Approximation Properties and Construction of Hermite Interpolants and Biorthogonal Multiwavelets. Bin Han

Transcription

1 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 ã of a such that ã 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 dual function vector with support [ 4, 4 of the piecewise Hermite cubics is given. 99 Mathematics Subject Classification. 4A25 4A5 65D5 46E35 4A63. 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 8544-, USA. bhan@math.princeton.edu, bhan Typeset by AMS-TEX

2 2 BIN HAN. Introduction In her celebrated paper [3, 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 [4. 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 [4. 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,, 4, 2, 2, 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 [4, 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 [ 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 [, 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 [,, 2, 6, 7, 25, 26, 27, 33, 38, 4, 4, 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

3 HERMITE INTERPOLANTS AND BIORTHOGONAL MULTIWAVELETS 3 M is called a dilation matrix if lim n M n =. 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: φ = β Z s a(β)φ(m β), (.) where φ = (φ,, φ r ) T is a r vector of functions, called a refinable function vector, and a is a finitely supported sequence of r r matrices on Z s, called the (matrix refinement) mask. When r =, φ is called a scalar refinable function and a is called a scalar refinement mask. By J a () we denote the following matrix associated with a mask a as J a () := β Z s a(β). If φ,, φ r are functions in L (R s ) with stable shifts and φ = (φ,, φ r ) T satisfies the refinement equation (.) with a mask a, then it was proved by Dahmen and Micchelli [2 that J a () has a simple eigenvalue m and all other eigenvalues in modulus < m. (.2) Conversely, if J a () satisfies the condition (.2), it was proved by Heil and Collella [26 and Cabrelli, Heil, and Molter [4 that there exists a unique vector φ of compactly supported distributions such that φ satisfies (.) and J a () φ() = φ() with φ() 2 = and the first nonzero component of φ() being positive. We call such solution the normalized solution of (.) and throughout this paper we denote the normalized solution of (.) with mask a by φ a. If φ is another distribution solution of (.), then we must have φ = cφ a 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 >, 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, 2, 27, 32, 34, 4, 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 l(z s ) denote the linear space of all sequences on Z s and l (Z s ) denote the linear space of all finitely supported sequences on Z s. For any positive integer r,

4 4 BIN HAN by ( l(z s ) ) r r we denote the linear space of all sequences of r r matrices on Z s and by ( l(z s ) ) r we denote the linear space of all sequences of r vectors on Z s. Similarly, we define ( l (Z s ) ) r r and ( l (Z s ) ) r. Given any compactly supported distribution vector f = (f,, f r ) T, we define the following linear operator c f on ( l(z s ) ) r by c f (λ) := β Z s λ(β) T f( β), λ ( l(z s ) ) r, (.3) where A T denotes the transpose of a matrix A. The shifts of f are said to be linearly independent if c f (λ) = for λ in ( l(z s ) ) r implies λ =. It was proved by Jia and Micchelli [3 that the shifts of a compactly supported distribution vector f = (f,, f r ) T are linearly independent if and only if the sequences ( fj (ξ + 2πβ) ) β Z s, j =,, r are linearly independent for all ξ C s. The shifts of f are stable if the sequences ( fj (ξ + 2πβ) ) β Z s, j =,, r are linearly independent for all ξ R s. Therefore, if the shifts of φ a are stable or linearly independent, then (.2) holds true. Before proceeding further, let us introduce some notation. Recall that M denotes a dilation matrix. Let Ω M be a complete set of representatives of the distinct cosets of Z s /MZ s. Without loss of generality, we assume that Ω M. For any mask a in ( l (Z s ) ) r r, we shall use the following notation throughout this paper J a ε (µ) := β Z s a(ε + Mβ)(M ε + β) µ /µ!, µ Z s +, ε Ω M (.4) and where J a (µ) := β Z s a(β)(m β) µ /µ! = ε Ω M J a ε (µ), µ Z s +, (.5) Z s + := {(β,, β s ) Z s : β j, j =,, s} and for any β = (β,, β s ) Z s and µ = (µ,, µ s ) Z s +, β µ := β µ βµ s s, µ! := µ! µ s! and β := β + + β s. For any ν = (ν,, ν s ), µ = (µ,, µ s ) Z s +, we say that ν µ if ν j µ j for all j =,, s, and we say that ν < µ if ν µ and ν µ. For any mask a in ( l (Z s ) ) r r, 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 vectors {y µ : µ Z s +, µ < k} with y such that ν µ ( ) ν J a ε (ν) T y µ ν = ν = µ where the numbers m µ ν are uniquely determined by m µ ν y ν µ Z s +, µ < k, ε Ω M, (.6) (M x) µ µ! = ν = µ m µ ν x ν ν!, x Rs. (.7) Given a vector φ = (φ,, φ r ) T of compactly supported distributions on R s, let S(φ) = {c φ (λ) : λ ( l(z s ) ) r } where the linear operator cφ is defined in (.3).

