leveldependent Hermite subdivision schemes, which act on vector valued data interpreting their components as function values and associated consecutive derivatives. In particular, we are interested in schemes preserving spaces of polynomials and exponentials. Such preservation property assures the existence of a cancellation operator in terms of which it is possible to obtain a factorization of the subdivision operators at each level. With the help of this factorization, we provide sufficient conditions for the convergence of the scheme based on some contractivity assumptions on the associated difference scheme. © 2017 IMACS. Published by Elsevier B.V. All rights reserved.

1. Introduction Subdivision schemes have become a useful tool for the approximation of discrete data in many areas such as computeraided geometric design, curve and surface reconstruction, signal/image processing. Essentially, they are iterative computational processes based on refinement rules, which can be can either level-independent or level-dependent, producing denser and denser sets of discrete data from an initial one. When dealing with subdivision schemes, it is natural to study their convergence and investigate the smoothness of the associated limit functions. This is especially true in case of Hermite subdivision schemes which act on vector data but with the special assumption that these vectors represent function values and consecutive derivatives up to a certain order. Convergence of Hermite subdivision schemes preserving polynomials has been addressed by Merrien and Sauer in [13] by means of the so-called Taylor factorization providing the difference scheme whose contractivity property is sufficient for proving convergence of the original scheme, see also [9,10,12]. In this paper, we further pursue the factorization approach for the convergence analysis of level-dependent Hermite subdivision schemes designed to preserve polynomials and exponentials rather than polynomials only. Indeed, for these wider class of Hermite subdivision schemes, in the recent paper [7] we have showed that preservation of elements in the space spanned by {1, x, . . . , x p , e ±λ1 x , . . . , e ±λr x } (which we denote by V p , for  := {λ1 , . . . , λr }) by level-dependent Hermite subdivision schemes allows us the derivation of the so called annihilator operator, a natural extension of the Taylor operator. In [7] we have also showed that the existence of the annihilator operator is strongly connected with the -factorization of the Hermite subdivision operator and with the so called -difference subdivision scheme. Using the latter results, in this paper we derive sufficient conditions of convergence of level-dependent Hermite subdivision schemes based on the so

*

Corresponding author. E-mail addresses: costanza.conti@unifi.it (C. Conti), [email protected] (M. Cotronei), [email protected] (T. Sauer).

http://dx.doi.org/10.1016/j.apnum.2017.02.011 0168-9274/© 2017 IMACS. Published by Elsevier B.V. All rights reserved.

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C. Conti et al. / Applied Numerical Mathematics 116 (2017) 119–128

called weak-contractivity property of the -difference subdivision operator. To our knowledge these sufficient conditions are the first conditions for showing convergence of level-dependent Hermite subdivision schemes. Our result is connected to the possibility of constructing polynomial and exponential reproducing Hermite schemes through an Hermite interpolation process in the space V p , . The paper is organized as follows. We start by fixing the notation in Section 2 and concentrate on the reproduction and factorization properties of level-dependent Hermite subdivision schemes in Section 3. Section 4 investigates the possibility of constructing polynomial and exponential reproducing Hermite schemes through an Hermite interpolation process in the space V p , . The core of the paper is Section 5 where we formulate our main convergence result which is based on the contractivity property of the -difference operators involved in the factorization of the subdivision operators. 2. Notation and background material We begin by fixing the notation and recalling some known facts about subdivision schemes. We denote by  (Z)m and  (Z)m×m the linear spaces of all sequences of m-vectors and m × m matrices, whose elements will be, respectively, denoted by boldface lower case and upper case letters. In particular, c ∈  (Z)m is c = (c (α ) : α ∈ Z), while A ∈  (Z)m×m stands for A = ( A (α ) : α ∈ Z), indexing a matrix A ∈ Rm×m as A = a jk : j , k = 0, . . . , m − 1 and a vector v ∈ Rm as   v = v j : j = 0, . . . , m − 1 . Operators acting on those spaces are denoted by capital calligraphic letter. The linear spaces of all bounded vector-valued or matrix-valued sequences, equipped with the infinity norm, are denoted by ∞ (Z)m and ∞ (Z)m×m , while 00 (Z)m and 00 (Z)m×m denote the subspaces of finitely supported sequences. The space of all uniformly continuous and uniformly bounded functions on R is written as C u (R) and equipped with the norm  f ∞ := supx∈R | f (x)| for f ∈ C u (R). In addition, C ud (R) stands for all function in C u (R) that have a d-th derivative in C u (R), d ∈ N. For A ∈ 00 (Z)m×m and c ∈ 00 (Z)m we define the associated symbols as the Laurent polynomials

A ∗ ( z) :=



c ∗ ( z) :=

A (α ) zα ,

α ∈Z



c (α ) zα ,

z ∈ C \ {0}.

