Approximation Properties and Construction of Hermite Interpolants ...
Approximation Properties and Construction of Hermite Interpolants and Biorthogonal Multiwavelets
Bin Han † Abstract. Multiwavelets are generated from refinable function vectors by using multiresolution analysis. In this paper we investigate the approximation properties of a multivariate refinable function vector associated with a general dilation matrix in terms of both the subdivision operator and the order of sum rules satisfied by the matrix refinement mask. Based on a fact about the sum rules of biorthogonal multiwavelets, a construction by cosets (CBC) algorithm is presented to construct biorthogonal multiwavelets with arbitrary order of vanishing moments. More precisely, to obtain biorthogonal multiwavelets, we have to construct primal and dual masks. Given any primal matrix mask a and a general dilation matrix M , the proposed CBC algorithm reduces the construction of all dual masks of a, which satisfy the sum rules of arbitrary order, to a problem of solving a well organized system of linear equations. We prove in a constructive way that for any given primal mask a with a dilation matrix M and for any positive integer k, we can always construct a dual mask e a of a such that e a satisfies the sum rules of order k. In addition, we provide a general way for the construction of Hermite interpolatory matrix masks in the univariate setting with any dilation factors. From such Hermite interpolatory masks, smooth Hermite interpolants, including the well known piecewise Hermite cubics as a special case, are obtained and are used to construct biorthogonal multiwavelets. As an example, a C 3 Hermite interpolant with support [−3, 3] is presented. Then we shall apply the CBC algorithm to such Hermite interpolatory masks to construct biorthogonal multiwavelets. Several examples of biorthogonal multiwavelets are provided to illustrate the general theory. In particular, a C 1 dual function vector with support [−4, 4] of the piecewise Hermite cubics is given.
1991 Mathematics Subject Classification. 41A25 41A05 65D05 46E35 41A63. Key words and phrases. Biorthogonal multivariate multiwavelets, Hermite interpolants, Hermite Interpolatory mask, accuracy order, sum rules, refinable function vectors. This research was supported in part by NSERC Canada under a postdoctoral fellowship. †Program in Applied and Computational Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544-1000, USA. [email protected], http://www.math.princeton.edu/∼bhan Typeset by AMS-TEX
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§1. Introduction In her celebrated paper [13], Daubechies constructed a family of compactly supported univariate orthogonal scaling functions and their corresponding orthogonal wavelets with the dilation factor 2. Since then wavelets with compact support have been widely and successfully used in various applications such as image compression and signal processing [14]. Each Daubechies wavelet is generated from one scaling function and therefore, is called a scalar wavelet. Though orthogonal wavelets have many desired properties such as compact support, good frequency localization and vanishing moments, they lack symmetry as demonstrated by Daubechies in [14]. However, symmetry of wavelets is a much desired property in applications. Such a property is claimed to produce less visual artifacts than non-symmetric wavelets. To achieve symmetry, several generalizations of scalar orthogonal wavelets have been studied in the literature. For example, biorthogonal wavelets achieve symmetry where orthogonality is replaced with biorthogonality, and multiwavelets achieve both orthogonality and symmetry where one scaling function is replaced with several scaling functions (i.e., a scaling function vector). Wavelets in multidimensional spaces with a general dilation matrix have also been extensively investigated in recent years since in many applications we have to deal with higher dimensional data such as images. Scalar biorthogonal wavelets in both univariate case and multivariate case have been extensively studied in the literature. See [2, 6, 7, 8, 9, 10, 14, 20, 21, 24, 35, 36, 46] and references therein for discussion on scalar biorthogonal wavelets. With symmetry and many other desired properties, scalar biorthogonal wavelets have been found to be more efficient and useful in many applications than the orthogonal ones [14, 43]. To achieve symmetry, another approach is to adopt multiwavelets where a scaling function vector instead of a single scaling function is used. To compare with scalar wavelets, multiwavelets have several advantages such as shorter support and higher vanishing moments. The success of wavelets largely contributes to the short support and high vanishing moments which are competing objectives in the design of wavelets. That is, to obtain a wavelet with higher vanishing moments, it is necessary to enlarge its support. Such advantages of multiwavelets provide new opportunities and choices in the wavelet theory which are impossible to achieve by using scalar wavelets. With more flexible trade-off between high vanishing moments and short support, multiwavelets are particularly attractive in the construction of wavelets on a bounded domain [11] to deal with problems arising from a finite domain with boundary conditions and are expected to be useful in many applications such as numerical solutions to partial differential equations and signal processing [11, 45]. As a generalization of the scalar wavelets, it is also of interest in its own right to investigate multiwavelets. The advantages of multiwavelets and their promising features in applications have attracted a great deal of interest and effort in recent years to extensively study them. To only mention a few references here, see [1, 11, 12, 16, 17, 25, 26, 27, 33, 38, 40, 41, 44, 47, 48] and references therein on discussion of various topics on multiwavelets and their applications. The generalization of scalar wavelets to multiwavelets is not trivial and the study of multiwavelets is much more complicated and involved than the study of the scalar wavelets which we shall see later. Before proceeding further, let us introduce some notation. An s×s integer matrix
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M is called a dilation matrix if limn→∞ M −n = 0. That is, all the eigenvalues of a dilation matrix M are greater than one in modulus. Throughout this paper, M denotes a dilation matrix and m := |detM |. In this paper, we are concerned with the following refinement equation: X φ= a(β)φ(M · −β), (1.1) β∈Zs
where φ = (φ1 , · · · , φr )T is a r × 1 vector of functions, called a refinable function vector, and a is a finitely supported sequence of r × r matrices on Zs , called the (matrix refinement) mask. When r = 1, φ is called a scalar refinable function and a is called a scalar refinement mask. By J a (0) we denote the following matrix associated with a mask a as X J a (0) := a(β). β∈Zs
If φ1 , · · · , φr are functions in L1 (Rs ) with stable shifts and φ = (φ1 , · · · , φr )T satisfies the refinement equation (1.1) with a mask a, then it was proved by Dahmen and Micchelli [12] that J a (0) has a simple eigenvalue m and all other eigenvalues in modulus < m. (1.2) Conversely, if J a (0) satisfies the condition (1.2), it was proved by Heil and Collella [26] and Cabrelli, Heil, and Molter [4] that there exists a unique vector φ of comb = φ(0) b with pactly supported distributions such that φ satisfies (1.1) and J a (0)φ(0) b b kφ(0)k2 = 1 and the first nonzero component of φ(0) being positive. We call such solution the normalized solution of (1.1) and throughout this paper we denote the normalized solution of (1.1) with mask a by φa . If φ is another distribution solution of (1.1), then we must have φ = cφa for some constant c. On the one hand, multiwavelets provide more flexibility and new features which are not possible for scalar wavelets. On the other hand, in the multiwavelet case r > 1, each element in the mask of the refinement equation becomes an r × r matrix comparing with a scalar number in the scalar wavelet case. The change from a mask of scalar numbers to a mask of matrices makes the analysis and investigation of multiwavelets far more complicated and involved than its scalar counterpart. For example, even in the univariate case, the sum rule and vanishing moment conditions of a matrix mask are much more complicated than the scalar case, see [3, 5, 12, 27, 32, 34, 40, 42]. The involvement of matrices in the mask makes the construction of multiwavelets with certain vanishing moments much more challenging than its scalar counterpart. To our best knowledge, no systematic method is proposed in the current literature to construct biorthogonal multiwavelets with arbitrary order of sum rules even for the simplest case — the univariate setting with the dilation factor 2. Many known approaches and constructions in the scalar wavelet case do not apply to the multiwavelet case. Since both biorthogonal multivariate wavelets and multiwavelets are of great interest in both theory and applications, one aim of this paper is to investigate the biorthogonal multiwavelets in multidimensional spaces and to propose a method to construct them systematically. Let `(Zs ) denote the linear space of all sequences on Zs and `0 (Zs ) denote the linear space of all finitely supported sequences on Zs . For any positive integer r,
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¡ ¢r×r by `(Zs ) we denote the linear space of all sequences of r × r matrices on Zs ¡ s ¢r and by `(Z ) we denote the linear space of all sequences of r × 1 vectors on Zs . ¡ ¢r×r ¡ ¢r Similarly, we define `0 (Zs ) and `0 (Zs ) . Given any compactly supported distribution vector f = (f1 , · · · , fr )T , we define the following linear operator cf on ¡ s ¢r `(Z ) by X ¡ ¢r cf (λ) := λ(β)T f (· − β), λ ∈ `(Zs ) , (1.3) β∈Zs
where AT denotes the transpose of a matrix shifts of f are said to be ¡ A. ¢The r linearly independent if cf (λ) = 0 for λ in `(Zs ) implies λ = 0. It was proved by Jia and Micchelli [31] that the shifts of a compactly supported distribution vector ¡ f = ¢(f1 , · · · , fr )T are linearly independent if and only if the sequences fbj (ξ + 2πβ) β∈Zs , j = 1, · · · , r are linearly independent for all ξ ∈ Cs . The shifts of f are ¡ ¢ stable if the sequences fbj (ξ + 2πβ) s , j = 1, · · · , r are linearly independent for β∈Z
all ξ ∈ Rs . Therefore, if the shifts of φa are stable or linearly independent, then (1.2) holds true. Before proceeding further, let us introduce some notation. Recall that M denotes a dilation matrix. Let ΩM be a complete set of representatives of the distinct cosets of Zs /M Zs . Without loss of generality, we assume that 0 ∈ ΩM . For any mask a ¡ ¢r×r in `0 (Zs ) , we shall use the following notation throughout this paper Jεa (µ) :=
X
a(ε + M β)(M −1 ε + β)µ /µ!,
µ ∈ Zs+ , ε ∈ ΩM
(1.4)
β∈Zs
and
J a (µ) :=
X
a(β)(M −1 β)µ /µ! =
β∈Zs
where
X
Jεa (µ),
µ ∈ Zs+ ,
(1.5)
ε∈ΩM
Zs+ := {(β1 , · · · , βs ) ∈ Zs : βj ≥ 0, j = 1, · · · , s}
and for any β = (β1 , · · · , βs ) ∈ Zs and µ = (µ1 , · · · , µs ) ∈ Zs+ , β µ := β1µ1 · · · βsµs , µ! := µ1 ! · · · µs ! and |β| := |β1 | + · · · + |βs |. For any ν = (ν1 , · · · , νs ), µ = (µ1 , · · · , µs ) ∈ Zs+ , we say that ν ≤ µ if νj ≤ µj for all j = 1, · · · , s, and we ¡ ¢r×r say that ν < µ if ν ≤ µ and ν 6= µ. For any mask a in `0 (Zs ) , we say that the mask a with a dilation matrix M satisfies the sum rules of order k if there exists a set of r × 1 vectors {yµ : µ ∈ Zs+ , |µ| < k} with y0 6= 0 such that X
(−1)|ν| Jεa (ν)T yµ−ν =
0≤ν≤µ
X
mµν yν
∀ µ ∈ Zs+ , |µ| < k, ε ∈ ΩM ,
(1.6)
|ν|=|µ|
where the numbers mµν are uniquely determined by X xν (M −1 x)µ = mµν , µ! ν!
x ∈ Rs .
(1.7)
|ν|=|µ|
Given a vector φ = (φ¡1 , · · · ,¢φr )T of compactly supported distributions on Rs , r let S(φ) = {cφ (λ) : λ ∈ `(Zs ) } where the linear operator cφ is defined in (1.3).
