A new family of non-stationary hermite subdivision schemes reproducing exponential polynomials

Applied Mathematics and Computation 366 (2020) 124763

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A new family of non-stationary hermite subdivision schemes reproducing exponential polynomials Byeongseon Jeong a, Jungho Yoon b,∗ a b

Institute of Mathematical Sciences, Ewha Womans University, Seoul 120-750, South Korea Department of Mathematics, Ewha Womans University, Seoul 120-750, South Korea

a r t i c l e

i n f o

Article history: Received 13 April 2019 Revised 23 July 2019 Accepted 16 September 2019

Keywords: Non-stationary hermite subdivision scheme Convergence Smoothness Exponential polynomial reproduction Approximation order

a b s t r a c t In this study, we present a new class of quasi-interpolatory non-stationary Hermite subdivision schemes reproducing exponential polynomials. This class extends and unifies the well-known Hermite schemes, including the interpolatory schemes. Each scheme in this family has tension parameters which provide design flexibility, while obtaining at least the same or better smoothness compared to an interpolatory scheme of the same order. We investigate the convergence and smoothness of the new schemes by exploiting the factorization tools of non-stationary subdivision operators. Moreover, a rigorous analysis for the approximation order of the non-stationary Hermite scheme is presented. Finally, some numerical results are presented to demonstrate the performance of the proposed schemes. We find that the quasi-interpolatory scheme can circumvent the undesirable artifacts appearing in interpolatory schemes with irregularly distributed control points. © 2019 Elsevier Inc. All rights reserved.

1. Introduction Subdivision schemes are very useful iterative algorithms for modeling curves and surfaces in computer-aided geometric design and manufacturing, and have a wide range of applications such as curve/surface reconstruction, multi-resolution modeling, character animation, and scientific visualization. Starting from an initial set of data, subdivision schemes recursively generate denser sets of data through local refinement rules. As a special type of vector subdivision schemes, Hermite subdivision schemes of order d + 1 refine collections of vector data where the components of each vector represent a function and its consecutive derivatives’ values up to degree d. A curve (or surface) and its derivatives are modeled as the limit of a sequence of such collections, which allows the mathematical description of shapes. In regards to the convergence of the subdivision scheme and the approximation order of its limit function, the preservation of a certain class of functions is a key property of Hermite subdivision schemes. Since Merrien [22] introduced the interpolatory Hermite subdivision schemes of order 2 which reproduce polynomials up to degree 3, the polynomial preservation property has been extensively explored in numerous subsequent studies including [4,6,10,13–17,19–29] and the references therein. Dyn and Levin [13,14] presented analysis techniques for interpolatory Hermite subdivision schemes reproducing polynomials. Dubuc and Merrien [16,17] introduced the so-called spectral condition which is strongly related to the polynomial preservation. The spectral condition guarantees the Taylor factorization of Hermite subdivision operators generating the associated Taylor subdivision scheme that is central to the convergence analysis of the Hermite subdivision scheme ∗

Corresponding author. E-mail addresses: [email protected] (B. Jeong), [email protected] (J. Yoon).

https://doi.org/10.1016/j.amc.2019.124763 0 096-30 03/© 2019 Elsevier Inc. All rights reserved.

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B. Jeong and J. Yoon / Applied Mathematics and Computation 366 (2020) 124763

[17,24,25]. Hermite subdivision schemes satisfying the sum rule condition which is equivalent to the spectral condition was studied in [20,21]. Conti et al. [6] constructed Hermite subdivision schemes by taking the de Rham transform [11,16] of the Merrien’s schemes [22,23], which preserve the spectral condition. In [29], Jeong and Yoon introduced families of Hermite subdivision schemes of order 2 reproducing polynomials. Merrien and Sauer [26], and Moosmüller and Dyn [28] further investigated the factorization property of Hermite subdivision operators to construct Hermite schemes with higher regularity from existing ones. Subsequently, Merrien and Sauer [27] proposed a generalized spectral condition, called spectral chain, which admits the analysis of wider class of Hermite subdivision schemes. Recently, the studies on the polynomial preserving and reproducing properties were extended to the case of exponential polynomials to provide the capability of modeling a wider class of shapes. In this case, the subdivision rules vary at each subdivision level, so that the schemes are called ‘non-stationary’ or ‘level-dependent’. Conti et al. [7] conducted a study on non-stationary interpolatory Hermite subdivision schemes of order 2 preserving trigonometric functions. Subsequently, Conti et al. [4] introduced the so-called Vp, -spectral condition generalizing the spectral condition which is used in analyzing the convergence of Hermite subdivision schemes reproducing exponential polynomials [5]. Moreover, Cotronei and Sissouno [9], and Cotronei et al. [8] explored the exponential polynomial preservation property of interpolatory Hermite subdivision schemes in regards to wavelet analysis. Notably, most of those studies on the non-stationary Hermite subdivision schemes are mainly focused on the interpolatory case. Interpolatory subdivision schemes refine data, at each step, by inserting values corresponding to intermediate points, while maintaining the old data, through linear combinations of neighboring points. Therefore, the rules for the newly inserted data points are crucial for delivering good shape of subdivision results. However, it is well-known that interpolatory schemes often generate undesirable artifacts (e.g., wiggle, undulation, or self-intersection) with irregularly distributed control points. In addition, the interpolatory scheme is generally less smooth than the non-interpolatory scheme of the same order. In this regard, to overcome these drawbacks, this paper aims to present a new large family of non-stationary Hermite subdivision schemes which unifies and extends the existing Hermite schemes, including interpolatory schemes [7,14]. Each scheme in this family reproduces a certain class of exponential polynomials such that it can represent geometrically important shapes such as spirals, circular shapes, or parts of conics. Also, the proposed scheme has tension parameters that provide design flexibility to accommodate the various design circumstances. When the values of these parameters are small, the scheme becomes nearly interpolatory while attaining at least the same or better smoothness compared to an interpolatory scheme of the same order (see Section 4.4). We investigate the smoothness of the new non-stationary Hermite subdivision schemes. Using the asymptotic behavior, jointly with the factorization tools [30], of non-stationary subdivision operators, we verify that the proposed Hermite subdivision schemes have the same smoothness as the stationary counterparts which reproduce algebraic polynomials [29]. Furthermore, a rigorous analysis is performed to show that a non-stationary Hermite subdivision scheme reproducing a set of N linearly independent exponential polynomials provides the approximation order N. Some numerical examples demonstrating the performance of the proposed non-stationary schemes are presented. We particularly observe that the new scheme can remove undesirable artifacts which often arise in interpolatory schemes with irregularly distributed control points. The rest of the paper is organized as follows. Section 2 provides some notation and terminologies together with the definitions of convergence of vector and Hermite subdivision schemes. We also discuss exponential polynomial reproducing properties and some basic requirements of the space spanned by exponential polynomials. In Section 3, we construct new families of non-stationary Hermite subdivision schemes reproducing exponential polynomials. Section 4 is dedicated to the investigation of smoothness of the proposed schemes by exploiting tools for the factorization of non-stationary subdivision operators and the asymptotically equivalent relation between two schemes. The approximation order of the proposed Hermite schemes is discussed in Section 5. Lastly, we demonstrate the performance of the proposed schemes with numerical examples in Section 6. 2. Preliminaries 2.1. Notation, vector and hermite subdivision schemes For a given d ∈ N, let d+1 (Z ) and (d+1 )×(d+1 ) (Z ) be the linear spaces of sequences of vectors in Rd+1 and (d + 1 ) × (d + 1 ) matrices, respectively. We set d+1 ∞ (Z ) the linear space of bounded sequences of (d + 1 )-vectors. The set of nonnegative integers is denoted by Z+ = {α ∈ Z : α ≥ 0}. For each m ∈ Z+ , the notation m stands for the space of algebraic polynomials of degree ≤ m and put

pn ( x ) =

xn , n!

n ∈ Z+ .

Obviously, { pn : n = 0, . . . , m} is a basis of m . We also use the notation Nm := {1, . . . , m} ⊂ N for any m ∈ N. In addition, depending on the context, we denote by  · ∞ the supremum norm of vectors, matrices, functions, sequences or operators. Starting with an initial sequence f0 ∈ d+1 (Z ), a vector subdivision operator SC[k] : d+1 (Z ) → d+1 (Z ) for k ∈ Z+ is an iterative algorithm which generates sequences of vectors fk+1 formally by the rule

fk+1 := SC[k] fk .