α ∈Z

The subdivision operator S A : ∞ (Z)m → ∞ (Z)m based on the matrix sequence or mask A ∈ 00 (Z)m×m is defined as

S A c (α ) :=



A (α − 2β) c (β),

α ∈ Z,

c ∈  (Z)m .

(1)

β∈Z

Alternatively, using symbol notation, we can also describe the action of the subdivision operator in the form

  (S A c )∗ ( z) = A ∗ ( z) c ∗ z2 ,

z ∈ C \ {0},

(2)

though, in strict formalism, (2) is only valid for c ∈ 00 (Z)m . A subdivision scheme consists of the successive application of (potentially different) subdivision operators S A [n] , con    structed from a sequence of masks A [n] : n ≥ 0 , where A [n] = A [n] (α ) : α ∈ Z ∈ 00 (Z)m×m is called the level n subdivision   mask and is assumed to be of finite support. Accordingly, a sequence of matrix symbols ( A [n] )∗ ( z) : n ∈ N0 characterizes such schemes. For some initial sequence c 0 ∈ ∞ (Z)m , the subdivision scheme S ( A [n] : n ≥ 0) iteratively produces a sequence of sequences

c n+1 := S A [n] c n ,

n ≥ 0,

(3)

whose elements can be interpreted as function values at 2−n−1 Z. Special type of vector subdivision schemes are Hermite subdivision schemes which are considered in this paper and therefore presented in more detail in the next section. The infinity norm of the operator is given by

S



A [n] ∞

:= max α ∈Z

 [n] A (α − 2β) ,

(4)

∞

β∈Z

while the norm for the vector-valued function g = [g 0 , g 1 , . . . , gd ] T ∈ C u (R)d+1 is set to  g ∞ := max  g j ∞ = j =0,...,d

max sup | g j (x)|. If g j =

j =0,...,d x∈R

dj f dx j

, j = 0, . . . , d for some function f ∈

dj f  f d,∞ := max j j =0,...,d dx

.

C ud (R),

this becomes the Sobolev norm

(5)

∞

Before concluding this preliminary section we give some definitions of contractivity for a level-dependent subdivision scheme, which generalize the notion given in the level-independent setting (often referred to as the stationary setting) and which are crucial for proving our main results.

C. Conti et al. / Applied Numerical Mathematics 116 (2017) 119–128

Definition 1. The scheme S ( A [n] : n ≥ 0) is said to be contractive if there exist

S A [n+N ] · · · S A [n]  ≤ ρ ,

121

ρ < 1 and N > 0 such that:

n ∈ N,

(6)

while it is said to be weakly contractive if ∞  S

S A [n−1] · · · S A [0] < ∞.

A [n]

(7)

n =0

Weak contractivity is a generalization of contractivity: in fact, if (6) holds true, then it is easily seen that ∞  S

∞  N +1 S A [n−1] · · · S A [0] ≤ C ( N + 1) ρn = C 1−ρ

A [n]

n =0

n =0

for some constant C . This particularly includes the case of level-independent schemes and the classical contractivity of the respective subdivision operators. 3. Hermite subdivision schemes and reproduction Hermite subdivision schemes act on vector valued data c ∈ (Z)d+1 , whose k-th component corresponds to a k-th derivative (see, for example, [4,6,11,13,15]). The exponential and polynomial preservation capabilities of subdivision schemes is interesting by itself and for convergence investigation, see [2,3]. A preliminary simple observation for considering Hermite dr −nr dr f (2−n ·), r = 0, . . . , d. This can be subdivision schemes is that for f ∈ C d (R) and for g := f (2−n ·) we have dx r g = 2 dxr written in vector form as



dj dx j





f (2 ·) : j = 0, . . . , d = D n f ( j ) (2−n ·) : j = 0, . . . , d , n



where D := diag 1, 12 , 14 , . . . ,

1 2d



(8)

. Since the sequence c n is related to evaluations on the grid 2−n Z, we consider Hermite

subdivision schemes with the n-th iteration of the following type:

D n +1 c n +1 =



A [n] (· − 2β) D n cn (β),

(9)

β∈Z

or, more compactly, D n+1 c n+1 = S A [n] D n c n , where in “usual” Hermite subdivision the mask is the same over all levels, i.e.,

[n] A [n] = A, n ∈ N0 . Setting  A := D −n−1 A [n] D n , formula (9) fits into the notation of Section 2 (with  A playing the role of A in that section) with the n-th subdivision operator of the form

f n+1 = S[n] f n =



A

[n]  A (· − 2β) f n (β).