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Following Heil, Strang and Strela [27], we say that φ has accuracy order k if Πk−1 ⊆ S(φ) where Πk−1 denotes the set of all polynomials of total degree less than k. We also agree that Π−1 = {0}. Accuracy order of φ has a close relation with both the approximation order provided by φ and the well known Strang-Fix conditions on φ. See Jia [30] for such concepts and related results. When φa is a refinable function vector with a mask a and a dilation matrix M , under the assumption that the shifts of φa are linearly independent, Cabrelli, Heil, and Molter [3, 4] characterized the accuracy order of φa in terms of the order of sum rules satisfied by the mask a. Also see [5, 27, 32, 34, 40] and references therein for discussion on accuracy order in the univariate setting. In Section 2, under a very mild condition, we shall provide another characterization (see Theorem 2.4) of the accuracy order of φa in terms of both the subdivision operator and sum rules. Accuracy order of a refinable function vector is also closely related to the concept of vanishing moments of biorthogonal multiwavelets (see [14, 43]). The success of a wavelet basis largely lies in the short support and high accuracy order of a refinable function vector. It is also known that if a refinable function vector φa ∈ C k has a linearly independent shifts, then ¡ ¢ it is necessary that φ has accuracy order k + 1. s r A function vector φ in L2 (R ) is called a primal function vector if φ satisfies ¡ ¢r×r the refinement equation (1.1) with a mask a in `0 (Zs ) and the shifts of φ e are ¡ linearly ¢r independent. A dual function vector φ of φ is a function vector in L2 (Rs ) such that φe satisfies the refinement equation (1.1) with a mask e a and Z Rs
e + β)T dx = δ(β)Ir φ(x) φ(x
∀ β ∈ Zs ,
(1.8)
where Ir denotes the r×r identity matrix, and δ(0) = 1, δ(β) = 0 for all β ∈ Zs \{0}. Clearly, if φ and φe satisfy the conditions in (1.8), then the shifts of φ and φe are linearly independent, respectively. A necessary condition for φ and φe to satisfy the conditions in (1.8) is the following well known discrete biorthogonal relations: X
a(β) e a(β + M α)T = mδ(α)Ir
∀ α ∈ Zs .
(1.9)
β∈Zs
¡ ¢r×r ¡ ¢r×r If a mask a in `0 (Zs ) satisfies (1.2) and there exists a sequence e a in `0 (Zs ) such that the conditions in (1.9) are satisfied, then we say that a is a primal mask and any such mask e a will be called a dual mask of a. Dahmen and Micchelli proved in [12] that if a is a primal mask with e a being a dual mask of a, then φa is a e a primal function vector with φ being a dual function vector of φa if and only if the subdivision schemes associated with a and e a converge in the L2 norm, respectively. Given a primal mask a, to construct a dual mask of a, we need to solve a system of linear equations given in (1.9). In the current literature, the lifting scheme is known to be a good method for constructing a dual mask of any given primal mask. For discussion on lifting scheme, the reader is referred to Sweldens [46], Daubechies and Sweldens [15], and Kovaˇ cevi´c and Sweldens [36]. We point out that in the univariate setting, Plonka’s factorization technique of a matrix symbol is very useful in studying refinable function vectors [38, 40] and was used by Plonka and Strela in [41] to construct smooth refinable function vectors. This factorization technique was also used by Strela in [44] to construct univariate
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multiwavelets. However, the factorization technique does not apply to the higher dimensions and the biorthogonal multiwavelets constructed by the factorization technique in [44] have very long support. As we mentioned before, the vanishing moments of a biorthogonal multiwavelet are important in both applications and construction of smooth biorthogonal multiwavelets. See [6, 14] for discussion on vanishing moments and their relation to sum rules. For a primal mask a, the lifting scheme can be used to solve the discrete biorthogonal relation (1.9), which is a system of linear equations, to get a dual mask e a of a. To achieve high vanishing moments of the resulting biorthogonal multiwavelet, the dual mask e a must satisfy the sum rules of high order. Even in the simplest case s = 1 and M = (2), it is not easy to use the definition of sum rules given in (1.6) to achieve desired order of sum rules satisfied by e a. The reason is that when r > 1, to obtain a dual mask e a of a given primal mask a, even the equations in the biorthogonal conditions (1.9) are linear, the equations given in (1.6) for sum rules are no longer linear equations since in general the vectors yν in (1.6) are determined by e a. Due to such difficulty, many methods on construction of scalar biorthogonal wavelets no longer hold in the multiwavelet case and not many examples of biorthogonal multiwavelets are available in the literature. For example, univariate multiwavelets were reported by Donovan, Geronimo, Hardin and Massopust [17], Dahmen, Han, Jia and Kunoth [11], He and Lai [25] and other examples were given in [7, 33, 44]. One purpose of this paper is to try to overcome such difficulty. In the scalar case, a construction by cosets (CBC) algorithm was proposed by Han in [21] to construct scalar biorthogonal wavelets with arbitrary vanishing moments. In this paper, we shall generalize the CBC algorithm in [21] to the multiwavelet case to overcome the above mentioned difficulty. For the advantages of the CBC algorithm over other known methods on construction of scalar biorthogonal wavelets, the reader is referred to [6, 21, 24]. In Section 3, we shall follow the line developed in [6, 21, 24] to discuss how to construct biorthogonal multiwavelets in the most general case. We propose a general construction by cosets (CBC) algorithm in Section 3. Such CBC algorithm reduces the construction of all dual masks, which satisfy the sum rules of arbitrary order, of a given primal mask to the problem of solving a well organized system of linear equations. Based on such algorithm, we shall demonstrate that for any given primal mask a with a dilation matrix M and for any positive integer k, we can construct a dual mask e a of a such that e a satisfies the sum rules of order k. In Section 4, we construct a special family of primal masks – Hermite interpolatory masks in the univariate setting with any dilation factor. The resulting refinable function vectors are Hermite interpolants which are useful in curve design in computer aided geometric design [18]. As an example, a C 3 Hermite interpolant is constructed with support [−3, 3]. In particular, such construction of Hermite interpolatory masks includes the piecewise Hermite cubics as a special case. Several new examples of biorthogonal multiwavelets are provided to illustrate the general theory developed in this paper. In particular, a C 1 dual function vector of the piecewise Hermite cubics is given and is supported on [−4, 4]. Finally, by the CBC algorithm, we construct a continuous dual function vector for the primal function vector which is a polynomial B-spline of order 6 with double knots in Plonka and Strela [41]. Such primal function vector belongs to C 4−η for any η > 0, has accuracy order 6 and has support [0, 3].
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§2. Accuracy Order of Refinable Function Vectors In this section, under a very mild condition we shall investigate the accuracy order satisfied by a mask in terms of both the subdivision operator and the sum rules. The motivation of our approach here lies in that it may be useful in the study of various properties associated with the subdivision operators. To do this, let us introduce some notation and several auxiliary results here. ¡ ¢r×r Given a mask a in `0 (Zs ) , the subdivision operator Sa associated with the mask a and a dilation matrix M is defined by X ¡ ¢r Sa λ(α) = a(α − M β)T λ(β), α ∈ Zs , λ ∈ `(Zs ) . (2.1) β∈Zs
Πrk
By we denote the set of r × 1 polynomial vectors with each component being r r a polynomial of degree at most k, Πr := ∪∞ k=0 Πk . For any p ∈ Π , we ¡ and ¢by r s s naturally have a sequence p|Zs ∈ `(Z ) given by p|Zs (α) = p(α), α ∈ Z . Since the mapping from Πr → Πr |Zs is one-to-one and onto, without confusion,¡we may ¢r use p ∈ Πr as both a polynomial vector in Πr and a polynomial sequence in `(Zs ) since it is easy to distinguish them in the In particular, for any p ∈ Πr , ¡ context. ¢r s if Sa (p|Zs ) is a polynomial sequence in `(Z ) , then we may use Sa p to represent ¡ ¢r both the polynomial vector in Πr and the polynomial sequence in `(Zs ) . Lemma 2.1. Let p ∈ Πr |Zs . Then Sa p ∈ Πr |Zs if and only if X X a(ε + M β)T p(x − M −1 ε − β) = a(M β)T p(x − β) β∈Zs
∀ ε ∈ ΩM , x ∈ Rs .
β∈Zs
If this is the case, then Sa p is given by the following polynomial vector X Sa p(x) = a(ε + M β)T p(M −1 x − M −1 ε − β), ε ∈ ΩM .
(2.2)
β∈Zs
Proof. By definition of the subdivision operator Sa , for any ε, α ∈ Zs , we have X Sa p(ε + M α) = aT (ε + M β)p(M −1 (ε + M α) − M −1 ε − β). β∈Zs
The claim in this lemma follows directly from the above equality.
¤
By Dj we denote the partial derivative with respect to the j-th unit coordinate. For any µ = (µ1 , · · · , µs ) ∈ Zs+ , Dµ denotes the differential operator D1µ1 · · · Dsµs . Moreover, we write D = (D1 , · · · , Ds )T . For any polynomial p ∈ Πr , we shall use the following convention: X p(x − iD)T := p(µ) (x)T (−iD)µ /µ!, µ∈Zs+
where i is the imaginary unit such that i2 = −1. For any p ∈ Πr , we observe X £ £ ¤ ¤£ ¤ p(x − iD)T f (·)g(·) = p(µ) (x − iD)T f (·) (−iD)µ g(·) /µ!. (2.3) µ∈Zs+
¡ ¢r×r , let For any mask a in `0 (Zs ) X a(β)e−iβ·ξ /m, H a (ξ) :=
ξ ∈ Rs
β∈Zs
and define
¡ ¢ ¡ ¢ Hka (ξ) := H a (M T )−1 ξ · · · H a (M T )−k ξ ,
k ∈ N.
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Proposition 2.2. If p ∈ Πr , then Sa p ∈ Πr if and only if p(x − iD)T H1a (2πβ0 ) = 0
∀ β0 ∈ Zs \M T Zs .
(2.4)
Moreover, if p ∈ Πr and Sa p ∈ Πr , then p(x − iD)T H1a (2πM T β) = (Sa p)(M x)T for all β ∈ Zs , and for any k > 1, a p(x − iD)T Hka (2πM T β) = (Sa p)(M x − iD)T Hk−1 (2πβ)
∀ β ∈ Zs .
Proof. By the definition of H1a , we have X X 1 £ ¤ −1 p(µ) (x)T a(β)(−iD)µ e−iM β·ξ |ξ=2πβ0 µ! s s
m p(x − iD)T H1a (2πβ0 ) =
β∈Z µ∈Z+
X X 1 −1 p(µ) (x)T a(β)(−M −1 β)µ e−2πiM β·β0 µ! β∈Zs µ∈Zs+ X −1 = p(x − M −1 β)T a(β)e−2πiM β·β0
=
β∈Zs
X
=
e−2πiM
ε∈ΩM
−1
ε·β0
X
pT (x − M −1 ε − β)a(ε + M β).
β∈Zs
Thus, it yields that (2.4) is equivalent to Sa p ∈ Πr by Lemma 2.1. Moreover, the above equality gives us p(x − iD)T¡ H1a (2πM T¢ β) = (Sa p)(M x)T for all β ∈ Zs . a (M T )−1 ξ . By (2.3), we have Note that Hka (ξ) = H1a (ξ)Hk−1 p(x − iD)T Hka (2πM T β) X 1 £ a ¡ T −1 ¢¤ = p(µ) (x − iD)T H1a (2πM T β)(−iD)µ Hk−1 (M ) · (2πM T β). µ! s µ∈Z+
On the other hand, by p(x − iD)T H1a (2πM T β) = (Sa p)(M x)T , we deduce that £ ¤ £ ¤ p(µ) (x − iD)T H1a (2πM T β) = Dµ p(x − iD)T H1a (2πM T β) = Dµ (Sa p)(M ·)T (x). So, in conclusion, we have p(x − iD)T Hka (2πM T β) =
X 1 £ ¤ a Dµ (Sa p)(M ·)T (x)(−iM −1 D)µ Hk−1 (2πβ) µ! s
µ∈Z+
a = (Sa p)(M x − iD)T Hk−1 (2πβ).