5 HERMITE INTERPOLANTS AND BIORTHOGONAL MULTIWAVELETS 5 Following Heil, Strang and Strela [27, we say that φ has accuracy order k if Π k S(φ) where Π k denotes the set of all polynomials of total degree less than k. We also agree that Π = {}. 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 [3 for such concepts and related results. When φ a is a refinable function vector with a mask a and a dilation matrix M, under the assumption that the shifts of φ a are linearly independent, Cabrelli, Heil, and Molter [3, 4 characterized the accuracy order of φ a in terms of the order of sum rules satisfied by the mask a. Also see [5, 27, 32, 34, 4 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 φ a 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 [4, 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 φ a C k has linearly independent shifts, then it is necessary that φ a has accuracy order k +. A function vector φ in ( L 2 (R s ) ) r is called a primal function vector if φ satisfies the refinement equation (.) with a mask a in ( l (Z s ) ) r r and the shifts of φ are linearly independent. A dual function vector φ of φ is a function vector in ( L2 (R s ) ) r such that φ satisfies the refinement equation (.) with a mask ã and R s φ(x) φ(x + β) T dx = δ(β)i r β Z s, (.8) where I r denotes the r r identity matrix, and δ() =, δ(β) = for all β Z s \{}. Clearly, if φ and φ satisfy the conditions in (.8), then the shifts of φ and φ are linearly independent, respectively. A necessary condition for φ and φ to satisfy the conditions in (.8) is the following well known discrete biorthogonal relations: β Z s a(β) ã(β + Mα) T = mδ(α)i r α Z s. (.9) If a mask a in ( l (Z s ) ) r r ( satisfies (.2) and there exists a sequence ã in l (Z s ) ) r r such that the conditions in (.9) are satisfied, then we say that a is a primal mask and any such mask ã will be called a dual mask of a. Dahmen and Micchelli proved in [2 that if a is a primal mask with ã being a dual mask of a, then φ a is a primal function vector with φã being a dual function vector of φ a if and only if the subdivision schemes associated with a and ã converge in the L 2 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 (.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 [5, and Kovačević 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, 4 and was used by Plonka and Strela in [4 to construct smooth refinable function vectors. This factorization technique was also used by Strela in [44 to construct univariate

6 6 BIN HAN 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, 4 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 (.9), which is a system of linear equations, to get a dual mask ã of a. To achieve high vanishing moments of the resulting biorthogonal multiwavelet, the dual mask ã must satisfy the sum rules of high order. Even in the simplest case s = and M = (2), it is not easy to use the definition of sum rules given in (.6) to achieve desired order of sum rules satisfied by ã. The reason is that when r >, to obtain a dual mask ã of a given primal mask a, even the equations in the biorthogonal conditions (.9) are linear, the equations given in (.6) for sum rules are no longer linear equations since in general the vectors y ν in (.6) are determined by ã. 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 [7, Dahmen, Han, Jia and Kunoth [, 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 [2 to construct scalar biorthogonal wavelets with arbitrary vanishing moments. In this paper, we shall generalize the CBC algorithm in [2 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, 2, 24. In Section 3, we shall follow the line developed in [6, 2, 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 ã of a such that ã 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 [8. 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 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 [4. Such primal function vector belongs to C 4 η for any η >, has accuracy order 6 and has support [, 3.

