Polynomial Reproduction of Vector Subdivision Schemes

Abstract

We discuss the polynomial reproduction of vector subdivision schemes with general integer dilation 𝑚≥2. We first present a simple algebraic condition for polynomial reproduction of such schemes with standard subdivision symbol. We then extend it to general subdivision symbol satisfying certain order of sum rules. We also illustrate our results with several examples. Our results show that such kind of scheme can produce exactly the same scalar polynomial from which the data is sampled by convolving with a finite nonzero sequence of vectors.

1. Introduction

Subdivision schemes are efficient algorithmic methods in approximation theory, computer aid geometric design, and wavelet construction; see [1, 2] and references therein. The subdivision scheme is called vector subdivision scheme if the corresponding subdivision symbol is a matrix of trigonometric polynomial. Note that the vector subdivision scheme plays an important role in multiwavelet and multichannel wavelets analysis; see [3–5] and references therein. In this paper we would like to derive simple algebraic conditions on the subdivision symbol that allow us to determine the degree of its polynomial reproduction, which is different from polynomial generalization.

The polynomial generalization is the capability of vector subdivision scheme to generate the full space of scalar polynomials with desired degree. This property is equivalent to approximation order (defined in 𝐿²-norm), accuracy, and sum rulers; see [6, 7]. The so-called polynomial reproduction is the capability of vector subdivision scheme to produce exactly the same scalar polynomial 𝑝(𝑥) from which the data is sampled by convolving with a special sequence of vectors 𝐡₀={𝐡_0,𝑗∈ℝ^𝑠,𝑗∈𝕊}, 𝐝^𝑇_0,𝑘∶=𝑝∗𝐡₀(𝑘)=_𝑗∈ℤ𝑝𝑘−𝑗𝐡_0,𝑗,𝑘∈ℤ, (1) where 𝕊 is some finite index set containing 0, ∗ denotes convolution operator, and 𝑇 denotes transpose operator.

There are many valuable works that were done when scalar subdivision scheme was considered [8–11]. Hormann and Sabin [8] presented some algebraic conditions which can be used to determine the degree of polynomial reproduction for a family of schemes. Polynomial reproduction was studied by Dyn et al. [9] for arbitrary prime and dual binary schemes and extended to univariate subdivision scheme of any arity by Conti and Hormann [10]. Further, Charina and Conti [11] yielded simple algebraic conditions on the subdivision symbol for the multivariate setting of scalar subdivision with dilation matrix 𝑚𝐼, |𝑚|>1; Charina and Romani [12] extended to the general expanding dilation matrix 𝑀∈ℤ^𝑠×𝑠 case. Motivated by all these works, we discuss the property of polynomial reproduction for vector subdivision schemes.

The remainder of this paper is organized as follows. Section 2 sets the notations concerning vector subdivision scheme with general integer dilation 𝑚≥2 and stresses the connection of convergence between the vector subdivision scheme and traditional cascade algorithm. In Section 3, we provide algebraic tools for determining the degree of polynomial reproduction of the vector subdivision scheme with standard subdivision symbol; basing on two scale similarity transform (TST), we also establish the relationship of vector subdivision schemes between general subdivision symbol and the standard subdivision symbol. In Section 4, several examples are provided to illustrate our results.

2. Background and Notation

A vector subdivision scheme 𝐒_{𝑃,𝐡0,Φ0} with integer dilation 𝑚≥2 is given by a finitely supported matrix sequence 𝑃={𝑃_𝑘∈ℝ^𝑠×𝑠,𝑘∈ℤ} (so-called subdivision mask), a constant column vector 𝐡₀={𝐡_0,𝑗∈ℝ^𝑠,𝑗∈𝕊}, and an initial function Φ₀(𝑥). The constant vector 𝐡₀ and the initial function Φ₀ are determined by 𝑃 and will be given by (39) or (55) in the next section. The symbols of 𝐡₀ and 𝑃 (denoted by 𝐡₀(𝑧) and 𝑃(𝑧), resp.) are given by the Laurent polynomials 𝐡₀(𝑧)=_𝑗∈𝕊𝐡_0,𝑗𝑧^𝑗,𝑃(𝑧)=_𝑘∈ℤ𝑃_𝑘𝑧^𝑘, (2) respectively.

In this paper, we use two vector spaces of Laurent polynomial, 𝔏(ℝ^𝑠) and 𝔏^′(ℝ^𝑠). We say 𝐡₀(𝑧)∈𝔏(ℝ^𝑠), if 𝐡^𝑇₀(𝑧)=𝐡^𝑇𝐴(𝑧)=𝐡^𝑇1_𝑗∈𝕊𝐫_𝑗𝑧^𝑗𝟎𝐼_𝑠−1, (3) with some 𝐡∈ℝ^𝑠 and 𝐡≠𝟎, where {𝐫_𝑗,𝑗∈𝕊} is some finite 1×(𝑠−1)-vector sequence and 𝐼_𝑠−1 is the (𝑠−1)×(𝑠−1) identity matrix. Similarly, we say 𝐡₀(𝑧) belongs to 𝔏^′(ℝ^𝑠) if 𝐡₀(𝑧)=𝐴(𝑧)𝐡.