¤
¡ ¢r×r and any positive integer k, we define a subspace For any mask a in `0 (Zs ) Pka of Πrk as Pka := {p ∈ Πrk : Saj p ∈ Πrk |Zs ∀ j ∈ N}. (2.5) It is evident that Pka is invariant under the subdivision operator Sa defined in (2.1). We are ready to prove the following result:
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Theorem 2.3. If p ∈ Pka , then cφa (p) ∈ Πk where Pka is defined in (2.5) and the operator cφa is defined in (1.3). In particular, cφa (p)(x) =
X
p(β)T φa (x−β) = p(x−iD)T φˆa (0) =
β∈Zs
X 1 p(µ) (x)T (−iD)φˆa (0). µ! s
µ∈Z+
¡ ¢ Conversely, if the sequences φˆj (2π(M T )−1 ε + 2πβ) β∈Zs , j = 1, · · · , r are linearly independent for all ε ∈ ΩM T where φa = (φ1 , · · · , φr )T is the normalized solution of (1.1) with the mask a, then for any q ∈ Πk ∩ S(φa ), there exists a unique p ∈ Pka such that cφa (p) = q. Proof. Since p ∈ Pka , by definition, Saj p ∈ Πr for all j ∈ N ∪ {0}. We claim that p(x − iD)T φˆa (2πβ) = 0
∀ β ∈ Zs \{0}.
(2.6)
For any β ∈ Zs \{0}, there is a positive integer k such that β ∈ Zs \(M T )k Zs . Note ¡ ¢ that φˆa (ξ) = Hka (ξ)φˆa (M T )−k ξ and p(x−iD)T φˆa (2πβ) =
X 1 £ ¤ £ ¡ ¢¤ Dµ p(x−iD)T Hka (2πβ) (−iD)µ φˆa (M T )−k · (2πβ). µ! s
µ∈Z+
Therefore, to prove (2.6), it suffices to prove that p(x − iD)T Hka (2πβ) = 0
∀ β ∈ Zs \(M T )k Zs , p ∈ Pka , x ∈ Rs .
(2.7)
By Proposition 2.2, (2.7) holds true for k = 1. Suppose is true for ¡ that (2.7) ¢ a k − 1. Note that Hka (ξ) = Hk−1 (ξ)f (ξ) where f (ξ) := H a (M T )−k ξ . By induction hypothesis, we deduce that for any β ∈ Zs \(M T )k−1 Zs , p(x − iD)T Hka (2πβ) =
X 1 £ ¤ a Dµ p(x − iD)T Hk−1 (2πβ) (−iD)µ f (2πβ) = 0. µ! s
µ∈Z+
For any β ∈ Zs \M T Zs , by Proposition 2.2 and induction hypothesis, ¡ ¢ a p(x − iD)T Hka (2π(M T )k−1 β) = (Sa p)(M x − iD)T Hk−1 2π(M T )k−2 β = 0. Therefore, (2.7) holds true and we proved (2.6). By Poisson summation formula (see [5]), we have cφa (p)(x) =
X
p(β)T φa (x − β) =
β∈Zs
= p(x − iD)T φˆa (0) =
X
e2πiβ·x p(x − iD)T φˆa (2πβ)
β∈Zs
X 1 p(µ) (x)T (−iD)µ φˆa (0). µ! s
µ∈Z+
¡ ¢ Conversely, let q ∈ Πk ∩ S(φa ). Since the sequences φˆj (2πβ) β∈Zs , j = 1, · · · , r are linearly independent, by [30, Lemma 8.2], there exists a unique polynomial vector p ∈ Πrk such that cφa (p) = q. Thus, to prove that p ∈ Pka , it suffices to prove
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that if p ∈ Πrk and cφa (p) ∈ Πk , then Sa p ∈ Πrk . By Poisson summation formula, we have X X e2πiβ·x p(x − iD)T φˆa (2πβ) = p(β)T φa (x − β) = cφa (p) ∈ Πk , β∈Zs
β∈Zs
¡ ¢ from which it follows that (2.6) holds true. Since φˆj (2π(M T )−1 ε + 2πβ) β∈Zs , j = 1, · · · , r are linearly¡ independent for all¢ ε ∈ ΩM¡T , there exits βε1 , · · ¢· βεr ∈ Zs such that the matrix [φˆa 2π(M T )−1 ε+2πβε1 , · · · , φˆa 2π(M T )−1 ε+2πβεr ] is invertible. Thus, for ξ in a neighborhood of 0, the following matrix £ ¡ ¢ ¡ ¢¤ F (ξ):= φˆa 2π(M T )−1 ε+2πβε1 +(M T )−1 ξ , · · · , φˆa 2π(M T )−1 ε+2πβεr +(M T )−1 ξ ¡ ¢ is invertible. By φˆa (ξ) = H1a (ξ)φˆa (M T )−1 ξ , we have £ a ¤ φˆ (2πε + 2πM T βεj + ξ) 1≤j≤r = H1a (2πε + ξ)F (ξ). Hence, the following identity is valid for ξ in a neighborhood of 0: ¤ £ H1a (2πε + ξ) = φˆa (2πε + 2πM T βεj + ξ) 1≤j≤r F (ξ)−1 . Therefore, by (2.3), £ ¤ p(x − iD)T H1a (2πε) = p(x − iD)T H1a (2πε + ·) (0) X £ ¡ ¢¤ £ ¤ = p(µ) (x − iD)T φˆa 2π(ε + M T βεj ) 1≤j≤r (−iD)µ F −1 (0). µ∈Zs+
¡ ¢ £ ¢¤ By (2.6), p(µ) (x − iD)T φˆa 2π(ε + M T β) = Dµ p(x − iD)T φˆa (2π(ε + M T β) = 0 for all ε ∈ ΩM T \{0} and β ∈ Zs . So p(x − iD)T H1a (2πε) = 0 for all ε ∈ ΩM T \{0}. By Proposition 2.2, Sa p ∈ Πrk . ¤ The main result in this section is the following theorem. ¡ ¢r×r Theorem 2.4. Let a be a mask in `0 (Zs ) and satisfy the condition (1.2) with a dilation matrix M . Let φa = (φ1 , · · · , φr )T be the normalized solution of the refinement equation (1.1) with the mask¢ a and the dilation matrix M . Suppose that ¡ the sequences φˆj (2π(M T )−1 ε + 2πβ) β∈Zs , j = 1, · · · , r are linearly independent for all ε ∈ ΩM T . Then the following statements are equivalent: (a) φa has accuracy order k; a a a (b) dim Pk−1 = dim Πk−1 where Pk−1 is defined in (2.5) and Pk−1 is invariant under Sa ; a a (c) The mapping cφa |Pk−1 : Pk−1 |Zs → Πk−1 is onto where cφa is defined in (1.3); (d) a satisfies the sum rules of order k. Moreover, if a satisfies the sum rules of order k given in (1.6) with {yµ : µ ∈ Zs+ , |µ| < k} and y0T φˆa (0) = 1, then X X βν xµ = y T φa (x − β) µ! ν! µ−ν s 0≤ν≤µ β∈Z
∀ |µ| < k, µ ∈ Zs+ .
HERMITE INTERPOLANTS AND BIORTHOGONAL MULTIWAVELETS
11
Proof. From Theorem 2.3, we see that for any positive integer k, the mapping a a cφa |Pk−1 : Pk−1 → Πk−1 is well defined. Under the assumption that the sequences ¡ ¢ ˆ a φj (2πβ) β∈Zs , j = 1, · · · , r are linearly independent, by [30, Lemma 8.2], cφa |Pk−1 a is one-to-one. Therefore, by Theorem 2.3, φ has accuracy order k if and only if a a is one-to-one and onto which is also equivalent to dim Pk−1 = dim Πk−1 . cφa |Pk−1 Thus, we proved that (a), (b) and (c) are equivalent. a We now discuss sum rules. Since cφa |Pk−1 is a one-to-one and onto mapping a a between Pk−1 carries the structure of and Πk−1 , the inverse mapping of cφa |Pk−1 a Πk−1 into Pk−1 . Define µ pµ := c−1 φa (x /µ!),
µ ∈ Zs+ , |µ| < k.
Since cφa (Dν pµ ) = Dν cφa (pµ ) for all ν, µ ∈ Zs+ with |µ| < k, we may assume that pµ =
X
yµ−ν
0≤ν≤µ
xν ν!
∀ µ ∈ Zs+ , |µ| < k,
(2.8)
for some r × 1 vectors yν , |ν| < k. Note that cφa (Sa pµ )(M x) = cφa (pµ )(x) = xµ /µ!. We have µ ¶ µ X ¶ ν −1 µ X x) −1 (M −1 µx mν (Sa pµ )(x) = cφa = cφa = mµν pν (x), µ! ν! |ν|=|µ|
|ν|=|µ|
where mµν are given in (1.7). By Lemma 2.1, we have X X mµν pν (M x) = Sa pµ (M x) = a(ε + M β)T pµ (x − M −1 ε − β)
∀ ε ∈ ΩM .
β∈Zs
|ν|=|µ|
Setting x = 0 in the above equality, we have (1.6). Hence, (c) implies (d). To prove that (d) implies (a), we define pµ as in (2.8). By the definition of sum rules and observing that Dν (pµ ) = pµ−ν where by convention pν ≡ 0 for P ν 6∈ Zs+ , it is not difficult to verify that Sa p ∈ Πrk and Sa p = |ν|=|µ| mµν pν for all a |µ| < k. Thus pµ ∈ Pk−1 , and by Theorem 2.3, cφa (pµ ) ∈ Πk−1 for all |µ| < k. Set qµ := cφa (pµ ). Then we have X qµ (M −1 x) = cφa (Sa pµ )(x) = mµν qν (x) ∀ µ ∈ Zs+ , |µ| < k. (2.9) |ν|=|µ|
Note that Dν qµ = qµ−ν for all ν ∈ Zs+ (see [4, Theorem 3.2]). We may assume qµ (x) =
X 0≤ν≤µ
lµ−ν
xν , ν!
µ ∈ Zs+ , |µ| < k,
for some lµ ∈ C, µ ∈ Zs+ with |µ| < k. We claim that lµ = 0 for any 0 < |µ| < k. Suppose that for some 0 < j < k, lν = 0 for all 0 < |ν| < j. Then qµ (x) = l0 xµ /µ! + lµ for any |µ| = j. By (2.9), we have l0
X X (M −1 x)µ xν + lµ = l0 mµν + mµν lν . µ! ν! |ν|=|µ|
|ν|=|µ|
12
BIN HAN
P Therefore, by (1.7), lµ = |ν|=|µ| mµν lν for all |µ| = j. But when j > 1, it was observed in [3] that all the eigenvalues of (mµν )|ν|=j,|µ|=j are less than 1 in modulus. Hence lµ = 0 for all |µ| = j. Now qµ = l0 xµ /µ! for all |µ| < k. By Theorem 2.3, l0 = y0T φˆ0 (0) 6= 0 follows directly from (1.2) and y0 6= 0. ¤ From the proof of Theorem 2.4, we see that in a sense the concept of sum rules provides us a convenient way of describing the subspace Pka of Πrk and Pka is an invariant subspace under Sa . When r = 1, the sum rules condition in (1.6) can be restated as follows: X X a(ε + M β)(ε + M β)µ = a(M β)(M β)µ ∀ |µ| < k, ε ∈ ΩM , (2.10) β∈Zs
β∈Zs
a which was given by Jia [29] and can be easily seen either by Lemma 2.1 and Pk−1 = Πk−1 or by (1.6). The equivalence between (a) and (d) in Theorem 2.4 was given by Cabrelli, Heil and Molter [3, 4] under a much stronger assumption that the shifts of φa are linearly independent. It was also given by Jia [29], Heil, Strang, and Strela [27] and Plonka [40] for the case s = 1 and M = (2). As in [4], from the proof of Theorem 2.4, (d) always implies (a) without the assumption in Theorem 2.4. In passing, we mention that Pka = {p ∈ Πrk : Saj p ∈ Πrk ∀ j = 1, · · · , dim Πrk } since Pka ⊆ Πrk , and if a satisfies the sum rules of order k + 1, then Sa |Pka = (cφa |Pka )−1 τM (cφa |Pka ) where τM : Πk → Πk is given by τM (p)(x) = p(M −1 x). Therefore, the structure of Sa restricted to the subspace Pka can be easily analyzed by using a much simpler operator τM .