7 HERMITE INTERPOLANTS AND BIORTHOGONAL MULTIWAVELETS 7 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. Given a mask a in ( l (Z s ) ) r r, the subdivision operator Sa associated with the mask a and a dilation matrix M is defined by S a λ(α) = β Z s a(α Mβ) T λ(β), α Z s, λ ( l(z s ) ) r. (2.) By Π r k we denote the set of r polynomial vectors with each component being a polynomial of degree at most k, and by Π r := k= Πr k. For any p Πr, we naturally have a sequence p Z s ( l(z s ) ) r given by p Z s(α) = p(α), α Z s. Since the mapping from Π r Π r Z s is one-to-one and onto, without confusion, we may use p Π r as both a polynomial vector in Π r and a polynomial sequence in ( l(z s ) ) r since it is easy to distinguish them in the context. In particular, for any p Π r, if S a (p Z s) is a polynomial sequence in ( l(z s ) ) r, then we may use Sa p to represent both the polynomial vector in Π r and the polynomial sequence in ( l(z s ) ) r. Lemma 2.. Let p Π r Z s. Then S a p Π r Z s if and only if a(ε + Mβ) T p(x M ε β) = a(mβ) T p(x β) ε Ω M, x R s. β Z s β Z s If this is the case, then S a p is given by the following polynomial vector S a p(x) = β Z s a(ε + Mβ) T p(m x M ε β), ε Ω M. (2.2) Proof. By definition of the subdivision operator S a, for any ε, α Z s, we have S a p(ε + Mα) = β Z s a T (ε + Mβ)p(M (ε + Mα) M ε β). The claim in this lemma follows directly from the above equality. By D j we denote the partial derivative with respect to the j-th unit coordinate. For any µ = (µ,, µ s ) Z s +, D µ denotes the differential operator D µ Dµ s s. Moreover, we write D = (D,, D s ) T. For any polynomial p Π r, we shall use the following convention: p(x id) T := p (µ) (x) T ( id) µ /µ!, µ Z s + where i is the imaginary unit such that i 2 =. For any p Π r, we observe p(x id) T [ f( )g( ) = [ p (µ) (x id) T f( ) [ ( id) µ g( ) /µ!. (2.3) µ Z s + For any mask a in ( l (Z s ) ) r r, let H a (ξ) := β Z s a(β)e iβ ξ /m, ξ R s and define H a k (ξ) := H a( (M T ) ξ ) H a( (M T ) k ξ ), k N.

8 8 BIN HAN Proposition 2.2. If p Π r, then S a p Π r if and only if p(x id) T H a (2πβ ) = β Z s \M T Z s. (2.4) Moreover, if p Π r and S a p Π r, then p(x id) T H a (2πM T β) = (S a p)(mx) T for all β Z s, and for any k >, p(x id) T H a k (2πM T β) = (S a p)(mx id) T H a k (2πβ) β Z s. Proof. By the definition of H a, we have m p(x id) T H a (2πβ ) = β Z s µ Z s + = β Z s µ Z s + µ! p(µ) (x) T a(β)( id) µ[ e im β ξ ξ=2πβ µ! p(µ) (x) T a(β)( M β) µ e 2πiM β β = β Z s p(x M β) T a(β)e 2πiM β β e 2πiM ε β ε Ω M = β Z s p T (x M ε β)a(ε + Mβ). Thus, it yields that (2.4) is equivalent to S a p Π r by Lemma 2.. Moreover, the above equality gives us p(x id) T H a (2πM T β) = (S a p)(mx) T for all β Z s. Note that H a k (ξ) = Ha (ξ)h a k ( (M T ) ξ ). By (2.3), we have p(x id) T Hk a (2πM T β) = µ! p(µ) (x id) T H a (2πM T β)( id) µ[ Hk a ( (M T ) ) (2πM T β). µ Z s + On the other hand, by p(x id) T H a (2πM T β) = (S a p)(mx) T, we deduce that p (µ) (x id) T H a (2πM T β) = D µ[ p(x id) T H a (2πM T β) = D µ[ (S a p)(m ) T (x). So, in conclusion, we have p(x id) T H a k (2πM T β) = µ Z s + µ! Dµ[ (S a p)(m ) T (x)( im D) µ H a k (2πβ) = (S a p)(mx id) T H a k (2πβ). For any mask a in ( l (Z s ) ) r r and any positive integer k, we define a subspace P a k of Πr k as P a k := {p Π r k : S j ap Π r k Z s j N}. (2.5) It is evident that P a k is invariant under the subdivision operator S a defined in (2.). We are ready to prove the following result:

