Polynomial Reproduction of Vector Subdivision Schemes

and Applied Analysis 3 By induction on n, it is easy to verify that Q n P Φ 0 (x) = ∑ k∈Z P n,k Φ 0 (m n x − k) , n ∈ N, (17) where Φ 0 is defined as (7), P n,k = ∑ j∈Z P 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 P Φ 0 converges to the normalized solutionΦ 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 δ(p) 0 , p = 1, . . . , s, we have f (p) 1,j = ∑ k∈Z δ (p) 1,k T Φ 0 (j − k) = ∑ k∈Z ∑ n∈Z δ (p) 0,n T P k−mn Φ 0 (j − k) = ∑ n∈Z δ (p) 0,n T ∑ k∈Z P k Φ 0 (m (m −1 j − n) − k) = ∑ n∈Z δ (p) 0,n T Q P Φ 0 (m −1 j − n) = eT p Q P Φ 0 (m −1 j) , (19)


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][4][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 2 -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 ( ) from which the data is sampled by convolving with a special sequence of vectors h 0 = {h 0, ∈ R , ∈ S}, where S is some finite index set containing 0, * denotes convolution operator, and denotes transpose operator.
There are many valuable works that were done when scalar subdivision scheme was considered [8][9][10][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 , | | > 1; Charina and Romani [12] extended to the general expanding dilation matrix ∈ Z × 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 ≥ 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.

Background and Notation
A vector subdivision scheme S ,h 0 ,Φ 0 with integer dilation ≥ 2 is given by a finitely supported matrix sequence = { ∈ R × , ∈ Z} (so-called subdivision mask), a constant column vector h 0 = {h 0, ∈ R , ∈ S}, and an initial function Φ 0 ( ). The constant vector h 0 and the initial function Φ 0 are determined by and will be given by (39) or (55) in the next section. The symbols of h 0 and (denoted by h 0 ( ) and ( ), resp.) are given by the Laurent polynomials respectively.
In this paper, we use two vector spaces of Laurent polynomial, L(R ) and L (R ). We say h 0 ( ) ∈ L(R ), if with some h ∈ R and h ̸ = 0, where {r , ∈ S} is some finite For a data sequence {d ∈ R , ∈ Z}, the vector subdivision operator acting on it is defined by Actually the vector subdivision scheme S ,h 0 ,Φ 0 is a recursive algorithm based on the vector subdivision operator and a convolution operator with the initial data d 0 = {d 0, ∈ R : ∈ Z} and the function where X( ) is the characteristic function of interval [0, 1) and y = {y ∈ R , ∈ S} is selected such that y( ) ∈ L (R ) and h 0 ( ) y( ) = 1.
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: We call { ℓ , ∈ Z, ℓ ∈ N} the sequence of parameter values associated with the vector subdivision scheme. Let us define continuous functions ℓ by linearly interpolating the data ℓ, to the parameter values ℓ , Definition 2. If the sequence of continuous functions { ℓ , ℓ ∈ N} converges (in uniform norm), then we define the limit function as We say that d 0 is the limit function of the vector subdivision scheme S ,h 0 ,Φ 0 acting on the initial data d 0 . If = 1 and h 0 = 1, one may find that the vector subdivision scheme is the same as the scalar one in [10]. The limit functions where e denotes th column of the × identity matrix. It is easy to check that Φ = ( 1 , . . . , ) is compactly supported and satisfies the refinement equation In this paper, we consider vector subdivision scheme that is convergent and nonsingular, so that d 0 = 0 if and only if d 0 = 0. Define the linear operator (which is called cascade operator) as follows: Abstract and Applied Analysis 3 By induction on , it is easy to verify that where Φ 0 is defined as (7), Under the convergent condition of cascade algorithm (see [13]), we can prove that Φ 0 converges to the normalized solution Φ of (15) in ( (R)) (1 ≤ ≤ ∞). 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 and ( ) 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, 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 the space of polynomials up to degree . Suppose ℓ = { ℓ, , ∈ Z} := { ( ℓ ), ∈ Z}, for ℓ ∈ N, and let (ℓ(Z)) be the space of -vector sequence indexed by Z.
Definition 6 (stepwise polynomial reproduction). A convergent stationary vector subdivision scheme S ,h 0 ,Φ 0 with parameter values (ℓ) , ℓ ∈ N is stepwise reproducing polynomials up to degree Noting basic limit function Φ defined by (14), it is easy to show that 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.

Algebraic Condition for Polynomial Reproduction
In this section we firstly consider the nonsingular subdivision scheme with standard subdivision symbol ( ) in (33) and provide algebraic conditions on ( ) 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 ( ) by and remark that the th derivative of a subsymbol is where ∈ is the polynomial We further assume that (i) The vector subdivision symbol ( ) takes the form of (33).

Standard Subdivision Symbol.
Suppose that subdivision symbol has the standard form (33) in this subsection. Then the constant vector h 0 and initial function Φ 0 ( ) in S ,h 0 ,Φ 0 can be chosen by respectively, where y satisfies h 0 y = 1. We provide a simple algebraic condition for determining ∈ R, which appears in (11) and guarantees the reproduction of linear polynomial. In Theorem 11, we then provide algebraic conditions on ( ) for checking the reproduction of polynomials of higher degree.
Theorem 11. Let ∈ N 0 . A convergent subdivision scheme with symbol ( ) in (33) and associated parameterization in (11), for some ∈ R, reproduces polynomials of degree up to if and only if Proof. The proof is induction. In the case = 0 the conclusion follows by Proposition 8. By Proposition 7, it suffices to prove the results for the step polynomial reproduction. "⇐:" Let ≤ and ℓ > 0. Suppose that ( ) = + ( ), with̃∈ Π −1 . We show that Actually by induction and stepwise polynomial reproducing, we have By Proposition 9, (43) is equivalent to Then, "⇒:" Assume that the vector subdivision scheme is Π reproducing. Let ( ) = ( ≤ ) and d ℓ = {h 0 ( ℓ ), ∈ Z}, with h 0 = e 1 . Using the same method as above, we have On the other hand, Combining the above two equations, we deduce By induction, we obtain As a result, by Proposition 9 the conclusion (43) follows.

Theorem 12. The vector subdivision scheme S̃,h
Proof. By Proposition 7, we only need to consider the equivalence under the condition of stepwise Π reproducing. We just show the sufficient part (the necessary part can be proved similarly).
Abstract and Applied Analysis 7

Examples
Example 1 (see [17,Example 4.2], GHM refinable mask). The vector subdivision scheme S̃,h 0 ,Φ 0 is based on the following mask symbol, It is not difficult to find that the subdivision masks atisfies the sum rules of order 2. The corresponding vector ( ) in (52) is selected by We construct a trigonometric matrix ( ) by Then, ( ) takes the form of (33), Using Proposition 10 we get that, for = 0, the scheme reproduces linear polynomials. Since (2) 11 (1) = −59/5 ̸ = 2 2 ( ) = 0, the scheme cannot reproduce polynomials of degree 2. For their graphs, see Figure 1.