§3. Construction of Dual Masks with Arbitrary Order of Sum Rules In this section, we shall discuss how to systematically construct dual masks with arbitrary order of sum rules for any given primal mask. More precisely, given a primal mask a, for any positive integer k, how to find all dual masks of a such that the dual masks satisfy the sum rules of order k. Even in the scalar and multivariate case, this is not a straightforward question and there are a lot of literature discussing it [6, 9, 10, 20, 21, 35, 36, 46]. This problem is also called filter design in the language of engineering [43]. As for the multiwavelet case, even in the univariate setting, to our best knowledge, no systematic method is available in the current literature to deal with this problem. As a matter of fact, it took a relatively long time in the wavelet community to find a continuous dual scaling function vector of the well known piecewise Hermite cubics [11]. In this paper, we shall demonstrate that designing multiwavelets with arbitrary vanishing moments can be reduced to solve a system of well organized linear equations by using a CBC algorithm. As an application of this method, a C 1 dual scaling function vector with support [−4, 4] of the piecewise Hermite cubics will be given in Section 4. More interesting is that a family of Hermite interpolatory masks will be constructed in Section 4 such that it includes the piecewise Hermite cubics as a special case. Several more new examples of biorthogonal multiwavelets will also be provided in Section 4. Due to the length of this paper, we shall discuss the smoothness and convergence problems associated with the refinable function vectors of our construction elsewhere. For any j ∈ N ∪ {0}, let Oj be the ordered set of {µ ∈ Zs+ : |µ| = j} in the lexicographic order. That is, (ν1 , · · · , νs ) is less than (µ1 , · · · , µs ) in lexicographic
HERMITE INTERPOLANTS AND BIORTHOGONAL MULTIWAVELETS
13
order if νj = µj for j = 1, · · · , i − 1 and νi < µi . By #Oj we denote the cardinality of the set Oj . The Kronecker product of two matrices A = (aij )1≤i≤l,1≤j≤n and B, written as A ⊗ B, is defined to be the following matrix
a11 B a21 B A ⊗ B := ...
a12 B a22 B .. .
··· ··· .. .
a1n B a2n B . .. .
al1 B
al2 B
···
aln B
¡ ¢r×r Theorem 3.1. Let a and e a in `0 (Zs ) be two masks satisfying the discrete biorthogonal relation (1.9) with a dilation matrix M . Suppose that J a (0) satisfies the condition (1.2) and e a satisfies the sum rules of order k for some positive integer k, i.e., there exist vectors yeµ , |µ| < k in Cr with ye0 6= 0 such that X
(−1)|ν| Jεea (ν)T yeµ−ν =
0≤ν≤µ
X
mµν yeν
∀ µ ∈ Zs+ , |µ| < k, ε ∈ ΩM ,
|ν|=|µ|
where mµν are given in (1.7) and ΩM is a complete set of representatives of the distinct cosets of Zs \M Zs with 0 ∈ ΩM . Let Oj be the ordered set of {µ ∈ Zs+ : |µ| = j} in the lexicographic order. Then yeµ , |µ| < k are uniquely determined up to a scalar multiplier constant by the following recursive relation: J a (0) ye0 = me y0 and for j = 1, · · · , k − 1, the vectors yeµ (µ ∈ Oj ) are determined by ¶ h i−1µ X X (e yµ )µ∈Oj = mIr(#Oj ) −(mµν )µ,ν∈Oj ⊗ J a (0) mην J a (µ − η) yeν . µ∈Oj
0≤η 4. 7741 2771 0 0 − 221184 184320 and e a(−β) = Ue a(β)U for all β ∈ N. Then φa is a Hermite interpolant such that a ν2 (φ ) ≈ 3.84745 and φa has accuracy order 7. φea is a dual function vector of φa such that ν2 (φea ) ≈ 0.91843 and φea has accuracy order 2. Therefore, φea is a continuous dual function vector of φa since ν∞ (φea ) ≥ 0.41843. Moreover, φa ∈ C 3 . Though the primal masks in all the above examples are interpolatory masks, the CBC algorithm can be easily applied to the general case. Let us take an example from Plonka and Strela [41] as the primal mask and then we use the CBC algorithm to construct a dual function vector for it. Example 4.8. The primal mask a in [41] is given by " 13 # " 51 # " 51 9 9 − 64 − 64 64 64 64 a(0) = , a(1) = , a(2) = 11 7 21 9 − 64 − 21 64 64 64 # " 6413 9 a(3) =
64
64
− 11 64
7 − 64
,
a(β) = 0
∀β 6= 0, 1, 2, 3.
9 64 9 64
# ,
28
BIN HAN
Then as pointed out by Plonka and Strela [41] that the refinable function vector φa is a polynomial B-spline of order 6 with double knots. νp (φa ) = 4 + 1/p for any 1 ≤ p ≤ ∞ and φa has accuracy order 6. By Theorem 3.1, we have ye0T = [1, 0],
ye1T = [3/2, −3/14],
ye3T = [39/56, −43/168],
ye2T = [17/14, −9/28],
ye4T = [529/1680, −1/7].
By using the CBC algorithm, we find a dual mask e a of a given by " e a(2) = " e a(4) = " e a(6) =
159239 122880
28291 122880
− 12198157 3317760
919531 3317760 1273 − 10240
1347 5120 847007 − 1105920
47627 − 368640 # 531
411 8192
8192
321 − 4096
63 − 512
,
#
" ,
e a(3) =
#
" ,
e a(5) = e a(7) =
#
− 18621 40960
7953 40960
5400373 3317760 1169 − 7680
− 1043381 3317760 # 551 − 30720
961993 " 3317760 19 − 8192 5 − 4096
331813 3317760 # 45 − 8192 3 1024
, ,
,
and e a(β) = Ue a(3−β)U for β = −4, −3, −2, −1, 0, 1, and otherwise, e a(β) = 0. Then φea is a dual function vector of φa with ν2 (φea ) ≈ 1.13543 and has accuracy order 5. Therefore, φea is a continuous dual function vector of φa since ν∞ (φea ) ≥ 0.63543. The graphs of all the above examples in this section are presented at the end of this paper. More examples of biorthogonal multiwavelets can be constructed by using the CBC algorithm studied in this paper and such algorithm can be easily implemented. Finally, we mention that the study of approximation order of biorthogonal multiwavelets in this paper is not only useful for construction of biorthogonal multiwavelets but also helpful for construction of orthogonal multiwavelets.
HERMITE INTERPOLANTS AND BIORTHOGONAL MULTIWAVELETS
29
0.15 1
0.1
0.8
0.05
0.6
0
0.4
−0.05 −0.1
0.2
−0.15 0 −1
−0.5
0 (a)
0.5
1
−1
1.5
10
1
5
0.5
0
−0.5
0 (b)
0.5
1
−2
0 (d)
2
4
−5
0
−10 −0.5 −4
−2
0 (c)
2
4
−4
Figure 1. (a), (b), (c) and (d) are the graphs of φb11 , φb21 , φea1 and φea2 in Example 4.5, respectively. φb1 is the piecewise Hermite cubics and φea is a C 1 dual function vector of φb1 .
0.2 1 0.1
0.8 0.6
0
0.4 −0.1
0.2 0 −3
−0.2 −2
−1
0 (a)
1
2
3
2.5
−3
−2
−1
0 (b)
1
2
3
10
2 5
1.5 1
0
0.5 0
−5
−0.5 −10
−1 −4
−2
0 (c)
2
4
−4
−2
0 (d)
Figure 2. (a), (b), (c) and (d) are the graphs of φb12 , φb22 , φea1 and φea2 in Example 4.6, respectively. φb2 is a C 2 Hermite interpolant and φea is a continuous dual function vector of φb2 .
2
4
30
BIN HAN
0.2
1 0.8
0.1
0.6
0
0.4 −0.1
0.2 0
−0.2
−3
−2
−1
0 (a)
1
2
3
−3
−2
−1
0 (b)
1
2
3
10
2 1.5
5
1 0 0.5 −5
0 −0.5
−10
−4
−2
0 (c)
2
4
−4
−2
0 (d)
2
4
Figure 3. (a), (b), (c) and (d) are the graphs of φa1 , φa2 , φea1 and φea2 in Example 4.7, respectively. φa is a C 3 Hermite interpolant and φea is a continuous dual function vector of φa . 0.4
1 0.8
0.2
0.6
0
0.4 −0.2 0.2 0
−0.4 0
1
2
3
0
1
2
(a)
3
(b)
2
10
1.5 5
1 0.5
0
0 −0.5
−5
−1 −1.5
−10 −4
−2
0
2 (c)
4
6
−4
−2
0
2
4
(d)
Figure 4. (a), (b), (c) and (d) are the graphs of φa1 , φa2 , φea1 and φea2 in Example 4.8, respectively. φa ∈ C 3 is a polynomial B-spline of order 6 with double knots and φea is a continuous dual function vector of φa .
6
HERMITE INTERPOLANTS AND BIORTHOGONAL MULTIWAVELETS
1.5
31
10
5
1
0 0.5
−5 0 −10
−0.5 −6
−4
−2
0 (c)
2
4
6
−6
φea1 1
−4
−2
0 (d)
2
of φea2 1 b1
Figure 5. (c) is the graph of and (d) is the graph Example 4.5. φea1 ∈ C 1 is a dual function vector of φ .
4
6
in
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
[19] [20]
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[21] B. Han, Analysis and construction of optimal multivariate biorthogonal wavelets with compact support, SIAM J. Math. Anal. (to appear). [22] B. Han and R. Q. Jia, Multivariate refinement equations and convergence of subdivision schemes, SIAM J. Math. Anal. 29 (1998), 1177–1199. [23] B. Han and R. Q. Jia, Optimal interpolatory subdivision schemes in multidimensional spaces, SIAM J. Numer. Anal. 36 (1998), 105–124. [24] B. Han and R. Q. Jia, Quincunx fundamental refinable functions and quincunx biorthogonal wavelets, preprint (1999). [25] W. He and M. J. Lai, Construction of bivariate nonseparable compactly supported orthonormal multiwavelets with arbitrarily high regularity, preprint (1998). [26] C. Heil and D. Colella, Matrix refinement equations and subdivision schemes, J. Fourier Anal. Appl. 2 (1996), 363–377. [27] C. Heil, G. Strang, and V. Strela, Approximation by translates of refinable functions, Numer. Math. 73 (1996), 75–94. [28] H. Ji, S. Riemenschneider, and Zuowei Shen, Multivariate compactly supported fundamental refinable functions, duals and biorthogonal wavelets, Stud. Appl. Math. 102 (1999), 173–204. [29] R. Q. Jia, Approximation properties of multivariate wavelets, Math. Comp. 67 (1998), 647– 665. [30] R. Q. Jia, Shift-invariant spaces and linear operator equations, Israel J. Math. 103 (1998), 259–288. [31] R. Q. Jia and C. A. Micchelli, On linear independence of integer translates of a finite number of functions, Proc. Edinburgh Math. Soc. 36 (1992), 69–85. [32] R. Q. Jia, S. Riemenschneider, and D. X. Zhou, Approximation by multiple refinable functions and multiple functions, Canadian J. Math. 49 (1997), 944–962. [33] R. Q. Jia, S. Riemenschneider, and D. X. Zhou, Smoothness of Multiple Refinable Functions and Multiple Wavelets, SIAM J. Matrix Anal. (1997) (to appear). [34] Q. T. Jiang, On the regularity of matrix refinable functions, SIAM J. Math. Anal. 29 (1998), 1157–1176. [35] J. Kovaˇ cevi´ c and M. Vetterli, Nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for Rn , IEEE Trans. on Information Theory 38 (1992), 533–555. [36] J. Kovaˇ cevi´ c and W. Sweldens, Wavelet family of increasing order in arbitrary dimensions, preprint (1997). [37] A. Logar and B. Sturmfels, Algorithms for the Quillen-Suslin Theorem, J. Algebra 145 (1992), 231–239. [38] C. A. Micchelli and T. Sauer, Regularity of multiwavelets, Adv. Comput. Math. 7 (1997), 455-545. [39] R. A. Nicolaides, On a class of finite elements generated by Lagrange interpolation, SIAM J. Numer. Anal. 9 (1972), 435–445. [40] G. Plonka, Approximation order provided by refinable function vectors, Constr. Approx. 13 (1997), 221–244. [41] G. Plonka and V. Strela, Construction of multi-scaling functions with approximation and symmetry, SIAM J. Math. Anal. 29 (1998), 481-510. [42] Zuowei Shen, Refinable function vectors, SIAM J. Math. Anal. 29 (1998), 235–250. [43] G. Strang and T. Nguyen, Wavelets and filter banks, Wellesley-Cambridge Press, 1996. [44] V. Strela, A note on construction of biorthogonal multi-scaling functions, Contemproary Mathematics 216, A. Aldroubi and E. B. Lin (eds.), AMS, 1998, pp. 149–157. [45] V. Strela, P. Heller, G. Strang, P. Topiwala and C. Heil, The application of multiwavelets filter banks to signal and image processing., IEEE Trans. on Image Processing (to appear). [46] W. Sweldens, The lifting scheme: a custom-design construction of biorthogonal wavelets, Appl. Comput. Harmon. Anal. 3 (1996), 186–200. [47] J. Z. Wang, Stability and linear independence associated with scaling vectors, SIAM J. Math. Anal. 29 (1998), 1140–1156. [48] D. X. Zhou, Multiple refinable Hermite interpolants, J. Apporx. Theory. (1998) (to appear).