9 HERMITE INTERPOLANTS AND BIORTHOGONAL MULTIWAVELETS 9 Theorem 2.3. If p Pk a, then c φ a(p) Π k where Pk a operator c φ a is defined in (.3). In particular, c φ a(p)(x) = β Z s p(β) T φ a (x β) = p(x id) T ˆφa () = is defined in (2.5) and the µ Z s + µ! p(µ) (x) T ( id) ˆφ a (). Conversely, if the sequences ( ˆφj (2π(M T ) ε + 2πβ) ) β Z s, j =,, r are linearly independent for all ε Ω M T where φ a = (φ,, φ r ) T is the normalized solution of (.) with the mask a, then for any q Π k S(φ a ), there exists a unique p P a k such that c φ a(p) = q. Proof. Since p P a k, by definition, Sj ap Π r for all j N {}. We claim that p(x id) T ˆφa (2πβ) = β Z s \{}. (2.6) For any β Z s \{}, there is a positive integer k such that β Z s \(M T ) k Z s. Note that ˆφ a (ξ) = H a k (ξ) ˆφ a( (M T ) k ξ ) and p(x id) T ˆφa (2πβ) = µ Z s + Therefore, to prove (2.6), it suffices to prove that µ! Dµ[ p(x id) T H a k (2πβ) ( id) µ[ ˆφa ( (M T ) k ) (2πβ). p(x id) T H a k (2πβ) = β Z s \(M T ) k Z s, p P a k, x R s. (2.7) By Proposition 2.2, (2.7) holds true for k =. Suppose that (2.7) is true for k. Note that Hk a(ξ) = Ha k (ξ)f(ξ) where f(ξ) := Ha( (M T ) k ξ ). By induction hypothesis, we deduce that for any β Z s \(M T ) k Z s, p(x id) T H a k (2πβ) = µ Z s + µ! Dµ[ p(x id) T H a k (2πβ) ( id) µ f(2πβ) =. For any β Z s \M T Z s, by Proposition 2.2 and induction hypothesis, p(x id) T H a k (2π(M T ) k β) = (S a p)(mx id) T H a k ( 2π(M T ) k 2 β ) =. Therefore, (2.7) holds true and we proved (2.6). By Poisson summation formula (see [5), we have c φ a(p)(x) = β Z s p(β) T φ a (x β) = β Z s e 2πiβ x p(x id) T ˆφa (2πβ) = p(x id) T ˆφa () = µ Z s + µ! p(µ) (x) T ( id) µ ˆφa (). Conversely, let q Π k S(φ a ). Since the sequences ( ˆφj (2πβ) ), j =,, r β Z s are linearly independent, by [3, Lemma 8.2, there exists a unique polynomial vector p Π r k such that c φ a(p) = q. Thus, to prove that p Pa k, it suffices to prove

10 BIN HAN that if p Π r k and c φ a(p) Π k, then S a p Π r k. By Poisson summation formula, we have e 2πiβ x p(x id) T ˆφa (2πβ) = p(β) T φ a (x β) = c φ a(p) Π k, β Z s β Z s from which it follows that (2.6) holds true. Since ( ˆφj (2π(M T ) ε+2πβ) ) β Z s, j =,, r are linearly independent for all ε Ω M T, there exits β ε, β r ε Z s such that the matrix [ ˆφ a( 2π(M T ) ε+2πβ ε),, ˆφa ( 2π(M T ) ε+2πβ r ε) is invertible. Thus, for ξ in a neighborhood of, the following matrix F (ξ):= [ ˆφa ( 2π(M T ) ε+2πβ ε+(m T ) ξ ),, ˆφ a( 2π(M T ) ε+2πβ r ε+(m T ) ξ ) is invertible. By ˆφ a (ξ) = H a (ξ) ˆφ a( (M T ) ξ ), we have [ ˆφa (2πε + 2πM T β j ε + ξ) j r = Ha (2πε + ξ)f (ξ). Hence, the following identity is valid for ξ in a neighborhood of : H a (2πε + ξ) = [ ˆφa (2πε + 2πM T βε j + ξ) F j r (ξ). Therefore, by (2.3), p(x id) T H a (2πε) = [ p(x id) T H a (2πε + ) () = [ p (µ) (x id) T ( ˆφa 2π(ε + M T βε) j ) [ j r ( id) µ F (). µ Z s + By (2.6), p (µ) (x id) T ˆφ a( 2π(ε + M T β) ) = D µ[ p(x id) T ˆφ a (2π(ε + M T β) ) = for all ε Ω M T \{} and β Z s. So p(x id) T H a (2πε) = for all ε Ω M T \{}. By Proposition 2.2, S a p Π r k. The main result in this section is the following theorem. Theorem 2.4. Let a be a mask in ( l (Z s ) ) r r and satisfy the condition (.2) with a dilation matrix M. Let φ a = (φ,, φ r ) T be the normalized solution of the refinement equation (.) with the mask a and the dilation matrix M. Suppose that the sequences ( ˆφj (2π(M T ) ε + 2πβ) ) β Z s, j =,, r are linearly independent for all ε Ω M T. Then the following statements are equivalent: (a) φ a has accuracy order k; (b) dim Pk a = dim Π k where Pk a is defined in (2.5) and Pa k is invariant under S a ; (c) The mapping c φ a P a k : Pk a Z s Π k is onto where c φ a is defined in (.3); (d) a satisfies the sum rules of order k. Moreover, if a satisfies the sum rules of order k given in (.6) with {y µ : µ Z s +, µ < k} and y T ˆφ a () =, then x µ µ! = ν µ β Z s βν ν! yt µ νφ a (x β) µ < k, µ Z s +.