(10)

β∈Z

A Hermite scheme is said to be interpolatory if ( A [n] )∗ ( z) + ( A [n] )∗ (− z) = 2D. With the same notation as in [7], based on a set

 := {±λ j : j = 1, . . . , r } ⊂ C \ {0},

λ j = λk ,

j = k,

we denote for d := p + 2# = p + 2r by V p , the space of polynomial and exponential functions:





V p , := span 1, x, . . . , x p , e ±λ1 x , . . . , e ±λr x ,

λ j ∈ C,

j = 1, . . . , r .

(11)

One of properties usually required in (scalar and vector) subdivision schemes is the preservations of certain polynomials. Indeed, preservation of polynomials is connected with both the regularity of the limit function and the approximation order of subdivision schemes (see [1], for example). Also, the reproduction of exponential-polynomials is linked to the approximation order of subdivision schemes and to their regularity [8] as well. In this paper we are interested in Hermite schemes preserving both polynomial and exponential data. Following [7], we say that the subdivision operator S A [n] , n ≥ 0, satisfies the V p , -spectral condition, if

S A [n] v n ( f ) = v n+1 ( f ),

f ∈ V p , , n ≥ 0,

(12)

where, for f ∈ C d (R), we denote by v n ( f ) the vector sequence with elements

v n ( f )(α ) := [ f (2−n α ), 2−n f  (2−n α ), . . . , 2−nd f (d) (2−n α )] T ,

α ∈ Z.

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C. Conti et al. / Applied Numerical Mathematics 116 (2017) 119–128

It has been shown in [13] that the polynomial preservation property is related to the factorization of the subdivision operator in terms of the so-called Taylor operator connected with the number of preserved polynomials. We recall that the symbol T ∗p ( z) ∈ R( p +1)×( p +1) of the complete Taylor operator for subdivision schemes preserving polynomials up to degree p takes the form

⎡ ⎢ ⎢ ⎢ ⎢ ∗ T p ( z) := ⎢ ⎢ ⎢ ⎣

( z −1 − 1)

−1

... . ( z −1 − 1) . . .. .

0

.. . 0

... ...

0

⎤ − p1! ⎥ .. ⎥ . ⎥ ⎥ .. ⎥. . −1 ⎥ ⎥ ⎦ ( z −1 − 1) −1 0 ( z −1 − 1) − ( p −11)! .. .

(13)

Recently in [7], we gave a similar result for Hermite schemes preserving both polynomial and exponential data. In particular, the so-called level-n cancellation operator for the function space spanned by V p , has been introduced as the convolution [n]

operator H p , : (Z)d+1 → (Z)d+1 such that [n]

H p , v n ( f ) =



[n]

H p , (· − α ) v n ( f )(α ) = 0,

f ∈ V p , .

(14)

α ∈Z [n]

In [7, Lemma 3] it has been shown that it can be obtained as H p , = H p ,2−n  , where H p , is the unique operator whose symbol H ∗p , ( z) ∈ R(d+1)×(d+1) has the following structure:

H ∗p , ( z) =



T ∗p ( z) 0

 Q ∗ ( z) R ∗ ( z)

(15)

.

Here, T ∗p ( z) is the symbol of the Taylor operator of order p and satisfies:



H ∗p , e ∓λ



1, ±λ, . . . , (±λ)d

T

= 0.

(16)

The remaining blocks Q ∗ ( z) ∈ R( p +1)×(2r ) , R ∗ ( z) ∈ R(2r )×(2r ) in (15) can be explicitly computed and it is easy to see from these identities that H ∗p , ( z) = z−1 I + H p ,2−n  (0), where H p ,2−n  (0) is a nonsingular block upper triangular matrix, see [7, Section 3] for details. In [7] it was proved that the operators H p ,2−n  reduce to Taylor operators as the frequencies tend to zero. As a consequence, the asymptotical behavior, as n tends to infinity, of such operators can be easily found. In fact one has:

lim H p ,2−n  = Td ,

n→∞

for d = p + 2#.

(17)

One of the main results proved in [7] was related to the possibility of factorizing level-dependent subdivision operators preserving polynomials and exponentials in terms of the level-n cancellation operator. Theorem 2. If the subdivision operator S A[n] satisfies the V p , -spectral condition or annihilates V p , , then there exists a finitely supported mask B [n] ∈ 00 (Z)(d+1)×(d+1) such that

H p ,2−(n+1)  S A [n] = S B [n] H p ,2−n 

or

S A [n] = S B [n] H p ,2−n  ,

(18)

respectively. This result relates the factorization to operators which preserve or annihilate the space V p , . We need a more quantitative version of it.