Bin Han † Abstract. Multiwavelets are generated from refinable function vectors by using multiresolution analysis. In this paper we investigate the approximation properties of a multivariate refinable function vector associated with a general dilation matrix in terms of both the subdivision operator and the order of sum rules satisfied by the matrix refinement mask. Based on a fact about the sum rules of biorthogonal multiwavelets, a construction by cosets (CBC) algorithm is presented to construct biorthogonal multiwavelets with arbitrary order of vanishing moments. More precisely, to obtain biorthogonal multiwavelets, we have to construct primal and dual masks. Given any primal matrix mask a and a general dilation matrix M , the proposed CBC algorithm reduces the construction of all dual masks of a, which satisfy the sum rules of arbitrary order, to a problem of solving a well organized system of linear equations. We prove in a constructive way that for any given primal mask a with a dilation matrix M and for any positive integer k, we can always construct a dual mask e a of a such that e a satisfies the sum rules of order k. In addition, we provide a general way for the construction of Hermite interpolatory matrix masks in the univariate setting with any dilation factors. From such Hermite interpolatory masks, smooth Hermite interpolants, including the well known piecewise Hermite cubics as a special case, are obtained and are used to construct biorthogonal multiwavelets. As an example, a C 3 Hermite interpolant with support [−3, 3] is presented. Then we shall apply the CBC algorithm to such Hermite interpolatory masks to construct biorthogonal multiwavelets. Several examples of biorthogonal multiwavelets are provided to illustrate the general theory. In particular, a C 1 dual function vector with support [−4, 4] of the piecewise Hermite cubics is given.
1991 Mathematics Subject Classification. 41A25 41A05 65D05 46E35 41A63. Key words and phrases. Biorthogonal multivariate multiwavelets, Hermite interpolants, Hermite Interpolatory mask, accuracy order, sum rules, refinable function vectors. This research was supported in part by NSERC Canada under a postdoctoral fellowship. †Program in Applied and Computational Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544-1000, USA. [email protected], http://www.math.princeton.edu/∼bhan Typeset by AMS-TEX
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§1. Introduction In her celebrated paper [13], Daubechies constructed a family of compactly supported univariate orthogonal scaling functions and their corresponding orthogonal wavelets with the dilation factor 2. Since then wavelets with compact support have been widely and successfully used in various applications such as image compression and signal processing [14]. Each Daubechies wavelet is generated from one scaling function and therefore, is called a scalar wavelet. Though orthogonal wavelets have many desired properties such as compact support, good frequency localization and vanishing moments, they lack symmetry as demonstrated by Daubechies in [14]. However, symmetry of wavelets is a much desired property in applications. Such a property is claimed to produce less visual artifacts than non-symmetric wavelets. To achieve symmetry, several generalizations of scalar orthogonal wavelets have been studied in the literature. For example, biorthogonal wavelets achieve symmetry where orthogonality is replaced with biorthogonality, and multiwavelets achieve both orthogonality and symmetry where one scaling function is replaced with several scaling functions (i.e., a scaling function vector). Wavelets in multidimensional spaces with a general dilation matrix have also been extensively investigated in recent years since in many applications we have to deal with higher dimensional data such as images. Scalar biorthogonal wavelets in both univariate case and multivariate case have been extensively studied in the literature. See [2, 6, 7, 8, 9, 10, 14, 20, 21, 24, 35, 36, 46] and references therein for discussion on scalar biorthogonal wavelets. With symmetry and many other desired properties, scalar biorthogonal wavelets have been found to be more efficient and useful in many applications than the orthogonal ones [14, 43]. To achieve symmetry, another approach is to adopt multiwavelets where a scaling function vector instead of a single scaling function is used. To compare with scalar wavelets, multiwavelets have several advantages such as shorter support and higher vanishing moments. The success of wavelets largely contributes to the short support and high vanishing moments which are competing objectives in the design of wavelets. That is, to obtain a wavelet with higher vanishing moments, it is necessary to enlarge its support. Such advantages of multiwavelets provide new opportunities and choices in the wavelet theory which are impossible to achieve by using scalar wavelets. With more flexible trade-off between high vanishing moments and short support, multiwavelets are particularly attractive in the construction of wavelets on a bounded domain [11] to deal with problems arising from a finite domain with boundary conditions and are expected to be useful in many applications such as numerical solutions to partial differential equations and signal processing [11, 45]. As a generalization of the scalar wavelets, it is also of interest in its own right to investigate multiwavelets. The advantages of multiwavelets and their promising features in applications have attracted a great deal of interest and effort in recent years to extensively study them. To only mention a few references here, see [1, 11, 12, 16, 17, 25, 26, 27, 33, 38, 40, 41, 44, 47, 48] and references therein on discussion of various topics on multiwavelets and their applications. The generalization of scalar wavelets to multiwavelets is not trivial and the study of multiwavelets is much more complicated and involved than the study of the scalar wavelets which we shall see later. Before proceeding further, let us introduce some notation. An s×s integer matrix
HERMITE INTERPOLANTS AND BIORTHOGONAL MULTIWAVELETS
3
M is called a dilation matrix if limn→∞ M −n = 0. That is, all the eigenvalues of a dilation matrix M are greater than one in modulus. Throughout this paper, M denotes a dilation matrix and m := |detM |. In this paper, we are concerned with the following refinement equation: X φ= a(β)φ(M · −β), (1.1) β∈Zs
where φ = (φ1 , · · · , φr )T is a r × 1 vector of functions, called a refinable function vector, and a is a finitely supported sequence of r × r matrices on Zs , called the (matrix refinement) mask. When r = 1, φ is called a scalar refinable function and a is called a scalar refinement mask. By J a (0) we denote the following matrix associated with a mask a as X J a (0) := a(β). β∈Zs
If φ1 , · · · , φr are functions in L1 (Rs ) with stable shifts and φ = (φ1 , · · · , φr )T satisfies the refinement equation (1.1) with a mask a, then it was proved by Dahmen and Micchelli [12] that J a (0) has a simple eigenvalue m and all other eigenvalues in modulus < m. (1.2) Conversely, if J a (0) satisfies the condition (1.2), it was proved by Heil and Collella [26] and Cabrelli, Heil, and Molter [4] that there exists a unique vector φ of comb = φ(0) b with pactly supported distributions such that φ satisfies (1.1) and J a (0)φ(0) b b kφ(0)k2 = 1 and the first nonzero component of φ(0) being positive. We call such solution the normalized solution of (1.1) and throughout this paper we denote the normalized solution of (1.1) with mask a by φa . If φ is another distribution solution of (1.1), then we must have φ = cφa for some constant c. On the one hand, multiwavelets provide more flexibility and new features which are not possible for scalar wavelets. On the other hand, in the multiwavelet case r > 1, each element in the mask of the refinement equation becomes an r × r matrix comparing with a scalar number in the scalar wavelet case. The change from a mask of scalar numbers to a mask of matrices makes the analysis and investigation of multiwavelets far more complicated and involved than its scalar counterpart. For example, even in the univariate case, the sum rule and vanishing moment conditions of a matrix mask are much more complicated than the scalar case, see [3, 5, 12, 27, 32, 34, 40, 42]. The involvement of matrices in the mask makes the construction of multiwavelets with certain vanishing moments much more challenging than its scalar counterpart. To our best knowledge, no systematic method is proposed in the current literature to construct biorthogonal multiwavelets with arbitrary order of sum rules even for the simplest case — the univariate setting with the dilation factor 2. Many known approaches and constructions in the scalar wavelet case do not apply to the multiwavelet case. Since both biorthogonal multivariate wavelets and multiwavelets are of great interest in both theory and applications, one aim of this paper is to investigate the biorthogonal multiwavelets in multidimensional spaces and to propose a method to construct them systematically. Let `(Zs ) denote the linear space of all sequences on Zs and `0 (Zs ) denote the linear space of all finitely supported sequences on Zs . For any positive integer r,
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¡ ¢r×r by `(Zs ) we denote the linear space of all sequences of r × r matrices on Zs ¡ s ¢r and by `(Z ) we denote the linear space of all sequences of r × 1 vectors on Zs . ¡ ¢r×r ¡ ¢r Similarly, we define `0 (Zs ) and `0 (Zs ) . Given any compactly supported distribution vector f = (f1 , · · · , fr )T , we define the following linear operator cf on ¡ s ¢r `(Z ) by X ¡ ¢r cf (λ) := λ(β)T f (· − β), λ ∈ `(Zs ) , (1.3) β∈Zs
where AT denotes the transpose of a matrix shifts of f are said to be ¡ A. ¢The r linearly independent if cf (λ) = 0 for λ in `(Zs ) implies λ = 0. It was proved by Jia and Micchelli [31] that the shifts of a compactly supported distribution vector ¡ f = ¢(f1 , · · · , fr )T are linearly independent if and only if the sequences fbj (ξ + 2πβ) β∈Zs , j = 1, · · · , r are linearly independent for all ξ ∈ Cs . The shifts of f are ¡ ¢ stable if the sequences fbj (ξ + 2πβ) s , j = 1, · · · , r are linearly independent for β∈Z
all ξ ∈ Rs . Therefore, if the shifts of φa are stable or linearly independent, then (1.2) holds true. Before proceeding further, let us introduce some notation. Recall that M denotes a dilation matrix. Let ΩM be a complete set of representatives of the distinct cosets of Zs /M Zs . Without loss of generality, we assume that 0 ∈ ΩM . For any mask a ¡ ¢r×r in `0 (Zs ) , we shall use the following notation throughout this paper Jεa (µ) :=
X
a(ε + M β)(M −1 ε + β)µ /µ!,
µ ∈ Zs+ , ε ∈ ΩM
(1.4)
β∈Zs
and
J a (µ) :=
X
a(β)(M −1 β)µ /µ! =
β∈Zs
where
X
Jεa (µ),
µ ∈ Zs+ ,
(1.5)
ε∈ΩM
Zs+ := {(β1 , · · · , βs ) ∈ Zs : βj ≥ 0, j = 1, · · · , s}
and for any β = (β1 , · · · , βs ) ∈ Zs and µ = (µ1 , · · · , µs ) ∈ Zs+ , β µ := β1µ1 · · · βsµs , µ! := µ1 ! · · · µs ! and |β| := |β1 | + · · · + |βs |. For any ν = (ν1 , · · · , νs ), µ = (µ1 , · · · , µs ) ∈ Zs+ , we say that ν ≤ µ if νj ≤ µj for all j = 1, · · · , s, and we ¡ ¢r×r say that ν < µ if ν ≤ µ and ν 6= µ. For any mask a in `0 (Zs ) , we say that the mask a with a dilation matrix M satisfies the sum rules of order k if there exists a set of r × 1 vectors {yµ : µ ∈ Zs+ , |µ| < k} with y0 6= 0 such that X
(−1)|ν| Jεa (ν)T yµ−ν =
0≤ν≤µ
X
mµν yν
∀ µ ∈ Zs+ , |µ| < k, ε ∈ ΩM ,
(1.6)
|ν|=|µ|
where the numbers mµν are uniquely determined by X xν (M −1 x)µ = mµν , µ! ν!
x ∈ Rs .