11 HERMITE INTERPOLANTS AND BIORTHOGONAL MULTIWAVELETS Proof. From Theorem 2.3, we see that for any positive integer k, the mapping c φ a P a k : Pk a Π k is well defined. Under the assumption that the sequences ( ˆφj (2πβ) ), j =,, r are linearly independent, by [3, Lemma 8.2, c β Z s φ a P a k is one-to-one. Therefore, by Theorem 2.3, φ a has accuracy order k if and only if c φ a P a k is one-to-one and onto which is also equivalent to dim Pk a = dim Π k. Thus, we proved that (a), (b) and (c) are equivalent. We now discuss sum rules. Since c φ a P a k is a one-to-one and onto mapping between Pk a and Π k, the inverse mapping of c φ a P a k carries the structure of Π k into Pk a. Define p µ := c φ a (x µ /µ!), µ Z s +, µ < k. Since c φ a(d ν p µ ) = D ν c φ a(p µ ) for all ν, µ Z s + with µ < k, we may assume that p µ = ν µ y µ ν x ν ν! µ Z s +, µ < k, (2.8) for some r vectors y ν, ν < k. Note that c φ a(s a p µ )(Mx) = c φ a(p µ )(x) = x µ /µ!. We have ( (M (S a p µ )(x) = c x) µ ) ( φ = c a φ m µ x ν ) µ! a ν = m µ ν p ν (x), ν! ν = µ ν = µ where m µ ν are given in (.7). By Lemma 2., we have m µ ν p ν (Mx) = S a p µ (Mx) = a(ε + Mβ) T p µ (x M ε β) ε Ω M. β Z s ν = µ Setting x = in the above equality, we have (.6). Hence, (c) implies (d). To prove that (d) implies (a), we define p µ as in (2.8). By the definition of sum rules and observing that D ν (p µ ) = p µ ν where by convention p ν for ν Z s +, it is not difficult to verify that S a p Π r k and S ap = ν = µ mµ ν p ν for all µ < k. Thus p µ Pk a, and by Theorem 2.3, c φ a(p µ) Π k for all µ < k. Set q µ := c φ a(p µ ). Then we have q µ (M x) = c φ a(s a p µ )(x) = m µ ν q ν (x) µ Z s +, µ < k. (2.9) ν = µ Note that D ν q µ = q µ ν for all ν Z s + (see [4, Theorem 3.2). We may assume q µ (x) = ν µ l µ ν x ν ν!, µ Zs +, µ < k, for some l µ C, µ Z s + with µ < k. We claim that l µ = for any < µ < k. Suppose that for some < j < k, l ν = for all < ν < j. Then q µ (x) = l x µ /µ! + l µ for any µ = j. By (2.9), we have l (M x) µ µ! + l µ = l ν = µ m µ ν x ν ν! + ν = µ m µ ν l ν.

12 2 BIN HAN Therefore, by (.7), l µ = ν = µ mµ ν l ν for all µ = j. But when j >, it was observed in [3 that all the eigenvalues of (m µ ν ) ν =j, µ =j are less than in modulus. Hence l µ = for all µ = j. Now q µ = l x µ /µ! for all µ < k. By Theorem 2.3, l = y T ˆφ () follows directly from (.2) and y. 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 Pk a of Πr k and Pa k is an invariant subspace under S a. When r =, the sum rules condition in (.6) can be restated as follows: a(ε + Mβ)(ε + Mβ) µ = a(mβ)(mβ) µ µ < k, ε Ω M, (2.) β Z s β Z s which was given by Jia [29 and can be easily seen either by Lemma 2. and Pk a = Π k or by (.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 φ a are linearly independent. It was also given by Jia [29, Heil, Strang, and Strela [27 and Plonka [4 for the case s = 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 Pk a = {p Πr k : Sj ap Π r k j =,, dim Πr k } since Pk a Πr k, and if a satisfies the sum rules of order k +, then S a P a k = (c φ a P a k ) τ M (c φ a P a k ) where τ M : Π k Π k is given by τ M (p)(x) = p(m x). Therefore, the structure of S a restricted to the subspace Pk a can be easily analyzed by using a much simpler operator τ M. 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,, 2, 2, 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 [. 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 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 {}, let O j be the ordered set of {µ Z s + : µ = j} in the lexicographic order. That is, (ν,, ν s ) is less than (µ,, µ s ) in lexicographic