Theorem 3. Suppose that any subdivision operator from the scheme S C [n] : n ∈ N0 annihilates V p , , that the supports of the masks C [n] , n ∈ N0 , are contained in some finite subset of Z and that there exists a mask C such that lim C [n] − C ∞ = 0. Then the scheme n→∞

  S B [n] : n ∈ N0 , defined by

SC [n] = S B [n] Hd,2−n  ,

n ∈ N0 ,

(19)

also has all supports contained in a finite subset of Z and there exists a constant K > 0 such that S B [n] ∞ ≤ K for all n ∈ N. Proof. To prove the support claim, we recall that (19) can be expressed in terms of symbols as



C [n]

∗

 ∗ ( z) = B [n] ( z) H ∗p ,2−n  ( z2 ).

(20)

C. Conti et al. / Applied Numerical Mathematics 116 (2017) 119–128

123

By assumption, the common support of all C [n] , n ∈ N0 , is contained in [1 , 2 ] ∩ Z for some 1 ≤ 2 . Taking into account that H ∗p ,2−n  ( z) = z−1 I + H p ,2−n  (0) and expressing (20) as 0 

C [n] (α ) =

B [n] (α − 2β) H p ,2−n  (β),

α = 1 , . . . , 2 ,

β=−1

yields the recursive property

B [n] (α ) = C [n] (α ) − B [n] (α − 2) H p ,2−n  (0),

α = 1 , . . . , 2 .

Since B [n] is finitely supported by Theorem 2, it follows from





B [n] (α − 2) = H p ,2−n  (0)−1 B [n] (α ) − C [n] (α ) that B [n] (α ) = 0 for α > 2 − 2 and forward induction yields

B [n] (α ) =

α −1

α < 1 , independently of n. With B [n] (1 + 2 +  ) = C [n] (1 +  ),  ∈ {0, 1}, a straight-

(−1) j C [n] (α − 2 j )( H p ,2−n  (0)) j ,

 1 ≤ α ≤ 2 ,

(21)

j =0

hence

S



B [n] ∞

  ≤ (2 − 1 ) max  H p ,2−n  (0)∞ ,  H p ,2−n  (0)∞2 −1 S C [n] ∞ ,

and the right hand side is bounded by some constant K since the C [n] are convergent.

2

The goal of this paper is to show how this factorization can be used to check convergence for polynomial and exponential reproducing Hermite schemes. First, we need to introduce the formal notion of convergence of Hermite subdivision schemes. Definition 4. The subdivision scheme S ( A [n] : n ≥ 0) is said to be convergent if for any vector sequence f 0 ∈ ∞ (Z)d and the corresponding sequence of refinements f n+1 = S A [n] f n , there exists a uniformly continuous vector field φ : R → Rd+1 , such that







lim sup φ 2−n α − f n (α ) ∞ = 0.

n→∞ α ∈Z

The scheme is said to be C d -convergent if in addition φ0 ∈ C ud (R) and

d j φ0 dx j

= φ j for j = 1, . . . , d.

A weaker form of convergence can be introduced, which considers, instead of the full vector sequence f n only its first k + 1 entries [ I k 0] f n , 0 ≤ k ≤ n; here, I k stands for the k × k identity matrix. Definition 5. The subdivision scheme S ( A [n] : n ≥ 0) is said to provide restricted C k -convergence if for any vector sequence f 0 ∈ ∞ (Z)d and the corresponding sequence of refinements f n , there exists a uniformly continuous vector field φ : R → Rk+1 , such that

 





lim sup φ 2−n α − [ I k 0] f n (α )∞ = 0,

n→∞ α ∈Z

with the property that φ0 ∈ C k (R) and

d j φ0 dx j

= φ j , j = 1, . . . , k.

Remark 6. Observe that any C k convergent scheme can be easily extended onto a restrictedly convergent scheme for d ≥ k by simply completing the matrix coefficients of the masks in a block diagonal way. In general, such extensions will not provide any higher order of restricted convergence. Indeed, the question of extending a (polynomial) Hermite scheme in such a way that the order of regularity increases has been considered recently in [14]. 4. Hermite schemes from piecewise Hermite interpolation in V p , Our convergence result is based on the possibility of constructing polynomial and exponential reproducing Hermite schemes through an Hermite interpolation process in the space V p , . This fact is shown in this section. Let us consider the vector sequence f n attached to the points of 2−n Z. From this sequence, we construct the piecewise Hermite interpolant in an extension of V p , on 2−n Z with the help of the following Lemma.