(1.7)
|ν|=|µ|
Given a vector φ = (φ¡1 , · · · ,¢φr )T of compactly supported distributions on Rs , r let S(φ) = {cφ (λ) : λ ∈ `(Zs ) } where the linear operator cφ is defined in (1.3).
HERMITE INTERPOLANTS AND BIORTHOGONAL MULTIWAVELETS
5
Following Heil, Strang and Strela [27], we say that φ has accuracy order k if Πk−1 ⊆ S(φ) where Πk−1 denotes the set of all polynomials of total degree less than k. We also agree that Π−1 = {0}. Accuracy order of φ has a close relation with both the approximation order provided by φ and the well known Strang-Fix conditions on φ. See Jia [30] for such concepts and related results. When φa is a refinable function vector with a mask a and a dilation matrix M , under the assumption that the shifts of φa are linearly independent, Cabrelli, Heil, and Molter [3, 4] characterized the accuracy order of φa in terms of the order of sum rules satisfied by the mask a. Also see [5, 27, 32, 34, 40] and references therein for discussion on accuracy order in the univariate setting. In Section 2, under a very mild condition, we shall provide another characterization (see Theorem 2.4) of the accuracy order of φa in terms of both the subdivision operator and sum rules. Accuracy order of a refinable function vector is also closely related to the concept of vanishing moments of biorthogonal multiwavelets (see [14, 43]). The success of a wavelet basis largely lies in the short support and high accuracy order of a refinable function vector. It is also known that if a refinable function vector φa ∈ C k has a linearly independent shifts, then ¡ ¢ it is necessary that φ has accuracy order k + 1. s r A function vector φ in L2 (R ) is called a primal function vector if φ satisfies ¡ ¢r×r the refinement equation (1.1) with a mask a in `0 (Zs ) and the shifts of φ e are ¡ linearly ¢r independent. A dual function vector φ of φ is a function vector in L2 (Rs ) such that φe satisfies the refinement equation (1.1) with a mask e a and Z Rs
e + β)T dx = δ(β)Ir φ(x) φ(x
∀ β ∈ Zs ,
(1.8)
where Ir denotes the r×r identity matrix, and δ(0) = 1, δ(β) = 0 for all β ∈ Zs \{0}. Clearly, if φ and φe satisfy the conditions in (1.8), then the shifts of φ and φe are linearly independent, respectively. A necessary condition for φ and φe to satisfy the conditions in (1.8) is the following well known discrete biorthogonal relations: X
a(β) e a(β + M α)T = mδ(α)Ir
∀ α ∈ Zs .
(1.9)
β∈Zs
¡ ¢r×r ¡ ¢r×r If a mask a in `0 (Zs ) satisfies (1.2) and there exists a sequence e a in `0 (Zs ) such that the conditions in (1.9) are satisfied, then we say that a is a primal mask and any such mask e a will be called a dual mask of a. Dahmen and Micchelli proved in [12] that if a is a primal mask with e a being a dual mask of a, then φa is a e a primal function vector with φ being a dual function vector of φa if and only if the subdivision schemes associated with a and e a converge in the L2 norm, respectively. Given a primal mask a, to construct a dual mask of a, we need to solve a system of linear equations given in (1.9). In the current literature, the lifting scheme is known to be a good method for constructing a dual mask of any given primal mask. For discussion on lifting scheme, the reader is referred to Sweldens [46], Daubechies and Sweldens [15], and Kovaˇ cevi´c and Sweldens [36]. We point out that in the univariate setting, Plonka’s factorization technique of a matrix symbol is very useful in studying refinable function vectors [38, 40] and was used by Plonka and Strela in [41] to construct smooth refinable function vectors. This factorization technique was also used by Strela in [44] to construct univariate
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multiwavelets. However, the factorization technique does not apply to the higher dimensions and the biorthogonal multiwavelets constructed by the factorization technique in [44] have very long support. As we mentioned before, the vanishing moments of a biorthogonal multiwavelet are important in both applications and construction of smooth biorthogonal multiwavelets. See [6, 14] for discussion on vanishing moments and their relation to sum rules. For a primal mask a, the lifting scheme can be used to solve the discrete biorthogonal relation (1.9), which is a system of linear equations, to get a dual mask e a of a. To achieve high vanishing moments of the resulting biorthogonal multiwavelet, the dual mask e a must satisfy the sum rules of high order. Even in the simplest case s = 1 and M = (2), it is not easy to use the definition of sum rules given in (1.6) to achieve desired order of sum rules satisfied by e a. The reason is that when r > 1, to obtain a dual mask e a of a given primal mask a, even the equations in the biorthogonal conditions (1.9) are linear, the equations given in (1.6) for sum rules are no longer linear equations since in general the vectors yν in (1.6) are determined by e a. Due to such difficulty, many methods on construction of scalar biorthogonal wavelets no longer hold in the multiwavelet case and not many examples of biorthogonal multiwavelets are available in the literature. For example, univariate multiwavelets were reported by Donovan, Geronimo, Hardin and Massopust [17], Dahmen, Han, Jia and Kunoth [11], He and Lai [25] and other examples were given in [7, 33, 44]. One purpose of this paper is to try to overcome such difficulty. In the scalar case, a construction by cosets (CBC) algorithm was proposed by Han in [21] to construct scalar biorthogonal wavelets with arbitrary vanishing moments. In this paper, we shall generalize the CBC algorithm in [21] to the multiwavelet case to overcome the above mentioned difficulty. For the advantages of the CBC algorithm over other known methods on construction of scalar biorthogonal wavelets, the reader is referred to [6, 21, 24]. In Section 3, we shall follow the line developed in [6, 21, 24] to discuss how to construct biorthogonal multiwavelets in the most general case. We propose a general construction by cosets (CBC) algorithm in Section 3. Such CBC algorithm reduces the construction of all dual masks, which satisfy the sum rules of arbitrary order, of a given primal mask to the problem of solving a well organized system of linear equations. Based on such algorithm, we shall demonstrate that for any given primal mask a with a dilation matrix M and for any positive integer k, we can construct a dual mask e a of a such that e a satisfies the sum rules of order k. In Section 4, we construct a special family of primal masks – Hermite interpolatory masks in the univariate setting with any dilation factor. The resulting refinable function vectors are Hermite interpolants which are useful in curve design in computer aided geometric design [18]. As an example, a C 3 Hermite interpolant is constructed with support [−3, 3]. In particular, such construction of Hermite interpolatory masks includes the piecewise Hermite cubics as a special case. Several new examples of biorthogonal multiwavelets are provided to illustrate the general theory developed in this paper. In particular, a C 1 dual function vector of the piecewise Hermite cubics is given and is supported on [−4, 4]. Finally, by the CBC algorithm, we construct a continuous dual function vector for the primal function vector which is a polynomial B-spline of order 6 with double knots in Plonka and Strela [41]. Such primal function vector belongs to C 4−η for any η > 0, has accuracy order 6 and has support [0, 3].
HERMITE INTERPOLANTS AND BIORTHOGONAL MULTIWAVELETS
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§2. Accuracy Order of Refinable Function Vectors In this section, under a very mild condition we shall investigate the accuracy order satisfied by a mask in terms of both the subdivision operator and the sum rules. The motivation of our approach here lies in that it may be useful in the study of various properties associated with the subdivision operators. To do this, let us introduce some notation and several auxiliary results here. ¡ ¢r×r Given a mask a in `0 (Zs ) , the subdivision operator Sa associated with the mask a and a dilation matrix M is defined by X ¡ ¢r Sa λ(α) = a(α − M β)T λ(β), α ∈ Zs , λ ∈ `(Zs ) . (2.1) β∈Zs
Πrk
By we denote the set of r × 1 polynomial vectors with each component being r r a polynomial of degree at most k, Πr := ∪∞ k=0 Πk . For any p ∈ Π , we ¡ and ¢by r s s naturally have a sequence p|Zs ∈ `(Z ) given by p|Zs (α) = p(α), α ∈ Z . Since the mapping from Πr → Πr |Zs is one-to-one and onto, without confusion,¡we may ¢r use p ∈ Πr as both a polynomial vector in Πr and a polynomial sequence in `(Zs ) since it is easy to distinguish them in the In particular, for any p ∈ Πr , ¡ context. ¢r s if Sa (p|Zs ) is a polynomial sequence in `(Z ) , then we may use Sa p to represent ¡ ¢r both the polynomial vector in Πr and the polynomial sequence in `(Zs ) . Lemma 2.1. Let p ∈ Πr |Zs . Then Sa p ∈ Πr |Zs if and only if X X a(ε + M β)T p(x − M −1 ε − β) = a(M β)T p(x − β) β∈Zs
∀ ε ∈ ΩM , x ∈ Rs .
β∈Zs
If this is the case, then Sa p is given by the following polynomial vector X Sa p(x) = a(ε + M β)T p(M −1 x − M −1 ε − β), ε ∈ ΩM .
(2.2)
β∈Zs
Proof. By definition of the subdivision operator Sa , for any ε, α ∈ Zs , we have X Sa p(ε + M α) = aT (ε + M β)p(M −1 (ε + M α) − M −1 ε − β). β∈Zs
The claim in this lemma follows directly from the above equality.
¤
By Dj we denote the partial derivative with respect to the j-th unit coordinate. For any µ = (µ1 , · · · , µs ) ∈ Zs+ , Dµ denotes the differential operator D1µ1 · · · Dsµs . Moreover, we write D = (D1 , · · · , Ds )T . For any polynomial p ∈ Πr , we shall use the following convention: X p(x − iD)T := p(µ) (x)T (−iD)µ /µ!, µ∈Zs+
where i is the imaginary unit such that i2 = −1. For any p ∈ Πr , we observe X £ £ ¤ ¤£ ¤ p(x − iD)T f (·)g(·) = p(µ) (x − iD)T f (·) (−iD)µ g(·) /µ!. (2.3) µ∈Zs+
¡ ¢r×r , let For any mask a in `0 (Zs ) X a(β)e−iβ·ξ /m, H a (ξ) :=
ξ ∈ Rs
β∈Zs
and define
¡ ¢ ¡ ¢ Hka (ξ) := H a (M T )−1 ξ · · · H a (M T )−k ξ ,
k ∈ N.
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Proposition 2.2. If p ∈ Πr , then Sa p ∈ Πr if and only if p(x − iD)T H1a (2πβ0 ) = 0
∀ β0 ∈ Zs \M T Zs .
(2.4)
Moreover, if p ∈ Πr and Sa p ∈ Πr , then p(x − iD)T H1a (2πM T β) = (Sa p)(M x)T for all β ∈ Zs , and for any k > 1, a p(x − iD)T Hka (2πM T β) = (Sa p)(M x − iD)T Hk−1 (2πβ)
∀ β ∈ Zs .