13 HERMITE INTERPOLANTS AND BIORTHOGONAL MULTIWAVELETS 3 order if ν j = µ j for j =,, i and ν i < µ i. By #O j we denote the cardinality of the set O j. The Kronecker product of two matrices A = (a ij ) i l, j n and B, written as A B, is defined to be the following matrix a B a 2 B a n B a A B := 2 B a 22 B a 2n B a l B a l2 B a ln B Theorem 3.. Let a and ã in ( l (Z s ) ) r r be two masks satisfying the discrete biorthogonal relation (.9) with a dilation matrix M. Suppose that J a () satisfies the condition (.2) and ã satisfies the sum rules of order k for some positive integer k, i.e., there exist vectors ỹ µ, µ < k in C r with ỹ such that ν µ ( ) ν J ãε (ν) T ỹ µ ν = ν = µ m µ ν ỹ ν µ Z s +, µ < k, ε Ω M, where m µ ν are given in (.7) and Ω M is a complete set of representatives of the distinct cosets of Z s \MZ s with Ω M. Let O j be the ordered set of {µ Z s + : µ = j} in the lexicographic order. Then ỹ µ, µ < k are uniquely determined up to a scalar multiplier constant by the following recursive relation: J a () ỹ = mỹ and for j =,, k, the vectors ỹ µ (µ O j ) are determined by [ ( (ỹ µ ) µ Oj = mi r(#oj) (m µ ν ) µ,ν Oj J a () η<µ ν = η m η νj a (µ η) ỹ ν )µ O j. In particular, when M = di s where d is an integer, we have J a () ỹ = d s ỹ and ỹ µ = ( ) d µ +s I r J a () ν<µ d µ ν J a (µ ν) ỹ ν, < µ < k, µ Z s +. Proof. From the discrete biorthogonal relation (.9), we have m α Z s δ(α)i r α ν ν! = a(β) ã(β + Mα) T αν ν!, ν Zs +. α Z s β Z s Note that ( α ν (M ν! = β + α) M β ) ν ν! Therefore, = ( ) η (M β) η η! η ν (M β + α) ν η. (ν η)! mδ(ν)i r = η ν α Z s = ε Ω M η ν ( ) η (M β)η a(β) ã(β + Mα) T (M β + α) ν η η! (ν η)! β Z s ( ) η a(ε + Mβ) (M ε + β) η J ãε (ν η) T. η! β Z s

14 4 BIN HAN Hence, we have mδ(ν)i r = ε Ω M η ν ( ) η J a ε (η)j ãε (ν η) T, ν Z s +. (3.) For any ν, µ Z s + such that µ < k and ν µ, multiplying ( ) ν ỹ µ ν with both sides of (3.) and taking sum, we have mỹ µ = ν µ = ( ) ν mδ(ν)ỹ µ ν ε Ω M ν µ η ν = ε Ω M η µ η ν µ = ε Ω M η µ J a ε (η) ( ) ν η J a ε (η) J ãε (ν η) T ỹ µ ν ( ) ν η J a ε (η) J ãε (ν η) T ỹ µ ν ν µ η ( ) ν J ãε (ν) T ỹ µ η ν. Since ã satisfies the sum rules of order k, by (.6) and the above equality, we have mỹ µ = ε Ω M η µ = J a ε (η) η µ ν = µ η Thus, we deduce that mỹ µ ν = µ m µ ν J a () ỹ ν = ν = µ η m µ η ν ỹ ν m µ η ν J a (η) ỹ ν = η<µ ν = η η µ ν = η m η ν J a (µ η) ỹ ν. m η νj a (µ η) ỹ ν, µ Z s +, µ < k. (3.2) Regarding (ỹ µ ) µ Oj as a r(#o j ) column vector where #O j is the cardinality of the set O j, the equation (3.2) can be rewritten in the following matrix form: [ ( mi r(#oj) (m µ ν ) µ,ν Oj J a () (ỹ µ ) µ Oj = η<µ ν = η m η νj a (µ η) ỹ ν )µ O j, for j =,,, k. When j =, we have mỹ = J a () ỹ. Since (.2) holds true, such vector ỹ is unique up to a scalar multiplier constant. When j =,, k, it is easy to see that all the eigenvalues of (m µ ν ) µ,ν Oj are σ µ, µ O j and therefore are less than in modulus where σ = (σ,, σ s ) T and σ j are all the eigenvalues of M. Since a satisfies the condition (.2), we conclude that the matrices mi r(#oj ) (m µ ν ) µ,ν Oj J a () are invertible for j > which completes the proof. When r =, Theorem 3. can be reduced to the following form which is a generalization of a result in [2.