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C. Conti et al. / Applied Numerical Mathematics 116 (2017) 119–128

Lemma 7. Let d = p + 2# = p + 2r. Let y 0 , y 1 ∈ Rd+1 be two vectors of data. The interpolation problem

f ( j ) ( ) = y j ,

j = 0, . . . , d ,

 ∈ {0, 1},

(22)

of Taylor conditions at 0 and 1 has a unique solution f in





 V := span 1, . . . , x p , e ±λ1 x , . . . , e ±λr x , xd+1 , . . . , x2d+1 ,

 V ⊃ V p , .

Moreover, for  → (0, . . . , 0), this solution converges in Sobolev norm to the polynomial Hermite interpolant with (d + 1)-fold nodes at 0 and 1. Proof. The collocation matrix of Taylor interpolation at 0 with respect to the basis {1, . . . , x p , e ±λ1 x , . . . , e ±λr x } of V p , is the block upper triangular matrix

⎡

1 0 ... ⎢ 0 1 ... ⎢

⎢ ⎢ ⎢ ⎢ V := ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

..

.

0 0

.. . p!

1

1

λ1 .. . p λ1

−λ1 .. . (−λ1 ) p

p +1

λ1 .. . λd1

1

... ... .. .

1

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ (−λr ) p ⎥ ⎥ (−λr ) p +1 ⎥ ⎥ ⎥ ⎥ ⎦ (−λr )d −λr

λr .. . p λr

...

⎤

p +1

(−λ1 ) p +1 . . . λr .. .. .. . . . (−λ1 )d ... λdr

which is also the transpose of the Vandermonde matrix for the polynomial Hermite interpolation (of degree d) at the ( p + 1)-fold point 0 and at . Since this problem is known to be unisolvent, V is nonsingular. Consequently, there exists a vector g  = ( g j , : j = 0, . . . , d) with g j , ∈ V p , , : j = 0, . . . , d such that ( g j , )(k) (0) = δ jk . On the other hand, it is easy

to see that there always exists a unique polynomial vector h 1 = (h1j : j = 0, . . . , d), with h1j ∈ span {xd+1+ j : j = 0, . . . , d}, solution of the Hermite problem

(h1j )(k) (1) = δ jk ,

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

which satisfies (h 1 )(k) (0) = 0 automatically. Therefore, setting

h0j , := g j , −

d  ( g j , )(k) (1) hk1 ,

j = 0, . . . , d ,

k =0

we have even constructed the vector fields h  = (h0, , . . . , hd, ) T : R → R(d+1) ,  ∈ {0, 1}, such that (hj , )(k) (  ) = δ jk δ ,  ,  . To prove the converj , k = 0, . . . , d,  ,   ∈ {0, 1}, which proves the solvability of the Hermite interpolation problem in V  gence claim for  → (0, . . . , 0), we only have to consider the subproblem of interpolating at 0. Here we first note that the basis of V p , can be written as

⎡

1

⎤

⎡

⎢ .. ⎥ ⎢ ⎢ . ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ xp ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ λ1 x ⎥ ⎢ ⎢ e ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ e −λ1 x ⎥ = ⎢ ⎢ ⎥ ⎢ ⎢ . ⎥ ⎢ ⎢ .. ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ λr x ⎥ ⎢ ⎣ e ⎦ ⎣ e −λr x

0!

0

..

⎤

0

. p!

1 1

.. . 1 1

p +1

p

... λ1 λ1 p . . . (−λ1 ) (−λ1 ) p +1 .. .. .. . . . ... ...

p λr

(−λr ) p

p +1 λr (−λr ) p +1

⎥ ⎥⎡ ⎥ ⎥ ⎥⎢ ⎥⎢ ... ⎥⎢ ⎥⎢ ⎢ ... ⎥ ⎥⎢ ⎣ .. ⎥ .⎥ ⎥ ⎥ ... ⎦

⎤

1 x ⎥ x2 2! x3 3!

.. .

⎥ ⎥ ⎥ ⎥ ⎥ ⎦

...

that is as



V T | M  V T | M 2 V T | . . .

 1, x,

x2 2!

,

x3 3!

T ,...

,

where M  := diag(0, . . . , 0, λd1+1 , λd1+1 , . . . , λdr +1 , (−λr )d+1 ). Since the coefficients of the interpolant with respect to

  [1, . . . , x p , e λ1 x , e −λ1 x , . . . , e λr x , e −λr x ] T are given as V −1 y 0 , with y 0 = y 0j : j = 0, . . . , d , the interpolant takes the form

C. Conti et al. / Applied Numerical Mathematics 116 (2017) 119–128

125

( y 0 )T V −T [1, . . . , x p , e λ1 x , e −λ1 x , . . . , e λr x , e −λr x ]T T

 x2 x3 = ( y 0 )T I | V −T M  V T | V −T M 2 V T | . . . 1, x, , , . . . 2! 3! =

d 

y 0j

j =0

xj

+ ( y 0 )T

j!

d ∞  

V − T M k V T e j

k =1 j =0

xk(d+1)+ j

(k(d + 1) + j )!