B. Jeong and J. Yoon / Applied Mathematics and Computation 366 (2020) 124763

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Specifically, for each refinement level k and α ∈ Z, the vector fk+1 (α ) is defined by



fk+1 (α ) := (SC[k] fk )(α ) :=

C[k] ( α − 2β )fk ( β ),

β ∈Z

where fk (α ) is attached to the point

tα := 2− α ,

or

2− (α − 1/2 ),

 ∈ Z+ ,

(1)

called the primal (or resp. dual) parameterization. The sequence of matrices C[k] = {C[k] (α ) : α ∈ Z} in (d+1 )×(d+1 ) (Z ) is referred to as the kth level subdivision mask of the vector subdivision scheme SC[k] and it is assumed throughout this paper that ∪k∈Z+ supp(C[k] ) are contained in a bounded subset of R. The support of SC[k] is the smallest interval [σ 1 , σ 2 ] containing

{α ∈ Z : C[k] (α ) = 0}. The supremum norm of the subdivision operator SC[k] is given by

SC[k] ∞ := max



C[k] ( 2β )∞ ,

β ∈Z

Also, for a matrix sequence C[k] ( z ) =





 C[k] ( 2β + 1 )∞ .

β ∈Z

C [k] ,

C[k] ( β )z β ,

k ∈ Z+ , we associate the symbol

z ∈ C \ {0}.

β ∈Z

which plays a crucial role in analyzing non-stationary Hermite subdivision schemes. Since C[k] is finitely supported, each component of the symbol C[k] (z ) turns into a Laurent polynomial. The symbol C[k] (z ) can be decomposed as C[k] (z ) = [k] [k] Ce (z ) + Co (z ) with [k]

Ce

(z ) :=



C[k ] ( 2 β ) z 2β ,

[k]

Co

(z ) :=

β ∈Z



C[k] (2β + 1 )z2β +1 .

β ∈Z

A Hermite subdivision scheme (of order d + 1) is a special type of vector subdivision used in computing functions and their d [k] derivatives. More specifically, for a given mask A[k] = {A[k] (α ) : α ∈ Z} with k ∈ Z+ , let  Am be a sequence of (d + 1 ) × (d + 1 ) matrices with the entries defined by

 A[mk] (α ) = D−m−1 A[k] (α )Dm ,

α ∈ Z,

where m ∈ Z+ and D is the (d + 1 ) × (d + 1 ) diagonal matrix defined by D = diag(1, 2−1 , . . . , 2−d ). Then, a Hermite subdivision operator H[k] : d+1 (Z ) → d+1 (Z ) is defined to be the form Am

(HA[mk] fk )(α ) :=



 A[mk] (α − 2β )fk (β ),

α ∈ Z, fk ∈ d+1 (Z ).

(2)

β ∈Z

Note that the Hermite subdivision operator H[k] can be written as H[k] = D−(m+1 ) SA[k] Dm using the vector subdivision operAm

Am

ator SA[k] with the mask A[k] . For simplicity, we use the notation HA[k] := H[k] . A Hermite (or resp. vector) subdivision scheme A

k

{HA[k] } (or resp. {SC[k] }) is said to be stationary if A[k] = A (or resp. C[k] = C) for all k ∈ Z+ . If the masks are dependent on the subdivision level, then the schemes are said to be non-stationary (or level-dependent).

Definition 2.1. A vector subdivision scheme {SC[k] } is said to be Cγ -convergent with γ ∈ Z+ if for any initial sequence f0 ∈ (0 ) , . . . , f(d ) ]T : R → Rd+1 with f( j ) ∈ C γ (R ) for j = 0, . . . , d such that d+1 ∞ (Z ), there exists a vector-valued function f := [f

lim sup fk (α ) − f(tαk )∞ = 0,

k→∞ α ∈Z

and f ≡ 0 for some initial sequence f0 . Definition 2.2. A Hermite subdivision scheme {HA[k] } is said to be CN -convergent with N ∈ N if for any initial sequence (0 ) , . . . , f(d ) ]T : R → Rd+1 with f(0 ) ∈ C N (R ) and f( j ) = f0 ∈ d+1 ∞ (Z ), there exists a vector-valued function f := [f 1, . . . , d such that

d j f (0 ) dx j

for j =

lim sup fk (α ) − f(tαk )∞ = 0,

k→∞ α ∈Z

and f ≡ 0 for some initial sequence f0 . The function f is called the limit function of the Hermite subdivision scheme. Particularly, the limit function [k] defined by

[k] = lim HA[mk+m] · · · HA[k] δ m→∞

0

(3)

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B. Jeong and J. Yoon / Applied Mathematics and Computation 366 (2020) 124763

with the initial sequence δ := {δ(α ) = δα ,0 Id+1 : α ∈ Z} for the Kronecker delta δ α ,0 is called the basic matrix limit function. The linearity of the rule (2) allows us to represent the limit function denoted by f∞ := limm→∞ H[k+m] · · · H[k] f0 for any Am

initial sequence f0 as

f∞ =



α ∈Z

[k] (· − α )f0 (α ).

A0

(4)

2.2. Reproduction of exponential polynomials It is important to relate the properties of the limit function f∞ with the initial sequence f0 when the sequence f0 is sampled from a special type of function which is practically of interest in the design of shapes in applications, such as conic sections, helix, or, generally, exponential polynomial functions. In this paper we use the term ‘reproduction’ to refer to the ability of a subdivision scheme to provide specific types of limit functions. Definition 2.3. We say that a Hermite subdivision scheme of order d + 1 reproduces a function f if for any initial data f0 (α ) = [ f (tα0 ), . . . , f (d ) (tα0 )]T , the scheme produces the data fk at level k ∈ Z+ such that fk (α ) = [ f (tαk ), . . . , f (d ) (tαk )]T for any α ∈ Z. As this study is interested in constructing Hermite subdivision schemes reproducing exponential polynomials, we begin by defining the space of exponential polynomials. Definition 2.4. For a given positive integer η, let λn , n ∈ Nη , be distinct numbers in C. Assume that μn ∈ N for n ∈ Nη such that μ1 + · · · + μη = N. The N-dimensional space spanned by N linearly independent exponential polynomials is given as

N := span{x−1 eλn x :  ∈ Nμn , n ∈ Nη }. Note that for each n = 1, . . . , η, μn denotes the multiplicity of the value λn . When λn = 0 for all n ∈ Nη , the space N becomes the space of algebraic polynomials N−1 . In what follows, for the sake of simplicity, we will write

N = span{φ j : j ∈ NN }. For a practical usage, throughout this paper, we assume that the space N in this paper satisfies the following basic requirements: • First, the space N should be ‘shift-invariant’ such that for any α ∈ R, f ∈ N implies f (· − α ) ∈ N . It makes the subdivision rule at each refinement level invariant under the shifting of evaluation point, resulting in a uniform non-stationary mask. • The space N should be symmetric in the sense that if a function f is in N , then f (−· ) is also in the space N . By virtue of this property, the masks of the non-stationary Hermite subdivision schemes can either be even or odd symmetric. • Lastly, the N × N Wronskian matrix of {φ j : j ∈ NN } defined by

W N (x ) :=

  φ (j i−1) (x ) : i, j ∈ NN ,

(5)

is invertible for any x in a neighborhood of zero. This condition indicates that the set {φ j : j ∈ NN } is linearly independent, and it further plays an important role in the analysis of the new Hermite subdivision schemes. 3. Construction of non-stationary Hermite subdivision schemes In this section, we introduce new non-stationary Hermite subdivision schemes of order 2 reproducing exponential polynomials for both (primal and dual) parameterizations. Their construction are dependent on the space of exponential polynomials which we desire to reproduce. For this purpose, it is convenient to use the following notation in order to make our presentation simple. For a given sequence C of matrices or vectors, and a (finite) set S ⊂ Z, we apply



C := [C(α ) : α ∈ S]. S

Letting {φ j : j ∈ NN } be a set of exponential polynomials, for a given x ∈ R, we then introduce the matrix:



φN (x ) := φN,k (x ) :=

φ1 ( x ) 2−k−1 φ1 (x )

... ...