For a data sequence {𝐝_𝑘∈ℝ^𝑠,𝑘∈ℤ}, the vector subdivision operator 𝑆_𝑃 acting on it is defined by 𝑆_𝑃𝑑^𝑇_𝑘=_𝑗∈ℤ𝐝^𝑇_𝑗𝑃_{𝑘−𝑚𝑗},𝑘∈ℤ. (4) Actually the vector subdivision scheme 𝐒_{𝑃,𝐡0,Φ0} is a recursive algorithm based on the vector subdivision operator 𝐝^𝑇_+1,𝑘=𝑆_𝑃𝐝^𝑇_,𝑘=_𝑗∈ℤ𝐝^𝑇_,𝑗𝑃_{𝑘−𝑚𝑗},𝑘∈ℤ,∈ℕ (5) and a convolution operator 𝑓_+1,𝑘=_𝑗∈ℤ𝐝^𝑇_+1,𝑗Φ₀𝑘−𝑗,𝑘∈ℤ,∈ℕ, (6) with the initial data 𝐝₀={𝐝_0,𝑘∈ℝ^𝑠∶𝑘∈ℤ} and the function Φ₀(𝑥)=_𝑗∈𝕊𝐲_𝑗𝒳𝑥−𝑗, (7) where 𝒳(𝑥) is the characteristic function of interval [0,1) and 𝐲={𝐲_𝑗∈ℝ^𝑠,𝑗∈𝕊} is selected such that 𝐲(𝑧)∈𝔏^′(ℝ^𝑠) and 𝐡₀(𝑧)^𝑇𝐲(𝑧)=1.

Lemma 1. If  𝐡₀(𝑧)∈𝔏(ℝ^𝑠), then there exist linearly independent vectors 𝐲¹(𝑧),…,𝐲^𝑠(𝑧)∈𝔏^′(ℝ^𝑠) such that 𝐡^𝑇₀(𝑧)𝐲^𝑗(𝑧)=1,𝑗=1,…,𝑠. (8)

Proof. Suppose that 𝐡^𝑇₀=𝐡^{𝑇1∑𝑗∈𝕊𝐫𝑗𝑧𝑗}_{𝟎𝐼𝑠−1}. It is not difficult to find that there exist linearly independent constant vectors 𝐲¹,…,𝐲^𝑠 satisfying 𝐡^𝑇𝐲^𝑗=1,𝑗=1,…,𝑠. (9) Let 𝐲^𝑗(𝑧)=1−_𝑗∈𝕊𝐫_𝑗𝑧^𝑗𝟎𝐼_𝑠−1𝐲^𝑗,𝑗=1,…,𝑠. (10) Then, by calculation the conclusion follows directly.

In order to define a sequence of continuous functions, we also need parameterization. As in [10] for the scalar case, we chose the associated parameter values: 𝑡_𝑘∶=𝑡₀+𝑘𝑚,𝑡₀=𝑡₀⁻¹−𝜏𝑚,𝑡⁰₀=0,𝑘∈ℤ,∈ℕ. (11) We call {𝑡_𝑘,𝑘∈ℤ,∈ℕ} the sequence of parameter values associated with the vector subdivision scheme. Let us define continuous functions 𝑓 by linearly interpolating the data 𝑓_,𝑘 to the parameter values 𝑡_𝑘, 𝑓𝑡_𝑘=𝑓_,𝑘,𝑘∈ℤ,∈ℕ. (12)

Definition 2. If the sequence of continuous functions {𝑓,∈ℕ} converges (in uniform norm), then we define the limit function as 𝑔_𝐝0=lim_→∞𝑓. (13)

We say that 𝑔_𝐝0 is the limit function of the vector subdivision scheme 𝐒_{𝑃,𝐡0,Φ0} acting on the initial data 𝐝₀. If 𝑠=1 and 𝐡₀=1, one may find that the vector subdivision scheme is the same as the scalar one in [10]. The limit functions 𝜙₀∶𝜙_𝜹0^(𝑝)∶=𝑔_𝜹0^(𝑝), 𝑝=1,…,𝑠, are called basic limit functions if the initial data is selected by 𝜹^(𝑝)_0,𝑘=𝐞_𝑝,𝑘=0,𝟎,otherwise,𝑘∈ℤ,𝑝=1,…,𝑠, (14) where 𝐞_𝑝 denotes 𝑝th column of the 𝑠×𝑠 identity matrix. It is easy to check that Φ=(𝜙₁,…,𝜙_𝑠)^𝑇 is compactly supported and satisfies the refinement equation Φ(𝑡)=_𝑘∈ℤ𝑃_𝑘Φ(𝑚𝑡−𝑘),𝑡∈ℝ. (15)

In this paper, we consider vector subdivision scheme that is convergent and nonsingular, so that 𝑔_𝐝0=0 if and only if 𝐝₀=𝟎. Define the linear operator 𝑄_𝑃 (which is called cascade operator) as follows: 𝑄_𝑃Φ=_𝑘∈ℤ𝑃_𝑘Φ(𝑚⋅−𝑘),Φ∈𝐿²(ℝ)^𝑠. (16) By induction on 𝑛, it is easy to verify that 𝑄^𝑛_𝑃Φ₀(𝑥)=_𝑘∈ℤ𝑃_𝑛,𝑘Φ₀𝑚^𝑛𝑥−𝑘,𝑛∈ℕ, (17) where Φ₀ is defined as (7), 𝑃_𝑛,𝑘=_𝑗∈ℤ𝑃_{𝑛−1,𝑗}𝑃_{𝑘−𝑀𝑗},𝑃_1,𝑘=𝑃_𝑘,𝑘∈ℤ. (18)

Under the convergent condition of cascade algorithm (see [13]), we can prove that 𝑄^𝑛_𝑃Φ₀ converges to the normalized solution Φ of (15) in (𝐿_𝑝(ℝ))^𝑠 (1≤𝑝≤∞).

Proposition 3. A vector subdivision scheme 𝐒_{𝑃,𝐡0,Φ0} is convergent and nonsingular if and only if (15) has compactly supported solution Φ and the integer translations of Φ are linearly independent.