Proof. By the definition of H1a , we have X X 1 £ ¤ −1 p(µ) (x)T a(β)(−iD)µ e−iM β·ξ |ξ=2πβ0 µ! s s
m p(x − iD)T H1a (2πβ0 ) =
β∈Z µ∈Z+
X X 1 −1 p(µ) (x)T a(β)(−M −1 β)µ e−2πiM β·β0 µ! β∈Zs µ∈Zs+ X −1 = p(x − M −1 β)T a(β)e−2πiM β·β0
=
β∈Zs
X
=
e−2πiM
ε∈ΩM
−1
ε·β0
X
pT (x − M −1 ε − β)a(ε + M β).
β∈Zs
Thus, it yields that (2.4) is equivalent to Sa p ∈ Πr by Lemma 2.1. Moreover, the above equality gives us p(x − iD)T¡ H1a (2πM T¢ β) = (Sa p)(M x)T for all β ∈ Zs . a (M T )−1 ξ . By (2.3), we have Note that Hka (ξ) = H1a (ξ)Hk−1 p(x − iD)T Hka (2πM T β) X 1 £ a ¡ T −1 ¢¤ = p(µ) (x − iD)T H1a (2πM T β)(−iD)µ Hk−1 (M ) · (2πM T β). µ! s µ∈Z+
On the other hand, by p(x − iD)T H1a (2πM T β) = (Sa p)(M x)T , we deduce that £ ¤ £ ¤ p(µ) (x − iD)T H1a (2πM T β) = Dµ p(x − iD)T H1a (2πM T β) = Dµ (Sa p)(M ·)T (x). So, in conclusion, we have p(x − iD)T Hka (2πM T β) =
X 1 £ ¤ a Dµ (Sa p)(M ·)T (x)(−iM −1 D)µ Hk−1 (2πβ) µ! s
µ∈Z+
a = (Sa p)(M x − iD)T Hk−1 (2πβ).
¤
¡ ¢r×r and any positive integer k, we define a subspace For any mask a in `0 (Zs ) Pka of Πrk as Pka := {p ∈ Πrk : Saj p ∈ Πrk |Zs ∀ j ∈ N}. (2.5) It is evident that Pka is invariant under the subdivision operator Sa defined in (2.1). We are ready to prove the following result:
HERMITE INTERPOLANTS AND BIORTHOGONAL MULTIWAVELETS
9
Theorem 2.3. If p ∈ Pka , then cφa (p) ∈ Πk where Pka is defined in (2.5) and the operator cφa is defined in (1.3). In particular, cφa (p)(x) =
X
p(β)T φa (x−β) = p(x−iD)T φˆa (0) =
β∈Zs
X 1 p(µ) (x)T (−iD)φˆa (0). µ! s
µ∈Z+
¡ ¢ Conversely, if the sequences φˆj (2π(M T )−1 ε + 2πβ) β∈Zs , j = 1, · · · , r are linearly independent for all ε ∈ ΩM T where φa = (φ1 , · · · , φr )T is the normalized solution of (1.1) with the mask a, then for any q ∈ Πk ∩ S(φa ), there exists a unique p ∈ Pka such that cφa (p) = q. Proof. Since p ∈ Pka , by definition, Saj p ∈ Πr for all j ∈ N ∪ {0}. We claim that p(x − iD)T φˆa (2πβ) = 0
∀ β ∈ Zs \{0}.
(2.6)
For any β ∈ Zs \{0}, there is a positive integer k such that β ∈ Zs \(M T )k Zs . Note ¡ ¢ that φˆa (ξ) = Hka (ξ)φˆa (M T )−k ξ and p(x−iD)T φˆa (2πβ) =
X 1 £ ¤ £ ¡ ¢¤ Dµ p(x−iD)T Hka (2πβ) (−iD)µ φˆa (M T )−k · (2πβ). µ! s
µ∈Z+
Therefore, to prove (2.6), it suffices to prove that p(x − iD)T Hka (2πβ) = 0
∀ β ∈ Zs \(M T )k Zs , p ∈ Pka , x ∈ Rs .
(2.7)
By Proposition 2.2, (2.7) holds true for k = 1. Suppose is true for ¡ that (2.7) ¢ a k − 1. Note that Hka (ξ) = Hk−1 (ξ)f (ξ) where f (ξ) := H a (M T )−k ξ . By induction hypothesis, we deduce that for any β ∈ Zs \(M T )k−1 Zs , p(x − iD)T Hka (2πβ) =
X 1 £ ¤ a Dµ p(x − iD)T Hk−1 (2πβ) (−iD)µ f (2πβ) = 0. µ! s
µ∈Z+
For any β ∈ Zs \M T Zs , by Proposition 2.2 and induction hypothesis, ¡ ¢ a p(x − iD)T Hka (2π(M T )k−1 β) = (Sa p)(M x − iD)T Hk−1 2π(M T )k−2 β = 0. Therefore, (2.7) holds true and we proved (2.6). By Poisson summation formula (see [5]), we have cφa (p)(x) =
X
p(β)T φa (x − β) =
β∈Zs
= p(x − iD)T φˆa (0) =
X
e2πiβ·x p(x − iD)T φˆa (2πβ)
β∈Zs
X 1 p(µ) (x)T (−iD)µ φˆa (0). µ! s
µ∈Z+
¡ ¢ Conversely, let q ∈ Πk ∩ S(φa ). Since the sequences φˆj (2πβ) β∈Zs , j = 1, · · · , r are linearly independent, by [30, Lemma 8.2], there exists a unique polynomial vector p ∈ Πrk such that cφa (p) = q. Thus, to prove that p ∈ Pka , it suffices to prove
10
BIN HAN
that if p ∈ Πrk and cφa (p) ∈ Πk , then Sa p ∈ Πrk . By Poisson summation formula, we have X X e2πiβ·x p(x − iD)T φˆa (2πβ) = p(β)T φa (x − β) = cφa (p) ∈ Πk , β∈Zs
β∈Zs
¡ ¢ from which it follows that (2.6) holds true. Since φˆj (2π(M T )−1 ε + 2πβ) β∈Zs , j = 1, · · · , r are linearly¡ independent for all¢ ε ∈ ΩM¡T , there exits βε1 , · · ¢· βεr ∈ Zs such that the matrix [φˆa 2π(M T )−1 ε+2πβε1 , · · · , φˆa 2π(M T )−1 ε+2πβεr ] is invertible. Thus, for ξ in a neighborhood of 0, the following matrix £ ¡ ¢ ¡ ¢¤ F (ξ):= φˆa 2π(M T )−1 ε+2πβε1 +(M T )−1 ξ , · · · , φˆa 2π(M T )−1 ε+2πβεr +(M T )−1 ξ ¡ ¢ is invertible. By φˆa (ξ) = H1a (ξ)φˆa (M T )−1 ξ , we have £ a ¤ φˆ (2πε + 2πM T βεj + ξ) 1≤j≤r = H1a (2πε + ξ)F (ξ). Hence, the following identity is valid for ξ in a neighborhood of 0: ¤ £ H1a (2πε + ξ) = φˆa (2πε + 2πM T βεj + ξ) 1≤j≤r F (ξ)−1 . Therefore, by (2.3), £ ¤ p(x − iD)T H1a (2πε) = p(x − iD)T H1a (2πε + ·) (0) X £ ¡ ¢¤ £ ¤ = p(µ) (x − iD)T φˆa 2π(ε + M T βεj ) 1≤j≤r (−iD)µ F −1 (0). µ∈Zs+
¡ ¢ £ ¢¤ By (2.6), p(µ) (x − iD)T φˆa 2π(ε + M T β) = Dµ p(x − iD)T φˆa (2π(ε + M T β) = 0 for all ε ∈ ΩM T \{0} and β ∈ Zs . So p(x − iD)T H1a (2πε) = 0 for all ε ∈ ΩM T \{0}. By Proposition 2.2, Sa p ∈ Πrk . ¤ The main result in this section is the following theorem. ¡ ¢r×r Theorem 2.4. Let a be a mask in `0 (Zs ) and satisfy the condition (1.2) with a dilation matrix M . Let φa = (φ1 , · · · , φr )T be the normalized solution of the refinement equation (1.1) with the mask¢ a and the dilation matrix M . Suppose that ¡ the sequences φˆj (2π(M T )−1 ε + 2πβ) β∈Zs , j = 1, · · · , r are linearly independent for all ε ∈ ΩM T . Then the following statements are equivalent: (a) φa has accuracy order k; a a a (b) dim Pk−1 = dim Πk−1 where Pk−1 is defined in (2.5) and Pk−1 is invariant under Sa ; a a (c) The mapping cφa |Pk−1 : Pk−1 |Zs → Πk−1 is onto where cφa is defined in (1.3); (d) a satisfies the sum rules of order k. Moreover, if a satisfies the sum rules of order k given in (1.6) with {yµ : µ ∈ Zs+ , |µ| < k} and y0T φˆa (0) = 1, then X X βν xµ = y T φa (x − β) µ! ν! µ−ν s 0≤ν≤µ β∈Z
∀ |µ| < k, µ ∈ Zs+ .
HERMITE INTERPOLANTS AND BIORTHOGONAL MULTIWAVELETS
11
Proof. From Theorem 2.3, we see that for any positive integer k, the mapping a a cφa |Pk−1 : Pk−1 → Πk−1 is well defined. Under the assumption that the sequences ¡ ¢ ˆ a φj (2πβ) β∈Zs , j = 1, · · · , r are linearly independent, by [30, Lemma 8.2], cφa |Pk−1 a is one-to-one. Therefore, by Theorem 2.3, φ has accuracy order k if and only if a a is one-to-one and onto which is also equivalent to dim Pk−1 = dim Πk−1 . cφa |Pk−1 Thus, we proved that (a), (b) and (c) are equivalent. a We now discuss sum rules. Since cφa |Pk−1 is a one-to-one and onto mapping a a between Pk−1 carries the structure of and Πk−1 , the inverse mapping of cφa |Pk−1 a Πk−1 into Pk−1 . Define µ pµ := c−1 φa (x /µ!),
µ ∈ Zs+ , |µ| < k.
Since cφa (Dν pµ ) = Dν cφa (pµ ) for all ν, µ ∈ Zs+ with |µ| < k, we may assume that pµ =
X
yµ−ν
0≤ν≤µ
xν ν!
∀ µ ∈ Zs+ , |µ| < k,
(2.8)
for some r × 1 vectors yν , |ν| < k. Note that cφa (Sa pµ )(M x) = cφa (pµ )(x) = xµ /µ!. We have µ ¶ µ X ¶ ν −1 µ X x) −1 (M −1 µx mν (Sa pµ )(x) = cφa = cφa = mµν pν (x), µ! ν! |ν|=|µ|
|ν|=|µ|
where mµν are given in (1.7). By Lemma 2.1, we have X X mµν pν (M x) = Sa pµ (M x) = a(ε + M β)T pµ (x − M −1 ε − β)
∀ ε ∈ ΩM .
β∈Zs
|ν|=|µ|
Setting x = 0 in the above equality, we have (1.6). Hence, (c) implies (d). To prove that (d) implies (a), we define pµ as in (2.8). By the definition of sum rules and observing that Dν (pµ ) = pµ−ν where by convention pν ≡ 0 for P ν 6∈ Zs+ , it is not difficult to verify that Sa p ∈ Πrk and Sa p = |ν|=|µ| mµν pν for all a |µ| < k. Thus pµ ∈ Pk−1 , and by Theorem 2.3, cφa (pµ ) ∈ Πk−1 for all |µ| < k. Set qµ := cφa (pµ ). Then we have X qµ (M −1 x) = cφa (Sa pµ )(x) = mµν qν (x) ∀ µ ∈ Zs+ , |µ| < k. (2.9) |ν|=|µ|
Note that Dν qµ = qµ−ν for all ν ∈ Zs+ (see [4, Theorem 3.2]). We may assume qµ (x) =
X 0≤ν≤µ
lµ−ν
xν , ν!