T φN ( x ) . 2−k−1 φN (x )

(6)

3.1. Primal schemes First, we present a new family of Hermite subdivision schemes {HA[k] } with the primal parameterization. Specifically, for each refinement level k ∈ Z+ , the values fk (α ) are attached to the parameter values tαk = 2−k α for α ∈ Z.

B. Jeong and J. Yoon / Applied Mathematics and Computation 366 (2020) 124763

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Case I: Odd Mask Let M be a positive integer. To construct the odd mask, we consider the 2M-point stencil

So := {−M + 1, . . . , M} ⊂ Z and formulate linear systems which reproduce the exponential polynomials {φ j : j ∈ N4M } and their derivatives on So in the following sense:

2(1−r )(k+1) φ (j r−1) (2−k−1 ) =



α ∈So

A[r1k] (1 − 2α )φ j (2−k α ) + 2−k



α ∈So

A[r2k] (1 − 2α )φ j (2−k α ),

r = 1, 2.

(7)

For each r = 1, 2, this linear system has 4M equations in 4M unknowns. In order to solve the linear systems, we first organize all the coefficients in the systems (7) into the matrix

A where (7) as

[k]





:= [A[1k] (1 − 2· )

A[2k] (1 − 2· ) ]T ∈ R4M×2 ,

So

[k] An ( · ),

Ek · A

[k]

So

n = 1, 2, is the nth column of A[k] ( · ). Then, using the matrix φ4M (x) in (6), we represent the linear systems

= φ4M (2−k−1 )

where Ek is the 4M × 4M matrix with entries defined by

Ek ( j, i ) = φ j (2−k (−M + i )),

Ek ( j, i + 2M ) = 2−k φ j (2−k (−M + i )),

i ∈ N 2M .

Here, the matrix Ek is a typical collocation matrix arising in the Hermite interpolation using the exponential polynomials {φ j : j ∈ N4M }. For the better readability of this paper, the non-singularity of the matrix Ek will be discussed later (see Remark 4.5). Case II: Even Mask To construct the even mask, for a given positive integer M, let us consider the (2M + 1 )-point stencil

Se := {−M, . . . , M} ⊂ Z. We then formulate linear systems which reproduce the exponential polynomials {φ j : j ∈ N4M } and their derivatives on the stencil Se as follows

2(1−r )(k+1) φ (j r−1) (0 ) =



α ∈Se

A[r1k] (−2α )φ j (2−k α ) + 2−k



α ∈Se

A[r2k] (−2α )φ j (2−k α )

r = 1, 2.

(8)

For a fixed r, this linear system is underdetermined because it has 4M equations in 4M + 2 unknowns. Hence, we set the tension parameters as k] A[12 (2M ) = −A[12k] (−2M ) = 2−(4M−4) θ ,

k] A[22 (±2M ) = 2−(4M−4) ω.

(9)

To solve the systems (8) for the rest of the unknowns, we express the systems in the matrix form. To do this, we arrange the remaining coefficients in the systems (8) into a matrix as

A

[k]



:= [A[1k] (−2· )

Se



A[2k] (−2· )

Se \{±M}

]T ∈ R4M×2 ,

where An (· ) with n = 1, 2 is the nth column of A[k] ( · ). Then, using the matrix φ4M (x) in (6) and the tension parameters (9), we rewrite the linear systems (8) as [k]

Ek · A

[k]



= φ 4M ( 0 ) − 2

−(k+4M−4 )

φ1 (−2−k M ) φ1 (2−k M )

... ...

T θ φ4 M (−2−k M ) . −θ φ4 M (2−k M )

ω , ω

(10)

where Ek is the 4M × 4M matrix with the entries defined by

Ek ( j, i ) = φ j (2−k (−M − 1 + i )), Ek ( j, i + 2M + 1 ) = 2−k φ j (2−k (−M + i )),

i ∈ N2M+1 , i ∈ N2M−1 . [k]

Remark 3.1. When θ = ω = 0 for all k ∈ Z+ , the Eq. (10) is reduced to Ek · A

A (2α ) = δα ,0 D, [k]

for D :=

diag(1, 2−1 )

= φ4M (0 ). Then, we can find the solution as

α = −M, . . . , M

indicating that the scheme {HA[k] } is interpolatory.

It is basic to require that the component sequence of the mask of a Hermite subdivision scheme is even or odd symmetric. Hence, we now show that the proposed primal subdivision mask is (even or odd) symmetric under the assumption on the space of exponential polynomials in Section 2.2.

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B. Jeong and J. Yoon / Applied Mathematics and Computation 366 (2020) 124763

Theorem 3.2. Let A[k] be the mask of the primal Hermite subdivision scheme {HA[k] }. Then, for r, s = 1, 2, we have Ars (α ) = [k]

(−1 )r+s A[rsk] (−α ),

α = 0, . . . , 2M.

Proof. First, we discuss the case of the odd mask of the Hermite scheme {HA[k] }. Note that since the space 4M = span{φ j : j ∈ N4M } is shift-invariant and symmetric (see Section 2.2), it is clear that φ j (2−k − · ) ∈ 4M . Hence, applying the basis function {φ j (2−k − · ) : j ∈ N4M } to the linear systems (7) and using the change of variables, we obtain

2(1−r )(k+1) φ (j r−1) (2−k−1 ) =

 α ∈So

(−1 )r−1 A[r1k] (2α − 1 )φ j (2−k α ) + 2−k

 α ∈So

(−1 )r A[r2k] (2α − 1 )φ j (2−k α ).

where So = {−M + 1, . . . , M}. Then the uniqueness of solution to the system induces that

A[r1k] (1 − 2α ) = (−1 )r−1 A[r1k] (2α − 1 ),

A[r2k] (1 − 2α ) = (−1 )r A[r2k] (2α − 1 ).

Similarly, the application of the basis functions {φ j (−x ) : j ∈ N4M } to the linear system (8) with fixed tension parameters yields that Ar1 (−2α ) = (−1 )r−1 Ar1 (2α ) and Ar2 (−2α ) = (−1 )r Ar2 (2α ) for α = 0, . . . , M. It finishes the proof. [k]

[k]

[k]

[k]



Example 3.3. Let M = 1 and consider the primal Hermite subdivision scheme {HA[k] } reproducing exponential polynomials in the space

4 = span{1, x, eλx , e−λx } which would be the most important and interesting example of the proposed Hermite subdivision scheme. The general form of its associated mask {A[k] (α ) : α = −2, . . . , 2} can be written as



1 − 2 τk θ

0

⎡



1 2

⎢ A (0 ) = 1 γk ω , A (1 ) = ⎣ 1 1 0 + − − 2 2 2 γk+1 A[rsk] (−α ) = (−1 )r+s A[rsk] (α ), r, s = 1, 2, α = 1, 2, [k]

[k]

1

⎤

4τk+1 ⎥ 1 ⎦, − 2γk+1

 τk θ A (2 ) =  1 γ  + k ω [k]

2

4

θ ω , 2

(11)

with

τk :=

λk (eλk + 1 ) (2eλk−1 − λk−1 eλk−1 − λk−1 − 2 ) , γk := , where λk := 2−k λ. λ (e k − 1 ) (λk−1 eλk − eλk−1 + 1 )

(12)

When θ = ω = 0, {HA[k] } turns the interpolatory Hermite scheme [7] corresponding to the mask

⎧⎡ 1 ⎪ ⎨ 2 ⎢ A[k] = ⎣ 1 1 ⎪ ⎩ + 2 γk+1

−

1

⎤



⎡

1

4τk+1 ⎥ , 1 ⎦ 0 − 2γk+1

1 0 ⎢ 2 1 ,⎣ 1 1 − − 2 2 γk+1

⎤⎫ ⎪ ⎬ 4τk+1 ⎥ . ⎦ 1 ⎪ ⎭ − 2γk+1 1

Fig. 1 shows the components of the basic matrix limit function [0] of the proposed Hermite scheme {HA[k] } with λ = iπ /2, i.e., 4 = span{1, x, cos(λx ), sin(λx )}, for the parameter choices θ = n/128 and ω = −n/20 with n = 0, . . . , 4. It demonstrates how the final limit functions are influenced by the choices of different tension parameters θ and ω. Remark 3.4. As discussed in Section 2.2, a special case of 4M is the space spanned by the algebraic polynomials

φn ( x ) =

xn−1 , ( n − 1 )!

n ∈ N 4M .

the proposed Hermite scheme then becomes the stationary Hermite subdivision scheme HM in [29] which reproduces polynomials in 4M−1 . Later, it is shown that these two schemes are asymptotically equivalent and have the same (integer) smoothness. 3.2. Dual schemes  [k] }) associated with the dual paWe construct a new family of Hermite subdivision schemes (hereafter, denoted by {H A rameterization. That is, the parameter values

tαk = 2−k (α − 1/2 ) are assigned to the refinements f[k] (α ) and new (refined) values are defined at location 14 and 34 between successive old points. We will see that for each M ∈ N, the corresponding dual Hermite scheme reproduces exponential polynomials in the (4M + 1 )-dimensional space 4M+1 .