(23)

.

Since the maximal eigenvalue in modulus of V − T M k V T is max j |λ j |k(d+1) , the series on the right converges in norm for || < 1 and converges to zero for  → (0, . . . , 0), leaving only the Taylor interpolant at 0. Taking derivatives of order up to d of (23), it follows that also the derivatives of this expressions converge to the derivatives of the Taylor interpolant, hence convergence takes place in the Sobolev norm  · d,∞ . 2  for a generic interval It becomes important to consider basis functions for the Hermite interpolation problem in V 

[a, b]. To this aim we consider the following

Lemma 8. The basis functions for the Hermite interpolation problem for a generic interval [a, b] are D t ht

 x−a  t

, t = b − a, where

 as constructed in Lemma 7 and D := diag(1, t , . . . , t d ). The space V h is the vector of cardinal functions in V  t p ,t  is reproduced by the interpolation.  is spanned by Proof. We simply observe that after the transformation x → x−t a , the exponential polynomial space V 



p

1, . . . , x , e

±t λ1 x−t a

,...,e

±t λr x−t a

 ,

x−a

d+1

 ,...,

t

x−a

2d+1 

,

t

so that the role of  is now played by t . Moreover, this space contains V p ,t  , hence unique interpolants reproduce this space. Since

(hj ,t  )(k) (a +   t ) =

1 tk

δ jk δ ,  ,

 ,   ∈ {0, 1},

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

we have to multiply the vector function ht by D t to obtain a vector of cardinal solutions.

2

It turns out that

the basis function vectors constructed in the proof of Lemma 7 allow us to perform Hermite interpolation on

α , α +1 2n

2n

by using D n h2−n  (2n · −α ).

Given the vector-valued sequence of data f n , the local interpolant qα ( f n ) takes the form

qα ( f n )(x) =





f nT (α +  ) D n h2−n  (2n x − α ),

x∈

 ∈{0,1}

which yields the global C d interpolant q( f n ) =

q( f n ) =

 α ∈Z

χ α , α + 1 2n

2n



α ∈Z χ



α , α +1 n

2n

α ∈Z

where

χ α , α + 1 2n

2n

2n

2

qα ( f

n)



,

(24)

as

(h2−n  )T (2n · −α ) D n f n (α +  )

 ∈{0,1}



q( f n ) =

, n

2

and finally the vector q( f n ) = q( f n ), q( f n ) , . . . , q( f n )(d)



α α+1



T

:

D −n N 2−n  (2n · −α ) D n f n (α +  ),

(25)

 ∈{0,1}



N 2−n  ( y ) := (hj ,2−n  )(k) ( y ) : j , k = 0, . . . , d , By interpolating at the midpoints n-th level mask G [n] is given by:

G [n] (1) = D N 02−n  (1/2) ,

2α +1 , 2n+1

y ∈ [0, 1],

 ∈ {0, 1}.

(26)

it is possible to derive an Hermite subdivision scheme S G [n] as in (9), where the

G [n] (0) = D ,

G [n] (−1) = D N 12−n  (1/2) .

The following proposition lists some of the properties of such scheme, which are easily verified directly from the construction.

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Proposition 9. The operator SG [n] satisfies the V p , spectral condition, the mask G [n] is obtained from the 0-th level one by substituting 2−n λ for λ, the matrix G [n] (−1) does not depend on n and

lim G [n] (1) = diag(1, −1, . . . , (−1)d ) G [n] (−1) diag(1, −1, . . . , (−1)d ).

n→∞





Moreover, the scheme S G [n] : n ∈ N0 is C d -convergent with basic limit function

⎧ 1 ⎪ ⎨ N  (x + 1), −1 ≤ x ≤ 0 F (x) = N 0 (x), 0≤x≤1 ⎪ ⎩ otherwise 0,

Moreover, in view of Lemma 8 the following result is valid.