Proof. Firstly, we claim that the convergence of vector subdivision scheme is the same as that of the cascade algorithm. Actually, by induction, for the initial data 𝜹₀^(𝑝),𝑝=1,…,𝑠, we have 𝑓^(𝑝)_1,𝑗=_𝑘∈ℤ𝜹^𝑇(𝑝)_1,𝑘Φ₀𝑗−𝑘=_{𝑘∈ℤ𝑛∈ℤ}𝜹^𝑇(𝑝)_0,𝑛𝑃_{𝑘−𝑚𝑛}Φ₀𝑗−𝑘=_𝑛∈ℤ𝜹^𝑇(𝑝)_{0,𝑛𝑘∈ℤ}𝑃_𝑘Φ₀𝑚𝑚⁻¹=𝑗−𝑛−𝑘_𝑛∈ℤ𝜹^𝑇(𝑝)_0,𝑛𝑄_𝑃Φ₀𝑚⁻¹𝑗−𝑛=𝐞^𝑇_𝑝𝑄_𝑃Φ₀𝑚⁻¹𝑗, (19) and 𝑓^(𝑝)_𝑛,𝑗=_𝑛∈ℤ𝜹^𝑇(𝑝)_0,𝑛𝑄^𝑛_𝑃Φ₀𝑗−𝑛=𝐞^𝑇_𝑝𝑄^𝑛_𝑃Φ₀𝑚^−𝑛𝑗. (20) Then, our declaration follows by a standard method in analysis. Note that the convergence of cascade algorithm with finite supported mask is equivalent to the existence of compactly supported solution of (15) (see [14]). We can obtain the desired result according to the proof of [11, Proposition 1.3].

Following [10, 11], we give the definitions of polynomial generation, polynomial reproduction, and stepwise polynomial reproduction with respect to vector subdivision scheme. We denote by Π_𝑘 the space of polynomials up to degree 𝑘. Suppose 𝑝={𝑝_,𝑘,𝑘∈ℤ}∶={𝑝(𝑡_𝑘),𝑘∈ℤ}, for ∈ℕ, and let ((ℤ))^𝑠 be the space of 𝑠-vector sequence indexed by ℤ.

Definition 4 (polynomial generation). A convergent stationary vector subdivision scheme 𝐒_{𝑃,𝐡0,Φ0} generates polynomials up to degree 𝑛_𝐺 (Π_𝑛𝐺-generating), if, for any polynomial 𝑝(𝑥)∈Π_𝑛𝐺, there exists some initial data 𝐝₀∈((ℤ))^𝑠 such that 𝑔_𝐝0=𝑝(𝑥).

Definition 5 (polynomial reproduction). A convergent stationary vector subdivision scheme 𝐒_{𝑃,𝐡0,Φ0} with parameter values 𝑡⁽⁾,∈ℕ is reproducing polynomials up to degree 𝑛_𝑅 (Π_𝑛𝑅-reproducing), if, for the initial data 𝐝₀={(𝑝∗𝐡₀)(𝑡_𝑘⁽⁰⁾),𝑘∈ℤ}, with any polynomial 𝑝(𝑥)∈Π_𝑛𝑅, the limit of the vector subdivision scheme satisfies 𝑔_𝐝0=𝑝(𝑥).

Definition 6 (stepwise polynomial reproduction). A  convergent stationary vector subdivision scheme 𝐒_{𝑃,𝐡0,Φ0} with parameter values 𝑡⁽⁾,∈ℕ is stepwise reproducing polynomials up to degree 𝑛_𝑅 (stepwise Π_𝑛𝑅-reproducing), if, for the data 𝐝={(𝑝∗𝐡₀)(𝑘),𝑘∈ℤ}, with any polynomial 𝑝(𝑥)∈Π_𝑛𝑅, 𝐝^𝑇₊₁=𝐒_𝑃𝐝^𝑇𝑝or,equivalently,₊₁∗𝐡^𝑇₀(𝑘)=_𝑗∈ℤ𝐝^𝑇𝑡_𝑗⁽⁾𝑃_{𝑘−𝑚𝑗},𝑘∈ℤ. (21)

Noting basic limit function Φ defined by (14), it is easy to show that 𝑔_𝐝0=_𝑘∈ℤ𝐝^𝑇_0,𝑘Φ(⋅−𝑘). (22) Thus the equivalence of polynomial generation and accuracy (approximation order, sum rulers) is clear; see [6, 7]. The following proposition shows that for a nonsingular convergent vector subdivision scheme the concepts of polynomial reproduction and stepwise polynomial reproduction are equivalent.

Proposition 7. A nonsingular convergent vector subdivision scheme is stepwise Π_𝑛𝑅 reproducing if and only if it is Π_𝑛𝑅 reproducing.