B. Jeong and J. Yoon / Applied Mathematics and Computation 366 (2020) 124763

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Fig. 1. Components of the basic limit function [0] of the proposed scheme {HA[k] } with the mask given in (11) for parameters θ = n/128 and ω = −n/20, n = 0, . . . , 4: from the top to the bottom at the origin ((a), (d)), x = −1 (b), and x = −0.5 (c).

Case I: Odd mask Let M be a positive integer. To construct the odd mask, we define a rule evaluating a new value at successive old points. To achieve this, we employ the (2M + 1 )-point stencil

3 4

location between

So := {−M + 1, . . . , M + 1} ⊂ Z and formulate linear systems reproducing the exponential polynomials {φ j : j ∈ N4M+1 } and their derivatives in the following sense

2(1−r )(k+1) φ (j r−1)

3 4



· 2−k =

 α ∈So

A[r1k] (1 − 2α )φ j (2−k α ) + 2−k

 α ∈So

A[r2k] (1 − 2α )φ j (2−k α ),

r = 1, 2.

(13)

For each fixed r, it is clear that this is an undetermined system of 4M + 1 equations and 4M + 2 unknowns. There is one degree of freedom that can be utilized as a tension parameter. Therefore, it can be set as k] A[12 (−2M − 1 ) = 2−(4M−3) θ ,

k] A[22 (−2M − 1 ) = 2−(4M−3) ω.

To solve the systems (13), we rewrite the systems in the matrix form. For this, we define a matrix

A

[k]



:= [A[1k] (1 − 2· )

So



A[2k] (1 − 2· )

So \{M+1}

]T ∈ R4M×2 .

(14)

8

B. Jeong and J. Yoon / Applied Mathematics and Computation 366 (2020) 124763

which is organized with the coefficients in the systems (13). Using the matrix φ4M+1 (x ) in (6) and the tension parameters (14), the linear systems (13) can expressed as the matrix form





[k] 3 −k  Ek · A = φ4M+1 2 − 2−(k+4M−3) 4



φ1 (2−k (M + 1 )) φ1 (2−k (M + 1 ))

... ...

T φ4 M+1 (2−k (M + 1 )) θ · 0 φ4 M+1 (2−k (M + 1 ))

0

ω

.

where  Ek is the 4M × 4M matrix consisting of entries defined by

 Ek ( j, i ) = φ j (2−k (−M + i )),  E ( j, i + 2M + 1 ) = 2−k φ (2−k (−M + i )), k

j

i ∈ N2M+1 , i ∈ N 2M .

Case II: Even mask To construct the even mask of the dual scheme, we use one-grid shifted stencil Se := {−M, . . . , M} ⊂ Z, such that the even mask is obtained in a mirror symmetric way to the odd mask in the following sense: for r, s = 1, 2,

A[rsk] (2α ) = (−1 )r+s A[rsk] (−1 − 2α ),

α ∈ Se .

Remark 3.5. Like the case of the primal Hermite schemes, the proposed dual scheme is reduced to the stationary dual  reproducing polynomials in  Hermite subdivision scheme H M 4M [29] if

φn ( x ) =

xn−1 , ( n − 1 )!

n ∈ N4M+1 .

Later, we show that these two (stationary and non-stationary) schemes are asymptotically equivalent and have the same (integer) smoothness.

4. Convergence and smoothness of the proposed Hermite scheme This section is devoted to the analysis of convergence and smoothness of the proposed non-stationary Hermite subdivision schemes. The analysis relies on the asymptotic behavior of non-stationary subdivision schemes, jointly with the tools for the factorization of non-stationary subdivision operators. We shall verify that the proposed Hermite subdivision schemes have the same smoothness as their stationary counterparts.

4.1. Asymptotic behavior In this section, we prove that the proposed non-stationary Hermite subdivision scheme {HA[k] } is asymptotically equivalent to its stationary counterpart which reproduces algebraic polynomials [29] (see also Remark 3.4 and 3.5). Recalling the formula of the subdivision operators HC[k] = D−k−1 SC[k] Dk , we define the asymptotic equivalence between two Hermite subdivision schemes as follows. Definition 4.1. A non-stationary Hermite subdivision scheme {HC[k] } is said to be asymptotically equivalent to a stationary Hermite scheme HC if



SC[k] − SC ∞ < ∞.

(15)

k∈Z +

In what follows, we show that the mask A[k] of the proposed scheme {HA[k] } converges to its stationary counterpart A at the rate O(2−k ) as k → ∞, which clearly meets the condition (15). Recall that the proposed Hermite schemes are categorized into two classes, namely, primal and dual schemes. The proof techniques in regards to the asymptotic equivalence relation are essentially similar in both of the cases. Therefore, our proof shall be focused on the class of the primal Hermite schemes which reproduce {φ j : j ∈ N4M }; the other case can be done similarly. To simplify our presentation, for the given function φ j , denote by Tj its Taylor polynomial of degree 4M − 1 around 0, i.e.,

T j (x ) := Tφ j (x ) :=

4 M−1 m=0

xm ( m ) φ ( 0 ). m! j

(16)

Moreover, let T0 and T1 indicate the 4M × 2M matrices defined respectively by

T0 := [T j ((n − M )2−k ) : j ∈ N4M , n ∈ N2M ], T1 := [T j ((n − M )2−k ) : j ∈ N4M , n ∈ N2M ].

(17)

B. Jeong and J. Yoon / Applied Mathematics and Computation 366 (2020) 124763

9

Theorem 4.2. Let {A[k] } be the mask of the proposed primal Hermite subdivision scheme {HA[k] } and let A be the mask of its stationary counterpart HA . Then we have the bound

A[k] − A∞ ≤ cM 2−k for some constant cM > 0 depending on M. Proof. Firstly, we shall compute the difference between the odd masks, that is, |A[k] (1 − 2α ) − A(1 − 2α )| as k tends to ∞. To this end, recall that the odd mask of the non-stationary Hermite scheme reproduces φ j with j ∈ N4M in the following sense: for r = 1, 2,



2(1−r )(k+1) φ (j r−1) (2−k−1 ) =

−k −k α ∈So Ar1 (1 − 2α )φ j (2 α ) + 2 [k]



[k]

−k α ∈So Ar2 (1 − 2α )φ j (2 α ),

(18)

where So = {−M + 1, . . . , M}. Now, letting Tj be the Taylor polynomial of degree 4M − 1 around 0 of φ j , each exponential polynomial φ j (around zero) is decomposed into two parts:

φ j = Tj + k R j , k = 2−k4M ,

(19)

where  k Rj is the remainder of the Taylor polynomial Tj . Using this expression, the linear system (18) can be represented as follows:





2(1−r )(k+1) T j(r−1) (2−k−1 ) + k R(jr−1) (2−k−1 ) =



α ∈So

A[r1k] (1 − 2α )(T j (2−k α ) + k R j (2−k α ))

+2−k

 α ∈So

A[r2k] (1 − 2α )(T j (2−k α ) + k R j (2−k α )).

(20)

On the other hand, since the stationary Hermite scheme HA reproduces algebraic polynomials in 4M−1 , it is immediate from (18) that



2(1−r )(k+1) T j(r−1) (2−k−1 ) =

α ∈So

Ar1 (1 − 2α )T j (2−k α ) + 2−k



α ∈So

Ar2 (1 − 2α )T j (2−k α ).

(21)

Let Ar be the coefficient vector obtained by rearranging the components of the stationary Hermite mask in (21): odd

Ar := Ar



:= [Ar1 (1 − 2· )

So



Ar2 (1 − 2· ) ]T , So

r = 1, 2.