Corollary 10. There exists a constant C > 0 such that N 2−n  ≤ C , n ≥ 0. 5. Convergence results In this section we formulate our main convergence result which is based on the contractivity property of a -difference operator involved in the factorization of the subdivision operators S A [n] , n ≥ 0. The hypotheses include also an asymptotical similarity assumption (see [5]) on the level dependent mask. This is quite a standard ingredient, used ever since level dependent schemes were considered. Theorem 11. Suppose that the subdivision operator S A[n] satisfies the V p , -spectral condition, that the supports of the masks A [n] , n ∈ N0 , are contained in some finite subset of Z and that there exists a mask A such that lim  A [n] − A ∞ = 0. Let S B [n] be as in n→∞

(18). If the sequence (2d S B [n] : n ≥ 0) produces a weakly contractive subdivision scheme, then S ( A [n] : n ≥ 0) is C d -convergent. [n] A = D −n−1 A [n] D n , let us consider the n-th level Hermite subdivision operator S Proof. For  A [n] in (10) producing the



T

vector sequence f n = f 0,n , f 1,n , . . . , f d,n attached to the points of 2−n Z. Let q( f n ) be the vector-valued Hermite interpolating function associated to it from (25). We will show that the sequence {q ( f n ) : n ≥ 0} is a Cauchy sequence in the Sobolev norm, therefore converging. We first observe that, for  f n+1 = S[n] f n and f n+1 = S[n] f n , it follows that qα ( f n+1 ) = qα ( f n ), where qα ( f n ) = G

A

 n n  ∈{0,1} N 2−n  (2 · −α ) D f n (α +  ) is the restriction of q( f n ) to the interval



α , α +1 2n

2n

as in (24). Hence, for x ∈



α , α +1 2n

2n

qα ( f n+1 )(x) − qα ( f n )(x) = qα ( f n+1 )(x) − qα ( f n+1 )(x)





=

 ∈{0,1}



D −(n+1) N n+1 (x) D n+1 (S[n] − S[n] ) f n (α +  ) A



= D −(n+1)





 ∈{0,1}

G

N n+1 (x) (S A [n] − SG [n] ) D n f n (α +  )

where, for abbreviation, we have set N n+1 := N 2−n−1  ,

 ∈ {0, 1}. Since S A [n] and SG [n] both satisfy the V p, -spectral

condition, their difference is an annihilator for V p , . Furthermore, from the assumptions on A [n] and from Proposition 9, it follows that all the hypotheses of Theorem 3 are satisfied. Thus, there exists SC [n] , with SC [n]  < K , such that S A [n] − SG [n] = SC [n] Hd,2−n  . Hence, expressing D n f n as S A [n−1] S A [n−2] · · · S A [0] f 0 , we obtain from above and from the factorization of S A [n] that

qα ( f n+1 )(x) − qα ( f n )(x)



= D −(n+1)

 ∈{0,1}

=D



−(n+1)

 ∈{0,1}

=2

−nd

D

−(n+1)

= diag(2







N n+1 (x) SC [n] 2d S B [n−1] · · · 2d S B [0] Hd, f 0 (α +  )

n(1−d)+1

,2





N n+1 (x) SC [n] S B [n−1] S B [n−2] · · · S B [0] Hd, f 0 (α +  )

 ∈{0,1} −nd





N n+1 (x) SC [n] Hd,2−n  S A [n−1] S A [n−2] · · · S A [0] f 0 (α +  )

  ,d 1,d , . . . , 2d ) F n0+ 1 (x) + F n+1 (x)

C. Conti et al. / Applied Numerical Mathematics 116 (2017) 119–128

where



 ,d



F n+1 (x) := N n+1 (x) SC [n] 2d S B [n−1] · · · 2d S B [0] Hd, f 0 (α +  ),

127

 ∈ {0, 1}.

Therefore,

qα ( f n+1 ) − qα ( f n )∞ ,d 1,d ≤ 2d  F n0+ 1 + F n+1 ∞

≤ 2d 2C SC [n] 2d S B [n−1] 2d S B [n−2] · · · 2d S B [0] Hd, f 0 ∞ ≤ C SC [n] 2d S B [n−1] 2d S B [n−2] · · · 2d S B [0] ∞ , and, since q( f n+1 ) − q( f n )∞ = supα ∈Z qα ( f n+1 ) − qα ( f n )∞ , we arrive at

q( f n+1 ) − q( f n )∞ ≤ C SC [n] ∞ 2d S B [n−1] 2d S B [n−2] · · · 2d S B [0] ∞ , hence, for m > 0,

q( f n+m ) − q( f n )∞ ≤ C K

∞ 

2d S B [ j−1] 2d S B [ j−2] · · · 2d S B [0] ∞

j =n

with K as in Theorem 3, which, by the weak contractivity assumption, converges to 0 uniformly in m for n → ∞. Hence q( f n ) is a Cauchy sequence of vectors that consist of a function and its derivatives, thus providing C d -convergence as claimed. 2 By weakening the assumption on S B [n] , another result can be formulated which involves restricted convergence. Corollary 12. Suppose that the subdivision operator S A[n] satisfies the V p , -spectral condition, that the supports of the masks A [n] , n ∈ N0 , are contained in some finite subset of Z and that there exists a mask A such that lim  A [n] − A ∞ = 0. Let S B [n] as in (18). n→∞