Proof. Since that 𝐡^𝑇₀(𝑧)𝐲(𝑧)=1, then by discrete Fourier transform we have ∑_𝑢∈𝕊𝐡^𝑇_{0,𝑛−𝑢}𝐲_𝑢=𝛿_𝑛. Thus under the stepwise polynomial reproduction, from (6) we can easily get, for 𝑝(𝑥)∈Π_𝑛𝑅, 𝑓_,𝑘=_𝑗∈𝕊𝐝^𝑇_,𝑘−𝑗Φ₀𝑗=_𝑗∈𝕊𝑝∗𝐡^𝑇₀𝑘−𝑗𝐲_𝑗=_𝑗∈𝕊𝑝𝑡_{𝑘−𝑗𝑢∈𝕊}𝐡^𝑇_0,𝑢𝐲_𝑗−𝑢=𝑝𝑡_𝑘,𝑘∈ℤ,∈ℕ. (23) As a result, 𝑓_,𝑘=𝑝(𝑡_𝑘) implies 𝑓_+1,𝑘=𝑝(𝑡_𝑘⁺¹). The rest of necessary part one may refer to [11, Proposition 1.7]. For the sufficient part, noting (6) and (7), we have 𝑓_,𝑘=_𝑗∈𝕊𝐝^𝑇_,𝑘−𝑗𝐲_𝑗,𝑘∈ℤ,∈ℕ. (24) Consider 𝑝(𝑥)∈Π_𝑛𝑅. Taking 𝑓_,𝑘=𝑝(𝑡_𝑘) and using the same methods of [11, Proposition 2.1], one obtains from (6) that 𝑓_+1,𝑘=𝑝𝑡_𝑘⁺¹. (25) Noting that 𝐡₀(𝑧)^𝑇𝐲(𝑧)=1, we deduce from 𝑓_,𝑘=𝑝(𝑡_𝑘) that 0=_𝑗∈𝕊𝐝^𝑇_,𝑘−𝑗𝐲_𝑗−_𝑛𝑝𝑡_𝑘−𝑛𝛿_𝑛=_𝑗∈𝕊𝐝^𝑇_,𝑘−𝑗𝐲_𝑗−_𝑛𝑝𝑡_{𝑘−𝑛𝑗∈𝕊}𝐡^𝑇_{0,𝑛−𝑗}𝐲_𝑗=_𝑗∈𝕊𝐝^𝑇_,𝑘−𝑗−𝑝∗𝐡^𝑇₀𝑘−𝑗𝐲_𝑗≜_𝑗∈𝕊𝐱_,𝑘−𝑗𝐲_𝑗, (26) where 𝐱_,𝑘=[𝑥_𝑘,1,𝑥_𝑘,2,…,𝑥_𝑘,𝑠]. Similarly, the equation 𝑓_+1,𝑘=𝑝(𝑡_𝑘⁺¹) implies _𝑗∈𝕊𝐝^𝑇_{+1,𝑘−𝑗}−𝑝₊₁∗𝐡^𝑇₀𝑘−𝑗𝐲_𝑗=0,𝑘∈ℤ. (27) Following Lemma 1, we select linearly independent vectors  𝐲^𝑡(𝑧) as (10) satisfying 𝐡₀(𝑧)^𝑇𝐲^𝑡(𝑧)=1,𝑡=1,…,𝑠. Then, [0,…,0]=_𝑗∈𝕊𝐱_,𝑘−𝑗𝐲¹_𝑗,…,𝐲^𝑠_𝑗=_{𝑗∈𝕊⧵{0}}𝐱_,𝑘−𝑗0𝐫_𝑗𝟎𝑂_𝑠−1𝑌+𝑥_𝑘,1,…,𝑥_𝑘,𝑠1𝐫₀𝟎𝐼_𝑠−1𝑌=𝑥_𝑘,1,𝑥_𝑘,2,…,𝑥_𝑘,𝑠𝑌+0,_𝑗∈𝕊𝑥_{𝑘−𝑗,1}𝐫_𝑗𝑌,𝑘∈ℤ, (28) where 𝑌=[𝐲¹,…,𝐲^𝑠]. Since 𝑌 is an invertible matrix, then 𝑓_,𝑘=𝑝(𝑡_𝑘) is equivalent to 𝑥_𝑘,1=𝑥_𝑘,2=⋯=𝑥_𝑘,𝑠=0. The conclusion follows directly.

3. Algebraic Condition for Polynomial Reproduction

In this section we firstly consider the nonsingular subdivision scheme with standard subdivision symbol 𝑃(𝑧) in (33) and provide algebraic conditions on 𝑃(𝑧) for checking the polynomial reproduction. With the two-scale similarity transform (TST) in the hand, we further discuss the properties of polynomial reproduction with respect to the general subdivision symbol satisfying certain order of sum rules. For TST, see [6, 13, 15, 16] and references therein.

We denote the subsymbols of a vector subdivision symbol 𝑃(𝑧) by 𝑃^𝛼(𝑧)=_𝑘∈ℤ𝑃_{𝑚𝑘+𝛼}𝑧^{𝑚𝑘+𝛼},𝛼=0,…,𝑚−1,𝑧∈ℂ⧵{0}, (29) and remark that the 𝑗th derivative of a subsymbol is 𝐷^(𝑗)𝑃^𝛼(𝑧)=_𝑘∈ℤ𝑞_𝑗(𝑚𝑘+𝛼)𝑧^{𝑚𝑘+𝛼−𝑗}, (30) where 𝑞_𝑗∈𝜋_𝑗 is the polynomial 𝑞_𝑗(𝑥)=^𝑗−1_𝑢=0(𝑥−𝑢). (31) We further assume that 𝑃(𝑧)=𝑃₁₁(𝑧)𝑃₁₂(𝑧)𝑃₂₁(𝑧)𝑃₂₂(𝑧), (32) where 𝑃₁₁(𝑧), 𝑃₁₂(𝑧), 𝑃₂₁(𝑧), and 𝑃₂₂(𝑧) are 1×1, 1×(𝑠−1), (𝑠−1)×1, and (𝑠−1)×(𝑠−1) matrices of Laurent polynomials, respectively. The subsymbols of 𝑃₁₁(𝑧), (𝑃₁₂(𝑧)) are defined by 𝑃^𝛼₁₁(𝑧), (𝑃^𝛼₁₂(𝑧)),  𝛼=0,…,𝑚−1. We define standard subvision symbol with sum rules of order 𝑛 by 𝑃(𝑧)=1+𝑧+⋯+𝑧^𝑚−1𝑛𝑄₁₁(𝑧)1−𝑧^𝑚𝑛𝑄₁₂(𝑧)𝑃₂₁(𝑧)𝑃₂₂(𝑧), (33) where 𝑄₁₁(1)=1/𝑚^𝑛−1 and 𝑄₁₁(𝑧), 𝑄₁₂(𝑧) are 1×1, 1×(𝑠−1) matrices, respectively. Similar to [10, Lemma 2.1] or [11, Proposition 2.1], we have the following proposition.