Then the linear system (21) can be written in the matrix form

T · Ar = b r

(22)

with the matrices T and br defined by

T = [T0 ; 2−k T1 ],

br := 2(1−r )(k+1) T j(r−1) (2−k−1 ) : j ∈ N4M

!T

.

With this equation, we return to the linear system (18). Letting



Ar := [A[r1k] (1 − 2· ) [k]

So



A[r2k] (1 − 2· ) ]T , So

r = 1, 2, [k]

in view of (19) and (20), one can rewrite the linear system (18) which uniquely defines the mask Ar

(T + k R )

[k] Ar

= br + k cr ,

k = 2−k4M ,

in the form

(23)

where R is a 4M × 4M matrix with the entries R j (2−k α ) and 2−k R j (2−k α ), and where cr ∈ R4M is the vector consisting of 2(1−r )(k+1 ) R(jr−1 ) (2−k−1 ) for j ∈ N4M . Note that the O( k ) perturbation of the non-singular matrix T leads to the O( k ) perturbation of its inverse [18]. Therefore, the equation (23) implies that [k] Ar = (T−1 + k R )(br + k cr )



= T−1 br + k T−1 cr +  R(br + k cr )



with  R∞ < ∞. Here, T−1 br = Ar and a direct calculation yields T−1 ∞ = O(2k(4M−1 ) ). Moreover, since k = 2−k4M , it fol[k] lows that k T−1 cr +  R(br + k cr )∞ = O(2−k ). It leads to the conclusion that Ar − Ar ∞ ≤ cM 2−k for any r = 1, 2. Likewise, the case of the even mask can be proved. It indeed verifies that A[k] − A∞ ≤ cM 2−k , which is the theorem’s claim.   [k] } and let A be the mask of its Theorem 4.3. Let {A[k] } be the mask of the non-stationary dual Hermite subdivision scheme {H A  . Then, we have the bound stationary counterpart H A

A[k] − A ≤ cM 2−k for some constant cM > 0 depending on M.

10

B. Jeong and J. Yoon / Applied Mathematics and Computation 366 (2020) 124763

Proof. This theorem holds immediately by applying the same technique in the proof of Theorem 4.2.



Remark 4.4. As a result of Theorems 4.2 and 4.3, it is clear that the proposed non-stationary Hermite schemes are asymptotically equivalent to their stationary counterparts reproducing algebraic polynomials for both primal and dual parameterizations. Remark 4.5. The uniqueness of the solution for the linear system to find the odd or even mask of the propose Hermite scheme reproducing exponential polynomials is yet to be proved. However, as discussed in (23), the Hermite matrix based on exponential polynomials can be understood as a perturbation of the classical Hermite matrix based on algebraic polynomials. It guarantees the non-singularity of the Hermite matrix based on exponential polynomials, at least for a sufficiently small 2−k . 4.2. Convergence analysis First, we highlight the tool for the convergence analysis of non-stationary Hermite subdivision schemes, introduced for the primal scheme [5], and extended it to the dual case [30]. Recalling the relation HA[k] = D−k−1 SA[k] Dk , we begin by presenting the definition of the spectral condition. Definition 4.6. A Hermite subdivision scheme {HA[k] } satisfies the spectral condition of order L + 1 if there exist polynomials pn (x), n = 0, . . . , L, independent of k such that

SA[k] pn = 2−n pn , where pn = {[ pn (α ), p n (α )]T : α ∈ Z} and pn (x ) =

n

m=0 cm x

m

with the leading coefficient cn = 1/n!.

From literature (e.g., see [25]), we know that if the Hermite scheme {HA[k] } of order 2 fulfills the spectral condition of order 2, we can obtain the so-called Taylor subdivision scheme {SB[k] } and the complete Taylor subdivision scheme {S } of B[k] {HA[k] } through factorization T SA[k] = 2−1 SB[k] T

 S [k] = S[k]  , and  B B

(24)

 are the Taylor operator and the difference operator defined respectively by where T and 



T =

δ

0



−1 , 1



= 1  0



0

δ

for the standard forward difference operator δ . The identities in (24) yields

 TSA[k] = 2−1 S B[k] T

(25)

 T . Moreover, if there exists a stationary Hermite scheme H with the mask A with the complete Taylor operator T :=  A [ k ] such that limk→∞ A − A∞ = 0, then it is easily checked that HA also satisfies the spectral condition, which enables us to have the stationary counterparts SB and S of {SB[k] } and {S }. The convergence of {HA[k] } is closely related to the B B[k] contractivity of the stationary scheme S . B Definition 4.7. A stationary vector subdivision scheme {SC } is said to be contractive if there exist μ < 1 and L > 0 such that

SCL ∞ ≤ μ. For the connection between the properties of S and {HA[k] }, we cite the following result in [30, Theorem 5.12]. B Theorem 4.8. Let {HA[k] } be a non-stationary Hermite subdivision scheme satisfying the spectral condition of order 2. Assume that there exist a stationary Hermite mask A and ρ ∈ (0, 1) such that A[k] − A∞ ≤ cρ k for all k ∈ Z+ with a constant c > 0. If the stationary complete Taylor scheme S associated with the scheme HA is contractive, then {HA[k] } is convergent, i.e., C1 . B 4.3. Smoothness analysis For the convenience of our presentation, we introduce the following notation. Let δ be the standard forward difference operator acting on a scalar sequence and denote by  a difference operator given by  := diag(δ , 1). In addition, let δ k be the generalized forward difference operator defined by

δk a(β ) = e−λ2 a(β + 1 ) − a(β ), a ∈ (Z ), −k

(26)

and let k indicate a difference operator given by k := diag(δ k , 1). The smoothness analysis of a non-stationary Hermite subdivision scheme {HA[k] } reproducing exponential polynomials can be done through analyzing the properties of the associated complete Taylor scheme {S }. The analysis is performed B[k] according to the factorization framework [30] of non-stationary subdivision operators in association with their asymptotic properties. Specifically, starting with a non-stationary vector subdivision scheme

SC[k] := 2SB[k] ,

B. Jeong and J. Yoon / Applied Mathematics and Computation 366 (2020) 124763

11

we assume that there exists a stationary mask C such that C[k] − C∞ = O(2−k ) as k → ∞. Let λ ∈ C and assume that the [k] [k] symbols Ce (z ) and Co (z ) associated with the scheme {SC[k] } satisfy the following property [k]

Cp

(e−λ2

−k−1

)



η (k ) 1



= ζ (k )

η (k + 1 ) 1

,

p = e, o,

(27)

where {η (k ) : k ∈ Z+ } is a scalar sequence converging to the limit η0 such that |η (k ) − η0 | = O(2−k ) as k → ∞, and {ζ (k ) : " k ∈ Z+ } is a sequence such that ∞ k=0 ζ (k ) converges to a nonzero number. Then we can obtain the scheme {S [k] } such C1

that

k+1 SC[k] = 2−1 ζ (k )SC[k] k

(28)

1

for the operator k := k (Vk )−1 where k := diag(δ k , 1) with δ k in (26) and Vk is a matrix given by

Vk =

η (k )



1 . 0

1

(29)

The assumption on the asymptotic behavior is sufficient to have C p (1 )



η0 1

=



η0 1

,

p = e, o,

(30)

for the symbols Ce (z ) and Co (z ) associated with the mask C, and k − ∞ = O(2−k ) for  := V −1 where

V =

η0 1



1 . 0

Considering the results in [1,31], the identity (30) leads us to obtain a stationary scheme SC1 such that

SC = 2−1 SC1 .