If the sequence (2k S B [n] : n ≥ 0) produces a weakly contractive subdivision scheme, with 0 ≤ k ≤ d, then S ( A [n] : n ≥ 0) provides restricted C k -convergence. Proof. Left multiplication with [ I k 0] in the proof of Theorem 11 yields that

  [ I k 0] qα ( f n+1 )(x) − qα ( f n )(x)

= 2−nk [ I k 0] D −(n+1)    · N n+1 (x) SC [n] 2k S B [n−1] 2k S B [n−2] · · · 2k S B [0] Hd, f 0 (α +  )  ∈{0,1}

  ,k 1,k = diag(2−nk , 2n(1−k)+1 , . . . , 2k , 0, . . . , 0) F n0+ 1 (x) + F n+1 (x) . Hence, using the same arguments as above,

  [ I k 0] qα ( f n+m )(x) − qα ( f n )(x) ≤ C SC [n] ∞

∞ 

∞

2k S B [ j−1] 2k S B [ j−2] · · · 2k S B [0] ∞ ,

j =n

from which the claim follows.

2

Examples of schemes satisfying the sufficient conditions given in Corollary 12 are those constructed in [7], which turn out to be C 0 convergent for specific values of the frequency λ. 6. Conclusion In this paper, we provided sufficient conditions for the convergence of Hermite subdivision schemes preserving polynomials and exponentials, under mild assumptions on the mask sequence and making use of the factorization property of the n-th level subdivision operator in terms of a cancellation and a -difference operator. To our knowledge, such results are the first in this direction and they complete the theoretical analysis of level-dependent Hermite subdivision schemes introduced in [7]. Future work includes an extension to the case of Hermite schemes preserving exponential polynomials and to the multivariate setting.

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Acknowledgements We thank the referees for their careful reading and their suggestions, and in particular for pointing out an incorrect formulation of Lemma 7. References [1] A.S. Cavaretta, W. Dahmen, C.A. Micchelli, Stationary subdivision, Mem. Am. Math. Soc. 93 (1991) 1–186. [2] M. Charina, C. Conti, L. Romani, Reproduction of exponential polynomials by multivariate non-stationary subdivision schemes with a general dilation matrix, Numer. Math. 127 (2014) 223–254. [3] C. Conti, L. Romani, Algebraic conditions on non-stationary subdivision symbols for exponential polynomial reproduction, J. Comput. Appl. Math. 236 (2011) 543–556. [4] C. Conti, J.-L. Merrien, L. Romani, Dual Hermite subdivision schemes of de Rham-type, BIT Numer. Math. 54 (2014) 955–977. [5] C. Conti, N. Dyn, C. Manni, M.-L. Mazure, Convergence of univariate non-stationary subdivision schemes via asymptotical similarity, Comput. Aided Geom. Des. 37 (2015) 1–98. [6] C. Conti, L. Romani, M. Unser, Ellipse-preserving Hermite interpolation and subdivision, J. Math. Anal. Appl. 426 (1) (2015) 211–227. [7] C. Conti, M. Cotronei, T. Sauer, Factorization of Hermite subdivision operators preserving exponentials and polynomials, Adv. Comput. Math. 45 (2017) 1055–1079. [8] C. Conti, L. Romani, J. Yoon, Approximation order and approximate sum rules in subdivision, J. Comput. Appl. Math. 207 (2016) 380–401. [9] S. Dubuc, J.-L. Merrien, Convergent vector and Hermite subdivision schemes, Constr. Approx. 23 (2006) 1–22. [10] S. Dubuc, J.-L. Merrien, Hermite subdivision schemes and Taylor polynomials, Constr. Approx. 29 (2009) 219–245. [11] J.-L. Merrien, A family of Hermite interpolants by bisection algorithms, Numer. Algorithms 2 (1992) 187–200. [12] J.-L. Merrien, T. Sauer, A generalized Taylor factorization for Hermite subdivision schemes, J. Comput. Appl. Math. 236 (4) (2011) 565–574. [13] J.-L. Merrien, T. Sauer, From Hermite to stationary subdivision schemes in one and several variables, Adv. Comput. Math. 36 (2012) 547–579. [14] J.-L. Merrien, T. Sauer, Extended Hermite subdivision schemes, J. Comput. Appl. Math. 317 (2017) 343–361. [15] U. Schwanecke, B. Juttler, A B-spline approach to Hermite subdivision, in: A. Cohen, C. Rabut, L.L. Schumaker (Eds.), Curve and Surface Fitting, SaintMalo, 1999, Vanderbilt University Press, Nashville, USA, 2000, pp. 385–392.