Proposition 8. The following are equivalent.(i)The vector subdivision symbol 𝑃(𝑧) takes the form of (33).(ii)The entries of 𝑃(𝑧) satisfy 𝑃₁₁(1)=𝑚,𝐷^(𝑗)𝑃₁₁(𝜖)=0,𝑗=0,…,𝑛−1,𝜖∈𝐸^′;𝐷^(𝑗)𝑃₁₂(𝜖)=𝟎,𝑗=0,…,𝑛−1,𝜖∈𝐸, (34) with 𝟎∈ℝ^𝑠−1, where 𝐸={𝑒^{−𝑖(2𝜋𝑘/𝑚)},𝑘=0,…,𝑚−1} and 𝐸^′=𝐸⧵{1}.(iii)The subsymbols of the entries of 𝑃(𝑧) satisfy 𝑚𝐷^(𝑗)𝑃^𝛼₁₁(1)=𝐷^(𝑗)𝑃₁₁(1)𝐷,𝑗=0,…,𝑛−1,𝛼=0,…,𝑚−1;_(𝑗)𝑃^𝛼₁₂(1)=𝟎,𝑗=0,…,𝑛−1,𝛼=0,…,𝑚−1. (35)(iv)The coefficients of the entries of 𝑃(𝑧) satisfy 𝑚_𝑘∈ℤ𝑞_𝑗(−𝑚𝑘)𝑃_{11,𝛼−𝑚𝑘}=𝐷^(𝑗)𝑃₁₁(1),_𝑘∈ℤ𝑞_𝑗(−𝑚𝑘)𝑃_{12,𝛼−𝑚𝑘}=𝟎,∈ℤ,𝑗=0,…,𝑛−1,𝛼=0,…,𝑚−1. (36)

Similar to [10, Lemma 4.2] or [11, Proposition 2.5], we also have the following proposition. It allows us to express the polynomial reproduction of  𝐒_{𝑃,𝐡0,Φ0} in terms of the properties of its symbol, where 𝑃(𝑧) takes the form of (33).

Proposition 9. Let 𝑛∈ℕ, 𝜏∈ℝ, and 𝑞_𝑗 as in (31). A vector subdivision symbol 𝑃(𝑧) satisfies 𝐷^(𝑗)𝑃₁₁(1)=𝑚𝑞_𝑗(𝜏),𝐷^(𝑗)𝑃₁₁(𝜖)=0,𝑗=0,…,𝑛,𝜖∈𝐸^′,𝐷^(𝑗)𝑃₁₂(𝜖)=𝟎,𝑗=0,…,𝑛,𝜖∈𝐸, (37) if and only if _𝑘∈ℤ𝑃_{11,𝑢−𝑚𝑘}𝑘^𝑗=𝑢−𝜏𝑚^𝑗,𝑢∈ℤ,𝑗=0,…,𝑛,_𝑘∈ℤ𝑃_{12,𝑢−𝑚𝑘}𝑘^𝑗=𝟎,𝑢∈ℤ,𝑗=0,…,𝑛. (38)

3.1. Standard Subdivision Symbol

Suppose that subdivision symbol has the standard form (33) in this subsection. Then the constant vector 𝐡₀ and initial function Φ₀(𝑥) in 𝐒_{𝑃,𝐡0,Φ0} can be chosen by 𝐡₀=𝐞₁,Φ₀(𝑥)=𝐲𝒳(𝑥), (39) respectively, where 𝐲 satisfies 𝐡^𝑇₀𝐲=1. We provide a simple algebraic condition for determining 𝜏∈ℝ, which appears in (11) and guarantees the reproduction of linear polynomial. In Theorem 11, we then provide algebraic conditions on 𝑃(𝑧) for checking the reproduction of polynomials of higher degree.

Proposition 10. Let 𝐡₀, Φ₀ be defined by (39) and let 𝑃 take the form of (33) with 𝑛=2. Let 𝐒_{𝑃,𝐡0,Φ0} be a nonsingular vector subdivision scheme. Then, 𝐒_{𝑃,𝐡0,Φ0} reproduces linear polynomials if and only if its parameter values are given by (11) with 𝜏=𝐷𝑃₁₁(1)𝑚. (40)

Proof. We follow the lines of Theorem 3.1 of [10] or Proposition 2.3 of  [11]. According to Proposition 7, polynomial reproduction is equivalent to stepwise polynomial reproduction. Hence it is sufficient to show that 𝐝_,𝑘=𝐡₀𝑝(𝑡_𝑘) implies 𝐝_+1,𝑘=𝐡₀𝑝(𝑡_𝑘⁺¹),𝑘∈ℤ, for any ∈ℕ. Note that any convergent vector subdivision scheme reproduces constants; we only consider the monomial 𝑝(𝑥)=𝑥. Let ∈ℕ and 𝐝_,𝑘=𝐡₀𝑝(𝑡_𝑘)=𝐡₀⋅(𝑡₀+(𝑗/𝑚)),𝑗∈ℤ. For any 𝑘∈ℤ and 𝛼=0,…,𝑚−1, we have 𝐝^𝑇_{+1,𝑚𝑘+𝛼}=_𝑗𝐝^𝑇_,𝑗𝑃_{𝑚𝑘+𝛼−𝑚𝑗}=_𝑗𝐝^𝑇_,𝑘−𝑗𝑃_{𝑚𝑗+𝛼}=_𝑗𝐡^𝑇₀𝑡₀+𝑘−𝑗𝑚𝑃_{𝑚𝑗+𝛼}=_𝑗𝐡^𝑇₀𝑡₀+𝑚𝑘+𝛼𝑚⁺¹𝑃_{𝑚𝑗+𝛼}−_𝑗𝐡^𝑇₀𝑡₀+𝑚𝑗+𝛼𝑚⁺¹𝑃_{𝑚𝑗+𝛼}=𝐡^𝑇₀𝑡₀+𝑚𝑘+𝛼𝑚⁺¹−𝐡^𝑇₀𝐃𝑃(1)𝑚⁺¹,𝐝^𝑇_{+1,𝑚𝑘+𝛼}=𝐡^𝑇₀𝑡₀−𝐡^𝑇₀𝜏𝑚⁺¹+𝐡^𝑇₀𝑚𝑘+𝛼𝑚⁺¹. (41)
As a result, the vector subdivision scheme reproduces linear polynomials stepwisely if and only if 𝐡^𝑇₀𝜏=𝐡^𝑇₀𝐷𝑃(1). (42) Then, (40) follows directly.