15 HERMITE INTERPOLANTS AND BIORTHOGONAL MULTIWAVELETS 5 Corollary 3.2. Let a and ã in l (Z s ) be two masks satisfying the discrete biorthogonal relation (.9) with a dilation matrix M. Suppose that β Zs a(β) = m := det M and ã satisfies the sum rules of order k. Let h a (µ) := β Z s ã(β)βµ /m for µ Z s +. Then h a (µ), µ < k are given by the following recursive relation: for µ Z s + and µ < k, h a (µ) = δ(µ) m ν<µ ( ) µ ν µ! ν!(µ ν)! ha (ν) a(β)β µ ν. β Z s In the following we shall study how to construct dual masks with arbitrary order of sum rules. For simplicity of exposition, we shall first investigate the case when the primal masks are interpolatory masks. This restriction is not essential and can be removed as demonstrated in [6. We shall outline such construction for the general case at the end of this section. CBC Algorithm. (Construction by Cosets Algorithm) Let a ( l (Z s ) ) r r be a mask such that J a () satisfies (.2) with a dilation matrix M, a() is invertible and a(mβ) = for all β Z s \{}. Let k be any positive integer. () Compute the vectors ỹ µ, µ Z s + with µ < k as in Theorem 3. such that ỹ ; (2) For any ε Ω M \{}, choose an appropriate subset E ε of Z s such that after setting ã(ε + Mβ) = for all β Z s \E ε, the following linear system ν µ ( ) ν J ãε (ν) T ỹ µ ν = ν = µ m µ ν ỹ ν µ Z s +, µ < k (3.3) has at least one solution for {ã(ε + Mβ) : β E ε } where J ãε (ν) is defined in (.4) as J ãε (ν) := β E ε ã(ε + Mβ)(M ε + β) ν /ν!; (3) Construct the coset of ã at as follows: for any α Z s, [ ã(mα) = mδ(α)i r ( ) ã(ε + Mβ) a(ε + M(β α)) T a() T. β E ε ε Ω M \{} Then the mask ã is a dual mask of the given mask a and ã satisfies the sum rules of order k. Proof. Since a(mβ) = for all β Z s \{}, we can rewrite (.9) as mδ(α)i r = a(ε + Mβ) ã(ε + Mβ + Mα) T ε Ω M β Z s = a()ã(mα) T + ε Ω M \{} β Z s a(ε + Mβ Mα) ã(ε + Mβ) T. Therefore, under the assumption that a is an interpolatory mask, the discrete biorthogonal relation (.9) is equivalent to the equality in Step (3). To prove

16 6 BIN HAN that ã satisfies the sum rules of order k, by the definition of sum rules in (.6) and equation (3.3), it suffices to verify that ν µ ( ) ν J ã (ν) T ỹ µ ν = ν = µ m µ ν ỹ ν µ Z s +, µ < k. (3.4) As in Theorem 3., from the discrete biorthogonal relation (.9), we have mỹ µ = ε Ω M η µ J a ε (η) ν µ η ( ) ν J ãε (ν) T ỹ µ η ν, µ Z s +. As we know from the proof of Theorem 3., {ỹ µ : µ < k} satisfies the following equation mỹ µ = Jε a (η) m µ η ν ỹ ν, µ < k. ε Ω M η µ ν = µ η Therefore, subtracting the last equality from the previous one and using equation (3.3), we end up with η µ J a (η) [ ν µ η ( ) ν J ã (ν) T ỹ µ η ν ν = µ η m µ η ν ỹ ν = µ < k. Since J a () = a() and therefore, is invertible, by induction, the above equality implies (3.4) which completes the proof. The advantage of the above algorithm lies in that we only need to deal with each coset ε Ω M \{} separately in Step (2). If ã is a dual mask of a such that ã satisfies the sum rules of order k, then ã must be from the above CBC algorithm by choosing some subset E ε and a solution in (3.3). The reader may wonder how to choose the subset E ε of Z s such that there is at least one solution to (3.3). In general, when E ε is large enough, there must be a solution to (3.3). In the scalar case r =, this fact was discussed in detail in [6, 2 and we shall sketch the ideas later in this section. For the multiwavelet case r >, let us demonstrate this fact for the univariate case. First, without loss of generality, we may assume that ỹ = [,,, T. For any ε Ω M \{}, let E ε be any subset of Z such that #E ε = k. Now set ã(ε + Mβ) = for any β Z\E ε. For any matrix A, let A[i, j denote its (i, j) entry. Set ã(ε + Mβ)[i, j = (or any numbers as you want) for all β E ε, i = 2,, r and j =,, r. Then (3.3) has a unique solution for {ã(ε + Mβ)[, j : β E ε, j =,, r} since in this case (3.3) is reduced to the following equation β E ε ã(ε + Mβ)[, j(m ε + β) µ = g ε,j (µ), µ < k where g ε,j (µ) are constants derived from (3.3). It is evident that the above equation has a unique solution since its coefficient matrix is a Vandermonde matrix. From the above CBC algorithm, it is not difficult to see that for any positive integer k, there always exists a dual mask which satisfies the sum rules of order k.