µ ∈ Zs+ , |µ| < k,
for some lµ ∈ C, µ ∈ Zs+ with |µ| < k. We claim that lµ = 0 for any 0 < |µ| < k. Suppose that for some 0 < j < k, lν = 0 for all 0 < |ν| < j. Then qµ (x) = l0 xµ /µ! + lµ for any |µ| = j. By (2.9), we have l0
X X (M −1 x)µ xν + lµ = l0 mµν + mµν lν . µ! ν! |ν|=|µ|
|ν|=|µ|
12
BIN HAN
P Therefore, by (1.7), lµ = |ν|=|µ| mµν lν for all |µ| = j. But when j > 1, it was observed in [3] that all the eigenvalues of (mµν )|ν|=j,|µ|=j are less than 1 in modulus. Hence lµ = 0 for all |µ| = j. Now qµ = l0 xµ /µ! for all |µ| < k. By Theorem 2.3, l0 = y0T φˆ0 (0) 6= 0 follows directly from (1.2) and y0 6= 0. ¤ From the proof of Theorem 2.4, we see that in a sense the concept of sum rules provides us a convenient way of describing the subspace Pka of Πrk and Pka is an invariant subspace under Sa . When r = 1, the sum rules condition in (1.6) can be restated as follows: X X a(ε + M β)(ε + M β)µ = a(M β)(M β)µ ∀ |µ| < k, ε ∈ ΩM , (2.10) β∈Zs
β∈Zs
a which was given by Jia [29] and can be easily seen either by Lemma 2.1 and Pk−1 = Πk−1 or by (1.6). The equivalence between (a) and (d) in Theorem 2.4 was given by Cabrelli, Heil and Molter [3, 4] under a much stronger assumption that the shifts of φa are linearly independent. It was also given by Jia [29], Heil, Strang, and Strela [27] and Plonka [40] for the case s = 1 and M = (2). As in [4], from the proof of Theorem 2.4, (d) always implies (a) without the assumption in Theorem 2.4. In passing, we mention that Pka = {p ∈ Πrk : Saj p ∈ Πrk ∀ j = 1, · · · , dim Πrk } since Pka ⊆ Πrk , and if a satisfies the sum rules of order k + 1, then Sa |Pka = (cφa |Pka )−1 τM (cφa |Pka ) where τM : Πk → Πk is given by τM (p)(x) = p(M −1 x). Therefore, the structure of Sa restricted to the subspace Pka can be easily analyzed by using a much simpler operator τM .
§3. Construction of Dual Masks with Arbitrary Order of Sum Rules In this section, we shall discuss how to systematically construct dual masks with arbitrary order of sum rules for any given primal mask. More precisely, given a primal mask a, for any positive integer k, how to find all dual masks of a such that the dual masks satisfy the sum rules of order k. Even in the scalar and multivariate case, this is not a straightforward question and there are a lot of literature discussing it [6, 9, 10, 20, 21, 35, 36, 46]. This problem is also called filter design in the language of engineering [43]. As for the multiwavelet case, even in the univariate setting, to our best knowledge, no systematic method is available in the current literature to deal with this problem. As a matter of fact, it took a relatively long time in the wavelet community to find a continuous dual scaling function vector of the well known piecewise Hermite cubics [11]. In this paper, we shall demonstrate that designing multiwavelets with arbitrary vanishing moments can be reduced to solve a system of well organized linear equations by using a CBC algorithm. As an application of this method, a C 1 dual scaling function vector with support [−4, 4] of the piecewise Hermite cubics will be given in Section 4. More interesting is that a family of Hermite interpolatory masks will be constructed in Section 4 such that it includes the piecewise Hermite cubics as a special case. Several more new examples of biorthogonal multiwavelets will also be provided in Section 4. Due to the length of this paper, we shall discuss the smoothness and convergence problems associated with the refinable function vectors of our construction elsewhere. For any j ∈ N ∪ {0}, let Oj be the ordered set of {µ ∈ Zs+ : |µ| = j} in the lexicographic order. That is, (ν1 , · · · , νs ) is less than (µ1 , · · · , µs ) in lexicographic
HERMITE INTERPOLANTS AND BIORTHOGONAL MULTIWAVELETS
13
order if νj = µj for j = 1, · · · , i − 1 and νi < µi . By #Oj we denote the cardinality of the set Oj . The Kronecker product of two matrices A = (aij )1≤i≤l,1≤j≤n and B, written as A ⊗ B, is defined to be the following matrix
a11 B a21 B A ⊗ B := ...
a12 B a22 B .. .
··· ··· .. .
a1n B a2n B . .. .
al1 B
al2 B
···
aln B
¡ ¢r×r Theorem 3.1. Let a and e a in `0 (Zs ) be two masks satisfying the discrete biorthogonal relation (1.9) with a dilation matrix M . Suppose that J a (0) satisfies the condition (1.2) and e a satisfies the sum rules of order k for some positive integer k, i.e., there exist vectors yeµ , |µ| < k in Cr with ye0 6= 0 such that X
(−1)|ν| Jεea (ν)T yeµ−ν =
0≤ν≤µ
X
mµν yeν
∀ µ ∈ Zs+ , |µ| < k, ε ∈ ΩM ,
|ν|=|µ|
where mµν are given in (1.7) and ΩM is a complete set of representatives of the distinct cosets of Zs \M Zs with 0 ∈ ΩM . Let Oj be the ordered set of {µ ∈ Zs+ : |µ| = j} in the lexicographic order. Then yeµ , |µ| < k are uniquely determined up to a scalar multiplier constant by the following recursive relation: J a (0) ye0 = me y0 and for j = 1, · · · , k − 1, the vectors yeµ (µ ∈ Oj ) are determined by ¶ h i−1µ X X (e yµ )µ∈Oj = mIr(#Oj ) −(mµν )µ,ν∈Oj ⊗ J a (0) mην J a (µ − η) yeν . µ∈Oj
0≤η 4. 7741 2771 0 0 − 221184 184320 and e a(−β) = Ue a(β)U for all β ∈ N. Then φa is a Hermite interpolant such that a ν2 (φ ) ≈ 3.84745 and φa has accuracy order 7. φea is a dual function vector of φa such that ν2 (φea ) ≈ 0.91843 and φea has accuracy order 2. Therefore, φea is a continuous dual function vector of φa since ν∞ (φea ) ≥ 0.41843. Moreover, φa ∈ C 3 . Though the primal masks in all the above examples are interpolatory masks, the CBC algorithm can be easily applied to the general case. Let us take an example from Plonka and Strela [41] as the primal mask and then we use the CBC algorithm to construct a dual function vector for it. Example 4.8. The primal mask a in [41] is given by " 13 # " 51 # " 51 9 9 − 64 − 64 64 64 64 a(0) = , a(1) = , a(2) = 11 7 21 9 − 64 − 21 64 64 64 # " 6413 9 a(3) =
64
64
− 11 64
7 − 64
,
a(β) = 0
∀β 6= 0, 1, 2, 3.
9 64 9 64
# ,
28
BIN HAN
Then as pointed out by Plonka and Strela [41] that the refinable function vector φa is a polynomial B-spline of order 6 with double knots. νp (φa ) = 4 + 1/p for any 1 ≤ p ≤ ∞ and φa has accuracy order 6. By Theorem 3.1, we have ye0T = [1, 0],
ye1T = [3/2, −3/14],
ye3T = [39/56, −43/168],
ye2T = [17/14, −9/28],
ye4T = [529/1680, −1/7].
By using the CBC algorithm, we find a dual mask e a of a given by " e a(2) = " e a(4) = " e a(6) =
159239 122880
28291 122880
− 12198157 3317760
919531 3317760 1273 − 10240
1347 5120 847007 − 1105920
47627 − 368640 # 531
411 8192
8192
321 − 4096
63 − 512
,
#
" ,
e a(3) =
#
" ,
e a(5) = e a(7) =
#
− 18621 40960
7953 40960
5400373 3317760 1169 − 7680
− 1043381 3317760 # 551 − 30720
961993 " 3317760 19 − 8192 5 − 4096
331813 3317760 # 45 − 8192 3 1024
, ,
,
and e a(β) = Ue a(3−β)U for β = −4, −3, −2, −1, 0, 1, and otherwise, e a(β) = 0. Then φea is a dual function vector of φa with ν2 (φea ) ≈ 1.13543 and has accuracy order 5. Therefore, φea is a continuous dual function vector of φa since ν∞ (φea ) ≥ 0.63543. The graphs of all the above examples in this section are presented at the end of this paper. More examples of biorthogonal multiwavelets can be constructed by using the CBC algorithm studied in this paper and such algorithm can be easily implemented. Finally, we mention that the study of approximation order of biorthogonal multiwavelets in this paper is not only useful for construction of biorthogonal multiwavelets but also helpful for construction of orthogonal multiwavelets.
HERMITE INTERPOLANTS AND BIORTHOGONAL MULTIWAVELETS
29
0.15 1
0.1
0.8
0.05
0.6
0
0.4
−0.05 −0.1
0.2
−0.15 0 −1
−0.5
0 (a)
0.5
1
−1
1.5
10
1
5
0.5
0
−0.5
0 (b)
0.5
1
−2
0 (d)
2
4
−5
0
−10 −0.5 −4
−2
0 (c)
2
4
−4
Figure 1. (a), (b), (c) and (d) are the graphs of φb11 , φb21 , φea1 and φea2 in Example 4.5, respectively. φb1 is the piecewise Hermite cubics and φea is a C 1 dual function vector of φb1 .
0.2 1 0.1
0.8 0.6
0
0.4 −0.1
0.2 0 −3
−0.2 −2
−1
0 (a)
1
2
3
2.5
−3
−2
−1
0 (b)
1
2
3
10
2 5
1.5 1
0
0.5 0
−5
−0.5 −10
−1 −4
−2
0 (c)
2
4
−4
−2
0 (d)
Figure 2. (a), (b), (c) and (d) are the graphs of φb12 , φb22 , φea1 and φea2 in Example 4.6, respectively. φb2 is a C 2 Hermite interpolant and φea is a continuous dual function vector of φb2 .
2
4
30
BIN HAN
0.2
1 0.8
0.1
0.6
0
0.4 −0.1
0.2 0
−0.2
−3
−2
−1
0 (a)
1
2
3
−3
−2
−1
0 (b)
1
2
3
10
2 1.5
5
1 0 0.5 −5
0 −0.5
−10
−4
−2
0 (c)
2
4
−4
−2
0 (d)
2
4
Figure 3. (a), (b), (c) and (d) are the graphs of φa1 , φa2 , φea1 and φea2 in Example 4.7, respectively. φa is a C 3 Hermite interpolant and φea is a continuous dual function vector of φa . 0.4
1 0.8
0.2
0.6
0
0.4 −0.2 0.2 0
−0.4 0
1
2
3
0
1
2
(a)
3
(b)
2
10
1.5 5
1 0.5
0
0 −0.5
−5
−1 −1.5
−10 −4
−2
0
2 (c)
4
6
−4
−2
0
2
4
(d)
Figure 4. (a), (b), (c) and (d) are the graphs of φa1 , φa2 , φea1 and φea2 in Example 4.8, respectively. φa ∈ C 3 is a polynomial B-spline of order 6 with double knots and φea is a continuous dual function vector of φa .
6
HERMITE INTERPOLANTS AND BIORTHOGONAL MULTIWAVELETS
1.5
31
10
5
1
0 0.5
−5 0 −10
−0.5 −6
−4
−2
0 (c)
2
4
6
−6
φea1 1
−4
−2
0 (d)
2
of φea2 1 b1
Figure 5. (c) is the graph of and (d) is the graph Example 4.5. φea1 ∈ C 1 is a dual function vector of φ .
4
6
in
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
[19] [20]
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