Theorem 11. Let 𝑛_𝑅∈ℕ₀. A convergent subdivision scheme with symbol 𝑃(𝑧) in (33) and associated parameterization in (11), for some 𝜏∈ℝ, reproduces polynomials of degree up to 𝑛_𝑅 if and only if 𝐷^(𝑘)𝑃₁₁(1)=𝑚𝑞_𝑘(𝜏),𝐷^(𝑘)𝑃₁₁(𝜖)=0,𝑘=0,…,𝑛_𝑅,𝜖∈𝐸^′,𝐷^(𝑘)𝑃₁₂(𝜖)=𝟎,𝑘=0,…,𝑛_𝑅,𝜖∈𝐸. (43)

Proof. The proof is induction. In the case 𝑘=0 the conclusion follows by Proposition 8. By Proposition 7, it suffices to prove the results for the step polynomial reproduction.
“⇐:” Let 𝑘≤𝑛_𝑅 and >0. Suppose that 𝑝(𝑥)=𝑐_𝑘𝑥^𝑘+𝑝(𝑥), with 𝑝∈Π_𝑘−1. We show that 𝐝=𝐡₀𝑝𝑡_𝑗=𝐡₀𝑐_𝑘𝑡_𝑗^𝑘+𝐡₀𝑝𝑡_𝑗,𝑗∈ℤ (44) satisfies 𝐝^𝑇₊₁=𝑆_𝑃𝐝^𝑇={𝐡^𝑇₀𝑝(𝑡_𝑗⁺¹),𝑗∈ℤ}. Actually by induction and stepwise polynomial reproducing, we have 𝐝^𝑇_+1,𝑗=_𝑢∈ℤ𝐝^𝑇_,𝑢𝑃_{𝑗−𝑚𝑢}=_𝑢∈ℤ𝐡^𝑇₀𝑐_𝑘𝑡₀+𝑢𝑚^𝑘𝑃_{𝑗−𝑚𝑢}+𝑝𝑡_𝑗⁺¹=_𝑢∈ℤ𝐡^𝑇₀𝑐_𝑘^𝑘_𝑝=0𝑘𝑝𝑢𝑚^𝑝𝑡₀^𝑘−𝑝𝑃_{𝑗−𝑚𝑢}+𝑝𝑡_𝑗⁺¹=𝐡^𝑇₀𝑐_𝑘^𝑘_𝑝=0𝑡₀^𝑘−𝑝_𝑢∈ℤ𝑘𝑝𝑢𝑚^𝑝𝑃_{𝑗−𝑚𝑢}+𝑝𝑡_𝑗⁺¹. (45) By Proposition 9, (43) is equivalent to 𝐡^𝑇_0𝑢∈ℤ𝑢^𝑝𝑃_{𝑗−𝑚𝑢}=_𝑢∈ℤ𝑢^𝑝𝑃_{11,𝑗−𝑚𝑢},𝑃_{12,𝑗−𝑚𝑢}=𝐡^𝑇₀𝑗−𝜏𝑚^𝑝. (46) Then, 𝐝^𝑇_+1,𝑗=𝐡^𝑇₀𝑐_𝑘^𝑘_𝑝=0𝑘𝑝𝑡₀^𝑘−𝑝𝑗−𝜏𝑚^+1𝑝+𝐡^𝑇₀𝑝𝑡_𝑗⁺¹=𝐡^𝑇₀𝑐_𝑘𝑡₀+𝑗−𝜏𝑚^+1𝑘+𝐡^𝑇₀𝑝𝑡_𝑗⁺¹=𝐡^𝑇₀𝑐_𝑘𝑡_𝑗^+1𝑘+𝐡^𝑇₀𝑝𝑡_𝑗⁺¹. (47)
“⇒:” Assume that the vector subdivision scheme is Π_𝑛𝑅 reproducing. Let 𝑝(𝑥)=𝑥^𝑘(𝑘≤𝑛_𝑅) and 𝐝={𝐡₀𝑝(𝑡_𝑗),𝑗∈ℤ}, with 𝐡₀=𝐞₁. Using the same method as above, we have 𝐝^𝑇_+1,𝑗=𝐡^𝑇₀^𝑘_𝑝=0𝑘𝑝𝑡₀^𝑘−𝑝_𝑢∈ℤ𝑢𝑚^𝑝𝑃_{𝑗−𝑚𝑢}. (48) On the other hand, 𝑝𝑡_𝑗⁺¹=^𝑘_𝑝=0𝑘𝑝𝑡₀^𝑘−𝑝𝑗−𝜏𝑚^+1𝑝. (49) Combining the above two equations, we deduce ^𝑘_𝑝=0𝑘𝑝𝑡₀^𝑘−𝑝1𝑚^𝑝_𝑢∈ℤ𝑢^𝑝𝐡^𝑇₀𝑃_{𝑗−𝑚𝑢}−𝐡^𝑇₀𝑗−𝜏𝑚^𝑝=0. (50) By induction, we obtain _𝑢∈ℤ𝑢^𝑘𝐡^𝑇₀𝑃_{𝑗−𝑚𝑢}−𝐡^𝑇₀𝑗−𝜏𝑚^𝑘=0. (51) As a result, by Proposition 9 the conclusion (43) follows.