(31)

By a direct application of the technique for the proof of [30, Proposition 5.11], it also follows from (28) and (31) that C1 − C1 ∞ = O(2−k ) as k → ∞. Then according to [30, Section 5], the contractivity of the scheme 2−1 SC1 associated with the mask C1 implies that the non-stationary scheme {SC[k] } is convergent, i.e., C0 . Applying the above process iteratively, we can generate the schemes {S [k] } satisfying [k]

Ci

k+1 SC[k] = 2−1 ζ (k )SC[k] k , i

i = 1, 2 . . . ,

i+1

where k and ζ (k) depend on both λ and Ci , and show that {S [k]

[k]

CL

} is C0 for some L > 0. This in fact implies that {SC[k] }

(i.e., {2S }) is CL and {SB[k] } becomes C L+1 (see [30, Theorem 5.9]). B[k] The above tool can be used to check the smoothness of the non-stationary Hermite scheme {HA[k] }. Due to [30, Theorem 4.4], if {SB[k] } is CN , then {HA[k] } is C N+1 . In this point of view, we have the following general framework to investigate the smoothness of the non-stationary Hermite subdivision scheme reproducing exponential polynomials. Theorem 4.9. [30] Let {HA[k] } be a convergent non-stationary Hermite subdivision scheme which reproduces exponential polynomials and satisfies the spectral condition of order 2. Assume that there exists a stationary Hermite mask A such that A[k] − A∞ ≤ c2−k for all k ∈ Z+ with a constant c > 0. Furthermore, assume that there exist masks {C[jk] }, j ∈ NN , satisfying (27), so that we have

k+1 SC[k] = 2−1 ζ (k )SC[k] k , j−1

j ∈ NN ,

j

(32)

with C0 := 2 B[k] where  B[k] is the mask of the complete Taylor scheme associated with {HA[k] }. Then, if the stationary counterpart 2−1 SCN of the vector scheme {2−1 S [k] } is contractive, then {HA[k] } is C N+1 . [k]

CN

Due to Theorem 4.9 together with Theorem 4.2 and 4.3, we can describe the equivalence of smoothness of the proposed Hermite schemes and their stationary counterparts as follows. Theorem 4.10. Let {HA[k] } be the proposed non-stationary Hermite subdivision scheme and let HA be its stationary counterpart reproducing algebraic polynomials. Assume that HA is C N+1 for N ∈ N, and {HA[k] } has the factorization (32) such that the stationary counterpart 2−1 SCN of {2−1 S [k] } is contractive. Then {HA[k] } is also C N+1 . CN

Remark 4.11. Although the stationary counterpart of the proposed non-stationary Hermite scheme is C N+1 , it is not easy to show that the non-stationary scheme satisfies the corresponding factorization (32). However, in the next section, we verify the factorization at least for the primal scheme {HA[k] } with M = 1

12

B. Jeong and J. Yoon / Applied Mathematics and Computation 366 (2020) 124763

4.4. Smoothness of the primal scheme with M = 1 In this section, we analyze the smoothness of the primal scheme {HA[k] } with M = 1. First, we discuss its convergence. According to the choice of the tension parameters in (9), the mask A of the stationary counterpart of {HA[k] } has the general form [29]:

⎧ ⎨ 2θ A= 3ω ⎩ −

⎡1 ω , ⎣2

2

−

3 4

2

⎤ 

1 8 ⎦, 1 − 8

−θ

⎡

⎤ 

1 0 2 ⎣ 1 + 4ω , 3 − 2 4

1 − 4θ 0

1 8 ⎦, 1 − 8

2θ 3ω 2

⎫ θ ⎬ . ω ⎭

(33)

2

As remarked in [30], since {HA[k] } reproduces at least {1, x}, it satisfies the spectral condition of order 2. Let {SB[k] } and {SB[k] } be the Taylor and the complete Taylor subdivision schemes associated with {HA[k] }, respectively. In order to relate to the stationary counterpart, we apply the similarity transform to  B[k] as

B

[k]



[k]

:= {B (α ) = S−1 B[k] ( α )S : α ∈ Z},

S=



0 1

1 . 0

As verified in [29, Section 5.2], the stationary counterpart of {S

[k]

B

} is contractive for the tension parameters (θ , ω) in a

certain subset of R2 . It indicates that the stationary counterpart S of {S } is also contractive for the same parameters (θ , B B[k] ω). Therefore, by Theorem 4.8, the proposed Hermite scheme {HA[k] } is at least C1 for those parameter values. Next, to analyze the smoothness of {HA[k] }, we iteratively factor the operator 2S [k] . Letting λk := λ2−k with λ ∈ C we use B

the notation



Δλk (z ) :=

eλk z−1 − 1 0



0 −1 V , 1 k

with Vk in (29). Moreover, set

ak := eλk − λk − 1,

bk := λk (eλk − 1 ),

ck := 1 + e−λk ,

dk := 2ak − bk ,

The matrices 2B p (e−λk+1 ), p = e, o, associated with the scheme {2S [k]

ζ (k ) =

4ak+1 ak ,

so that we obtain the scheme {S

Δ−λk+1 (z ) · 2B

[k]

[k]

C1

} with the symbol

[k]

a2

c d k+1 k+1 k

, yielding the scheme {S

Δλk+1 (z )C1

[k]

[k]

C2

[k]

C1

and ζ (k ) =

k k

[k]

C2

} satisfy the property (27) for the choice of the tension

bk+1 dk+1 4λk+1 λk dk

λ e k−1 bk dk ak ek

[k]

ak ck (ak −bk ) , eλk λ d

(z ) = 2−1 ζ (k )C[2k] (z )Δλk (z2 ).

parameter

Δ0 ( z ) C 2

and

} such that

[k]

with η (k ) = −

bk ak

satisfying

} satisfy the property (27) with η (k ) =

Furthermore, the matrices C2,p (1 ), p = e, o, corresponding to {S

θ = θk :=

} satisfies the property (27) with η (k ) =

B [k] C1 ( z )

(z ) = 2−1 ζ (k )C[1k] (z )Δ−λk (z2 ).

Next, the matrices C1,p (eλk+1 ), p = e, o, associated to {S a2k dk+1

[k]

ek := ak−1 − 2bk .

and ζ (k ) =

(34) 2eλk ak+1 ck+1 dk ek+1 . ak dk+1 ek

Thus we obtain the scheme {S

[k]

C3

} satisfying

(z ) = 2−1 ζ (k )C[3k] (z )Δ0 (z2 ).

1 It can be easily checked for θ = θk in (34) that |θ − 64 | ≤ c2−k with some constant c > 0. According to [29, Section 5.2], 1 −1 −1 the stationary counterpart {2 SC3 } of {2 S [k] } is contractive for θ = 64 and ω in a certain set  ⊂ R. It verifies that the C3

stationary scheme HA becomes C4 for the same range of θ and ω (see Fig. 2). Hence, we can derive from Theorem 4.10 that the non-stationary scheme {HA[k] } also becomes C4 for θ = θk in (34) and ω ∈ . Fig. 2 presents the ranges of the tension parameters associated to the smoothness of the subdivision schemes HA with M = 1 [29]. For each N = 1, . . . , 3, the schemes {HA[k] } and HA are CN for (θ , ω) in the area N . 5. Approximation order A Hermite subdivision scheme of order d + 1 which fulfills the polynomial reproduction property up to degree N − 1 provides the approximation order N for N ≥ d. It means that if the initial data is sampled from a CN function f with the (d ) T density 2−τ (e.g., f (2−τ n )) for some τ ≥ 0, the Hermite subdivision scheme yields a vector limit function f∞ = [ f∞ , . . . , f∞ ]

B. Jeong and J. Yoon / Applied Mathematics and Computation 366 (2020) 124763

13

Fig. 2. Ranges of tension parameters corresponding to the smoothness of HA (and {HA[k] }) with M = 1. That is, HA (also, {HA[k] }) is CN for (θ , ω) ∈ N with N = 1, . . . , 4 [29].

which is close to the original vector function f = [ f, . . . , f (d ) ]T in the following sense

 f (r ) − f∞(r ) L∞ () ≤ c f 2−τ (N−r ) , 0 ≤ r ≤ d, for a constant cf > 0 depending on f but independent of τ . The objective of this section is to verify that a non-stationary Hermite subdivision scheme reproducing a set of N exponential polynomials provides the same approximation order as the stationary Hermite scheme. Although the proposed Hermite scheme is of order 2, this section considers the arbitrary n ( R ), order d + 1 of a Hermite scheme. This study especially focuses on approximating functions in the Sobolev space W∞ n (R ), the n ∈ N, which is the set of all functions f ∈ L∞ (R ) whose derivatives f(r) , 0 ≤ r ≤ n, is also in L∞ (R ). For each f ∈ W∞ associated norm is defined as

 f n,∞ :=

n 

 f ( r ) L∞ ( R ) .

r=0

The proof of the approximation order depends on the properties of the basic limit functions of the Hermite subdivision scheme.

, . . . , f (d ) ]T be the vector limit function of a Cd -convergent Hermite subdivision scheme {H Lemma 5.1. Let f∞ = [ f∞ , f∞ ∞ A[k] : k ≥ τ } starting with an initial sequence f0 parameterized with {tατ : α ∈ Z} for some τ ∈ Z+ . Let [τ ] be the basic matrix limit function of {HA[k] : k ≥ τ }. Then we can represent f∞ as

f∞ =



[τ ]

D −τ 

α ∈Z

(2τ · −α )Dτ f0 (α ).

Proof. For α ∈ Z, let δα be the sequence defined by δα := {δα (β ) = δα ,β Id+1 : β ∈ Z} for the Kronecker delta function δ α ,β .  It follows from the identities H[τ +m] = D−τ H[τ +m] Dτ and f0 = α ∈Z δα f0 (α ) that Aτ +m

f∞ = lim HA[τ +m] · · · HA[τ ] f0 m→∞

= lim D m→∞

=

 α ∈Z

τ

τ +m

−τ

D −τ

HA[τ +m] · · · HA[τ ] Dτ m



0

Am

 α ∈Z

δα f0 (α ) 

lim HA[τ +m] · · · HA[τ ] δα Dτ f0 (α ).

m→∞

m

0

14

B. Jeong and J. Yoon / Applied Mathematics and Computation 366 (2020) 124763

Since δα is parameterized with {tβτ : β ∈ Z}, by the definition (3), we see that limm→∞ H[τ +m] · · · H[τ ] δα = [τ ] (2τ · −α ), Am A0 which proves the claim.  Lemma 5.2. Let {HA[k] } and {HA } be Cd -convergent Hermite subdivision schemes and let [k] and  be their basic matrix limit functions respectively. Assume that the schemes {HA[k] } and {HA } are asymptotically equivalent. Then [k] converges uniformly to  as k tends to ∞. Proof. The lemma can be proved by applying the same technique for the proof of [12, Lemma 15].



Now, we turn to the proof of the approximation order of the proposed scheme. To do this, we introduce an auxiliary function ψ which is useful for our further analysis. For an arbitrary fixed point x¯ in R, define ψ as a combination of φ j with j ∈ NN , i.e.,



ψ (s ) := ψx¯ (s ) :=

c j φ j (s − x¯ )

(35)

j∈NN

with the coefficient vector c = [c j : j ∈ NN ]T obtained by solving the linear system

ψ (−1) (x¯ ) = f (−1) (x¯ ),  ∈ NN .

(36)

Equivalently, this linear system can be represented in the matrix form W N (0 ) · c = f with

f = [ f (−1) (x¯ ) :  ∈ NN ]T ,

! φ (j ν −1) (0 ) : ν, j ∈ NN ,

W N (0 ) :=

By assumption, the Wronskian matrix W N (0 ) is invertible, which implies that the coefficient vector c is uniquely determined. Theorem 5.3. Assume that the initial data f0 is sampled as f0 = {f0 (α ) = [ f (tατ ), . . . , f (d ) (tατ )]T : α ∈ Z} with a function f ∈ N (R ), where t τ indicates the parametrization in (1). Let {H d W∞ α A[k] } be a C -convergent Hermite subdivision scheme reproducing a set of exponential polynomials {φ j : j ∈ NN }. Then, we have the estimate

 f (r ) − f∞(r ) ∞ ≤ c f N,∞ 2−τ (N−r ) , 0 ≤ r ≤ d, (d ) where f∞ = [ f∞ , . . . , f∞ ] is the vector limit function of the subdivision scheme {HA[k] }.

Proof. Let x¯ ∈ R be a fixed point and ψ be the function defined as in (35). Since the Hermite subdivision scheme {HA[k] } is exact for the exponential polynomials φ j with j ∈ NN , owing to Lemma 5.1, it is apparent that

ψ (r ) =

d  n=0

2(r−n )τ

 (r ) ϕn,τ (2τ · −α )ψ (n) (tατ ),

0 ≤ r ≤ d,

α ∈Z

where ϕ0,τ , . . . , ϕd,τ are the entries in the first row of the basic matrix limit function of {HA[k] }. Recall that by the construction of ψ , f (r ) (x¯ ) = ψ (r ) (x¯ ) for r = 0, . . . , d. It clearly leads to the equation (r ) ¯ f (r ) (x¯ ) − f∞ (x ) = ψ (r ) (x¯ ) −

d 

2(r−n )τ

n=0

=

d 

 (r ) ϕn,τ (2τ x¯ − α ) f (n) (tατ )

α ∈Z

 (r )   2(r−n )τ ϕn,τ (2τ x¯ − α ) ψ (n) (tατ ) − f (n) (tατ ) .

n=0

α ∈Z

Now, we use the Taylor expansion arguments for the functions ψ and f. Letting Tg,x¯ be the degree-(N − 1 ) Taylor polynomial of a smooth function g around x¯, it can be easily induced from (35) and (36) that Tψ ,x¯ = T f,x¯ . Hence, by applying the explicit formula of the remainder of the Taylor polynomial, we obtain

| f (r ) (x¯ ) − f∞(r ) (x¯ )| ≤

d  n=0

2(r−n )τ

   (r ) ( ψ − f ) ( N ) ( ξα )  ϕn,τ (2τ x¯ − α )(tατ − x¯ )N−n (N − n )! 

α ∈Z

with ξ α between x¯ and tατ . By the construction of c > 0 independent of tαk and x¯. Hence, we obtain

| f (r ) (x¯ ) − f∞(r ) (x¯ )| ≤ cN 2−τ (N−r )  f N,∞

ψ in (35), it holds immediately that |ψ (N) (ξ α )| ≤ cfN,∞ for a constant

 d    (r ) τ ¯  ϕn,τ (2 x − α )(2τ x¯ − tα0 )N−n 

(37)

n=0 α ∈Z

(r ) for some cN > 0. By Lemma 5.2, each function ϕn, τ is (uniformly) bounded since it converges uniformly, as τ → ∞, to the stationary counterpart which is continuous. Moreover, there exists a number c0 > 0 such that (r ) τ #{α ∈ Z : ϕn, τ (2 x¯ − α ) = 0} < c0

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Fig. 3. Limit curves generated by using the primal Hermite scheme {HA[k] } with M = 1 for λ = i (first), 4i (second) and 4 (third): Tension parameters θ = n/64 and ω = −n/10 for n = 0 (red), 1 (green) and 2 (blue) are used. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 4. Limit curves of the proposed scheme {HA[k] } with different pairs of tension parameters θ and ω for the initial polygon with irregularly spaced vertices. From left to right, (θ , ω ) = (0, 0 ) (interpolatory scheme), (5/128, −1/4 ), and (5/64, −1/4 ).

(r ) for any x¯ and any τ ∈ Z+ since ϕn, τ is compactly supported. Therefore, the claim of this theorem holds immediately.



Corollary 5.4. Let {HA[k] } be the non-stationary Hermite subdivision scheme of order 2 reproducing exponential polynomials {φ j : j ∈ N4M } with the primal parametrization. Then {HA[k] } provides the approximation order 4M.  [k] } be the non-stationary Hermite subdivision scheme of order 2 reproducing exponential polynomials Corollary 5.5. Let {H A  [k] } provides the approximation order 4M + 1. {φ j : j ∈ N4M+1 } with the dual parametrization. Then {H A 6. Numerical examples In this section, we present some numerical examples to demonstrate the performance of the proposed non-stationary Hermite subdivision scheme. We apply the primal Hermite subdivision {HA[k] } with M = 1 reproducing exponential polynomials in the space