3.2. General Subdivision Symbol

We devote this section to discussing the polynomial reproduction of vector subdivision scheme with general subdivision symbol. Suppose that subdivision symbol 𝑃(𝑧) has sum rules of order 𝑛 (see [6, 7]), that is, if there exists a row vector 𝐲(𝜉) of trigonometric polynomials such that 𝐲(0)≠0 and [𝐲(𝑑⋅)𝑃(⋅)]^(𝑗)2𝜋𝑘𝑚=𝛿_𝑘𝐲^(𝑗)(0),𝑗=0,…,𝑛−1;𝑘=0,…,𝑚−1. (52)

In [16, Theorem 2.2], they showed that subdivision symbol 𝑃(𝜉) satisfies the sum rules of order 𝑛, if and only if there exists an invertible 𝑠×𝑠 matrix 𝑈(𝑧)=1_𝑗∈𝑆𝐫_𝑗𝑧^𝑗𝟎𝐼_𝑠−1𝑈⁰ (53) such that 𝑃(𝑧)=𝑈(𝑧^𝑚)𝑃(𝑧)𝑈⁻¹(𝑧) with 𝑃(𝑧) taking the form (33), where 𝑈⁰ is a constant invertible matrix. It is evident that 𝑈⁻¹(𝑧)=𝑉(𝑧)∶=𝑉⁰1−_𝑗∈𝑆𝐫_𝑗𝑧^𝑗𝟎𝐼_𝑠−1 (54) with 𝑉⁰𝑈⁰=𝐼_𝑠.

Let the constant vector 𝐡₀ and initial function Φ₀(𝑥) in 𝐒_{𝑃,𝐡0,Φ0} be defined by (39). Then we choose 𝐡₀ and Φ₀(𝑥) in 𝐒_{𝐡𝑃,0,Φ0} by 𝐡^𝑇₀(𝑧)=𝐡^𝑇₀𝑉(𝑧),Φ₀(𝑥)=_𝑗∈𝕊𝐲_𝑗𝒳𝑥−𝑗, (55) with 𝐲(𝑧)=𝑈(𝑧)𝐲. It is easy to see that 𝐡₀(𝑧)∈𝔏(ℝ^𝑠),  𝐲(𝑧)∈𝔏^′(ℝ^𝑠), and 𝐡^𝑇₀(𝑧)𝐲(𝑧)=1.

Theorem 12. The vector subdivision scheme 𝐒_{𝐡𝑃,0,Φ0} is Π_𝑛𝑅 reproducing, if and only if  𝐒_{𝑃,𝐡0,Φ0} is Π_𝑛𝑅 reproducing.

Proof. By Proposition 7, we only need to consider the equivalence under the condition of stepwise Π_𝑛𝑅 reproducing. We just show the sufficient part (the necessary part can be proved similarly).
Suppose 𝑝(𝑥)∈Π_𝑛𝑅 and 𝐝={𝐝_,𝑘∶=𝐡₀𝑝(𝑡_𝑘⁽⁾),𝑘∈ℤ}, where ∈ℕ and parameter values 𝑡_𝑘⁽⁾ are defined by (11). By the definition of stepwise polynomial reproduction, we have 𝐝_+1,𝑘=_𝑗∈ℤ𝐝^𝑇𝑡_𝑗⁽⁾𝑃_{𝑘−𝑚𝑗},𝑘∈ℤ. (56) Since 𝑉(𝑧)𝑈(𝑧)=1, then we get ∑_𝑢∈𝕊𝑉_𝑢𝑈_𝑗−𝑢=𝛿_𝑗𝐼_𝑠, where 𝑈_𝑗,𝑗∈ℤ are the coefficients of  𝑈(𝑧) with 𝑈_𝑗=𝑂_𝑠,𝑗∉𝕊 and 𝑉_𝑗,𝑗∈ℤ are defined similarly. For the new scheme 𝐒_{𝐡𝑃,0,Φ}, we have 𝐝^𝑇_,𝑗=𝐩∗𝐡^𝑇₀𝑗=_𝑞∈𝕊𝐡^𝑇₀𝑝𝑡_𝑗−𝑞𝑉_𝑞. (57) Then 𝐒_𝑃𝑑^𝑇_𝑘=_𝑗∈ℤ𝐝^𝑇𝑡_𝑗⁽⁾𝑃_{𝑘−𝑚𝑗}=_𝑗∈ℤ𝐝^𝑇𝑡_𝑗⁽⁾_{𝑣∈ℤ𝑢∈𝕊}𝑈_𝑢𝑃_{𝑣−𝑚𝑢}𝑉_{𝑘−𝑚𝑗−𝑣}=_{𝑣∈ℤ𝑢∈𝕊𝑗∈ℤ𝑞∈𝕊}𝐡^𝑇₀𝑝𝑡_𝑗−𝑞𝑉_𝑞𝑈_𝑢𝑃_{𝑣−𝑚𝑢}𝑉_{𝑘−𝑚𝑗−𝑣}=_{𝑣∈ℤ𝑢∈ℤ𝑗∈ℤ}𝐡^𝑇₀𝑝𝑡_{𝑗𝑞∈𝕊}𝑉_𝑞𝑈_{𝑢−𝑗−𝑞}𝑃_{𝑣−𝑚𝑢}𝑉_𝑘−𝑣=_{𝑣∈ℤ𝑢∈ℤ}𝐡^𝑇₀𝑝𝑡_𝑢𝑃_{𝑣−𝑚𝑢}𝑉_𝑘−𝑣. (58) Here, necessary variable substitution is used. Following the stepwise polynomial reproduction of 𝐒_{𝑃𝐡0,Φ0}, we know that 𝐒_𝑃𝑑_𝑘=_𝑣∈ℤ𝐡^𝑇₀𝑝𝑡_𝑣⁺¹𝑉_𝑘−𝑣=𝐝₊₁𝑡_𝑘⁺¹. (59) Thus the proof is completed.