4 = span{1, x, exp(λx ), exp(−λx )}

(38)

for λ ∈ R ∪ iR. If λ ∈ iR, the space 4 is clearly equivalent to 4 = span{1, x, cos(λx ), sin(λx )}. The corresponding subdivision masks are given in (11). Example 6.1. The first example illustrates how the shape of the limit curve of {HA[k] } varies depending on the tension parameters θ and ω. Given a fixed λ, we generate three limit curves with θ = n/64 and ω = −n/10 for n = 0, 1, 2. As remarked in Example 3.3, when n = 0, the proposed scheme {HA[k] } becomes interpolatory. As n increases, the limit curves get farther away from the initial control polygon. Fig. 3 shows three sets of limit curves for λ = i, 4i, 4. The arrows indicate the directions of the initial tangent vectors at the given control points. We can observe the design flexibility of the proposed Hermite scheme. Example 6.2. If the initial control points are very irregular, an interpolatory scheme may produce undesired artifacts (e.g., self-intersection) at its final limit curve. However, as we see in Fig. 4, the proposed scheme {HA[k] } can circumvent this obstacle by choosing proper tension parameters. In this example, we set λ = iπ /2 in the exponential polynomial (38) and use the tension parameters (θ , ω ) = (5/128, −1/4 ) (middle) and (ω, θ ) = (5/64, −1/4 ) (right) respectively for the computation of the proposed scheme {HA[k] }. of the stationary scheme. In Fig. 5, the limit curves were generated using the tension parameters (θ , ω ) = (0.055, −0.3 ) (left) for the stationary scheme HA and (θ , ω ) = (0.055, −0.3 cos(2−k+1 π /7 )) (right) for the proposed scheme {HA[k] }. As observed in the zoomed area, the stationary scheme HA yields a twisting artifact in the irregular area of the initial polygon. This demonstrates an advantage of the proposed scheme. Example 6.3. (Surface of revolution) The surface of revolution is a commonly used technique in various CAGD applications. We present surfaces of revolution formed by revolving the limit curves of the scheme {HA[k] }. Fig. 6 displays the initial

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Fig. 5. Limit curves of the stationary scheme HA with tension parameters (θ , ω ) = (0.055, −0.3 ) (left) and the proposed scheme {HA[k] } with (θ , ω ) = (0.055, −0.3 cos(2−k+1 π /7 )) (right) for the initial polygon with irregularly spaced vertices.

Fig. 6. Surfaces of revolution generated by the proposed scheme {HA[k] }. From left to right, initial polygonal curve with tangent vectors, and surfaces with the parameters (θ , ω ) = (n/64, −n/10 ) for n = 0, 1, 2, respectively.

control polygonal curve with tangent vectors, and three surfaces of revolution whose generating curves are produced from the initial polygonal curve by using λ = iπ /2 and the parameters θ = n/64 and ω = −n/10 for n = 0, 1, 2. Remark 6.4. Subdivision schemes are very useful tools for geometrical modeling in a wide range of applications such as animation, computer graphics, computer vision and computer aided geometric design. Besides, it is worthwhile to investigate applicability of subdivision schemes to other systems such as stochastic systems (e.g, see [2,3] and the references therein). Acknowledgments Byeongseon Jeong was supported by the grant NRF-2019R1I1A1A01060757 and NRF-2019R1A6A1A11051177 funded by the Ministry of Education. Jungho Yoon was supported by the grant NRF-2015R1A5A1009350 and NRF-2019R1F1A1060804 through the NRF (National Research Foundation). References [1] M. Charina, C. Conti, T. Sauer, Regularity of multivariate vector subdivision schemes, Numer. Algor. 39 (1–3) (2005) 97–113. [2] J. Cheng, J. Park, J. Cao, W. Qi, Hidden Markov model-based nonfragile state estimation of switched neural network with probabilistic quantized outputs, IEEE Trans. Cybern. (2019), doi:10.1109/TCYB.2019.2909748. [3] J. Cheng, C.K. Ahn, H.R. Karimi, J. Cao, W. Qi, An event-based asynchronous approach to Markov jump systems with hidden mode detections and missing measurements, IEEE Trans. Syst. Man Cybern. (2019), doi:10.1109/TSMC.2018.2866906. [4] C. Conti, M. Cotronei, T. Sauer, Factorization of hermite subdivision operators preserving exponentials and polynomials, Adv. Comput. Math. 42 (5) (2016) 1055–1079. [5] C. Conti, M. Cotronei, T. Sauer, Convergence of level-dependent hermite subdivision schemes, Appl. Numer. Math. 116 (2017) 119–128. [6] C. Conti, J.-L. Merrien, L. Romani, Dual hermite subdivision schemes of de RHAM-type, BIT Numer. Math. 54 (4) (2014) 955–977. [7] C. Conti, L. Romani, M. Unser, Ellipse-preserving hermite interpolation and subdivision, J. Math. Anal. Appl. 426 (1) (2015) 211–227. [8] M. Cotronei, C. Moosmüller, T. Sauer, N. Sissouno, Level-dependent interpolatory hermite subdivision schemes and wavelets, Constr. Approx. (2018), doi:10.10 07/s0 0365- 018- 9444-4. [9] M. Cotronei, N. Sissouno, A note on hermite multiwavelets with polynomial and exponential vanishing moments, Appl. Numer. Math. 120 (2017) 21–34. [10] S. Dubuc, Scalar and hermite subdivision schemes, Appl. Comput. Harmon. Anal. 21 (3) (2006) 376–394. [11] S. Dubuc, De RHAM transforms for subdivision schemes, J. Approx. Theory 163 (8) (2011) 966–987. [12] N. Dyn, D. Levin, Analysis of asymptotically equivalent binary subdivision schemes, J. Math. Anal. Appl. 193 (2) (1995) 594–621. [13] N. Dyn, D. Levin, Analysis of hermite-type subdivision schemes, in: C.K. Chui, L.L. Schumaker (Eds.), Approximation Theory VIII, Wavelets and Multilevel Approximation (College Station, TX), 2, World Sci., River Edge, NJ, 1995, pp. 117–124. [14] N. Dyn, D. Levin, Analysis of hermite-interpolatory subdivision schemes, in: S. Dubuc, G. Deslauriers (Eds.), Spline Functions and the Theory of Wavelets, Providence R. I. Am. Math. Soc., 1999. 105–113 [15] S. Dubuc, J.L. Merrien, Convergent vector and hermite subdivision schemes, Constr. Approx. 23 (1) (2005) 1–22.

B. Jeong and J. Yoon / Applied Mathematics and Computation 366 (2020) 124763

17

[16] S. Dubuc, J.L. Merrien, de rham transform of a hermite subdivision scheme, in: M. Neamtu, L.L. Schumaker (Eds.), Approximation Theory XII, San Antonio, Nashboro Press, Nashville, 20 08. 20 07, pp. 121–132 [17] S. Dubuc, J.L. Merrien, Hermite subdivision schemes and taylor polynomials, Constr. Approx. 29 (2) (2009) 219–245. [18] G.H. Golub, C.F.V. Loan, Matrix Computations, John Hopkins University Press, Baltimore, 1996. [19] N. Guglielmi, C. Manni, D. Vitale, Convergence analysis of c2 hermite interpolatory subdivision schemes by explicit joint spectral radius formulas, Linear Algebra Appl. 434 (4) (2011) 884–902. [20] B. Han, Approximation properties and construction of hermite interpolants and biorthogonal multiwavelets, J. Approx. Theory 110 (1) (2001) 18–53. [21] B. Han, T.P. Yu, Y. Xue, Noninterpolatory hermite subdivision schemes, Math. Comput. 74 (251) (2005) 1345–1367. [22] J.L. Merrien, A family of hermite interpolants by bisection algorithms, Numer. Algor. 2 (2) (1992) 187–200. [23] J.L. Merrien, Interpolants d’hermite c2 obtenus par subdivision, M2AN Math.Model. Numer. Anal. 33 (1) (1999) 55–65. [24] J.-L. Merrien, T. Sauer, A generalized taylor factorization for hermite subdivision schemes, J. Comput. Appl. Math. 236 (4) (2011) 565–574. [25] J.-L. Merrien, T. Sauer, From hermite to stationary subdivision schemes in one or several variables, Adv. Comput. Math. 36 (4) (2012) 547–579. [26] J.-L. Merrien, T. Sauer, Extended hermite subdivision schemes, J. Comput. Appl. Math. 317 (2017) 343–361. [27] J.-L. Merrien, T. Sauer, Generalized taylor operators and polynomial chains for hermite subdivision schemes, Numer. Math. 142 (1) (2019) 167–203. [28] C. Moosmüller, N. Dyn, Increasing the smoothness of vector and hermite subdivision schemes, IMA J. Numer. Anal. 39 (2) (2019) 579–606. [29] B. Jeong, J. Yoon, Construction of hermite subdivision schemes reproducing polynomials, J. Math. Anal. Appl. 451 (1) (2017) 565–582. [30] B. Jeong, J. Yoon, Analysis of non-stationary hermite subdivision schemes reproducing exponential polynomials, J. Comput. Appl. Math. 349 (2019) 452–469. [31] T. Sauer, Stationary vector subdivision: quotient ideals, differences and approximation power, Rev. R. Acad. Cien. Ser. A Mat. 96 (2) (2002) 257–277.