4. Examples

Example 1 (see [17, Example 4.2], GHM refinable mask). The vector subdivision scheme 𝐒_{𝐡𝑃,0,Φ0} is based on the following mask symbol, 𝑃(𝑧)=12³_𝑘=0𝑃_𝑘𝑧^𝑘=1()√40121+𝑧162−√2(𝑧+1)𝑧²−10𝑧+1−6+20𝑧−6𝑧². (60)

It is not difficult to find that the subdivision mask 𝑃 satisfies the sum rules of order 2. The corresponding vector 𝑦(𝑧) in (52) is selected by 𝑦(𝑧)=√221+𝑧⁻¹𝑧⁻¹. (61) We construct a trigonometric matrix 𝑈(𝑧) by 𝑈(𝑧)=√22()1√3−𝑧2−𝑧2. (62) Then, 𝑃(𝑧) takes the form of (33), (1+𝑧)²𝑄₁₁(𝑧)1−𝑧²²𝑄₁₂(𝑧)𝑄₂₁(𝑧)𝑄₂₂(𝑧) (63) with 𝑄₁₁(𝑧)=1𝑧10³−14𝑧²;𝑄+20𝑧−2₁₂(𝑧)=√2𝑧20²;𝑄−14𝑧+22₂₁(𝑧)√=−2𝑧10³−12𝑧²;𝑄−5𝑧₂₂(𝑧)1=−𝑧10⁴−14𝑧³+22𝑧²−5. (64) Using Proposition 10 we get that, for 𝜏=0, the scheme reproduces linear polynomials. Since 𝐷⁽²⁾𝑃₁₁(1)=−59/5≠2𝑞₂(𝜏)=0, the scheme cannot reproduce polynomials of degree 2. For their graphs, see Figure 1.

(a)

(b)

Figure 1 

Polynomial reproduction of scheme 𝐒𝐡𝑃,0,Φ0 with scheme mask symbol (60); solid line denotes the true polynomial and asterisk point denotes the discrete data using 5 times iteration of subdivision operator; (a) 𝑝(𝑡)=𝑡; (b) 𝑝(𝑡)=𝑡2.

Example 2 (see [17, Example 5]). The vector subdivision scheme 𝐒_{𝐡𝑃,0,Φ0} is based on the mask symbol 𝑃(𝑧) with 𝑃₁₁(𝑧)=10−3√1080(1+𝑧)1+38+12√10𝑧+𝑧²,𝑃₁₂(𝑧)=5√6−2√1580(1−𝑧)(1+𝑧)²,𝑃₂₁(𝑧)=5√6−3√1580(1−𝑧)1−103+√3𝑧+𝑧²,𝑃₂₂(𝑧)=5−3√101040(1+𝑧)13−10+6√10𝑧+13𝑧². (65)

The subdivision mask 𝑃 satisfies the sum rules of order 3; see [17]. The corresponding vector 𝑦(𝑧) in (52) is selected by 𝑦(𝑧)=−2+√10√121+10𝑧²+4+√10𝑧×√−16−2+√10𝑧12⁻²−𝑧⁻¹. (66) If 𝑈(𝑧) is chosen by 𝑈(𝑧)=1√15−√6()12,𝑧−1)(𝑧−301 (67) then 𝑃(𝑧)=𝑈(𝑧^𝑚)𝑃(𝑧)𝑈⁻¹(𝑧) takes the form of (33), with  𝑛=3,𝑚=2. Using Proposition 10 we get that, for 𝜏=3/2, the scheme reproduces linear polynomials. Since 𝐷⁽²⁾𝑃₁₁(1)=√10−2≠2𝑞₂(𝜏)=3/2, the scheme cannot reproduce polynomials of degree 2. For their graphs, see Figure 2.

(a)

(b)

Figure 2 

Polynomial reproduction of scheme 𝐒𝐡𝑃,0,Φ0 with scheme mask symbol (65); solid line denotes the true polynomial and asterisk point denotes the discrete data using 5 times iteration of subdivision operator; (a) 𝑝(𝑡)=𝑡; (b) 𝑝(𝑡)=𝑡2.

Example 3 (see [16, Example 5.2]). The vector subdivision scheme 𝐒_{𝐡𝑃,0,Φ0} is based on the following mask symbol: 𝑃(𝑧)=12+𝑧4+134𝑧−8𝑧3𝑧8𝑧−116116𝑧4−1−𝑧16𝑧.16 (68)

The subdivision mask 𝑃 satisfies the sum rules of order 4; see [16] or [18]. If 𝑈(𝑧) is chosen by 𝑈(𝑧)=11𝑧12²−𝑧⁻²−23𝑧−𝑧⁻¹,01 (69) then 𝑃(𝑧) takes the form of (33), (1+𝑧)⁴𝑄₁₁(𝑧)1−𝑧²⁴𝑄₁₂(𝑧)𝑄₂₁(𝑧)𝑄₂₂(𝑧), (70) with 𝑄₁₁(𝑧)=196𝑧⁵𝑧⁶−4𝑧⁵+𝑧⁴+16𝑧³+𝑧²;𝑄−4𝑧+1₁₂(𝑧)=11152𝑧⁷1−𝑧²𝑧⁴−8𝑧³+8𝑧²;𝑄−8𝑧+1₂₁(𝑧)√=−1();𝑄8𝑧𝑧−1)(𝑧+1₂₂(𝑧)1=−96𝑧³𝑧⁶−8𝑧⁵+11𝑧⁴−32𝑧³+11𝑧²−8𝑧+1. (71) Using Proposition 10 we get that, for 𝜏=0, the scheme reproduces linear polynomials. Since 𝐷⁽³⁾𝑃₁₁(1)=𝐷⁽²⁾𝑃₁₁(1)=0, the scheme also can reproduce polynomials up to degree 3. For their graphs, see Figure 3.

(a)

(b)

(c)

(d)

Figure 3 

Polynomial reproduction of scheme 𝐒𝐡𝑃,0,Φ0 with scheme mask symbol (68); solid line denotes the true polynomial and asterisk point denotes the discrete data using 5 times iteration of subdivision operator; (a) 𝑝(𝑡)=𝑡; (b) 𝑝(𝑡)=𝑡2; (c) 𝑝(𝑡)=𝑡3; (d) 𝑝(𝑡)=𝑡4.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant no. 11071152) and the Natural Science Foundation of Guangdong Province (Grant nos. 10151503101000025 and S2011010004511).