17 HERMITE INTERPOLANTS AND BIORTHOGONAL MULTIWAVELETS 7 We shall prove this fact for the general case at the end of this section. In the case r =, the equation (3.3) becomes β Z s ã(ε + Mβ)(ε + Mβ) µ = h a (µ) µ < k, µ Z s + where h a (µ) are calculated from Corollary 3.2. See [2 for more details. When r =, in [6, we demonstrated that given any primal mask a, the CBC algorithm in [6, 2 reduces the construction of all the dual masks of a such that the dual masks satisfy the sum rules of order k to a problem of obtaining a sequence b in l (Z s ) such that b() =, b(mβ) = for all β Z s \{}, and for each ε Ω M \{}, β Z s b(ε + Mβ)(ε + Mβ) µ = g ε (µ) µ < k, µ Z s + where g ε (µ), µ < k can be similarly computed as h a (µ). In the scalar case r =, an algorithm is proposed in [6, 2, 24 to concretely implement the general CBC algorithm. Let a be any given scalar primal mask a which is symmetric about the origin. For any positive integer k, such algorithm gives a unique dual mask of a such that the dual mask satisfies the sum rules of order 2k and is symmetric about the origin. See [6, 2, 24 for some examples of scalar biorthogonal wavelets constructed by such CBC algorithm. We mention that in the scalar case, if a is an interpolatory mask (i.e., a() = and a(mβ) = for all β Z s \{}), then a convolution method is proposed in the literature to construct scalar biorthogonal wavelets. Let us recall this method from Han [2. For any positive integers n and r, we define a multivariate polynomial q n,r as follows: q n,r (x,, x n ) := r j = r j n= (n + j + + j n )! x j (n )!j! j n! x jn n, x,, x n R. It is evident that q n,r (x σ(),, x σ(n) ) = q n,r (x,, x n ) for any permutation σ on (,, n). If x + +x n =, then it is well known from the probability theory that n x r i q n,r (x,, x i, x i+,, x n ) = n, r N. (3.5) i= For any mask a in ( l (Z s ) ) r r, by a(z) we denote its symbol given by where a(z) := β Z s a(β)z β, z T s, T s := {(z,, z s ) C s : z = = z s = }. Observe that a sequence a is an interpolatory mask with a dilation matrix M if and only if a ( e iξ+2πiε) /m = ξ R s, ε Ω M T where m := det M and Ω M T is a set of complete representatives of the cosets of Z s /M T Z s with Ω M T. Therefore, a new interpolatory mask can be obtained from a known one using convolution and the probability identity (3.5). Let us recall the convolution method proposed in [2 to build new interpolatory masks from known ones. For a proof of the following theorem, see Proposition 3.7 in [2.

18 8 BIN HAN Theorem 3.3 (Convolution Method). Let M be a dilation matrix with m := det M and a be an interpolatory mask in l (Z s ). For any positive integer k, let the sequence b be defined by its symbol as b(e iξ ) := m 2 k[ a(e iξ ) k qm,k (( a(e iξ+2πiε )/m ) ε Ω M T \{}), ξ R s. Then b is a dual mask of a. If a satisfies the sum rules of order r, then b satisfies the sum rules of order (k )r. Later on, it was demonstrated by Ji, Riemenschneider and Shen in [28 that if a scalar refinable function φ has certain regularity with an interpolatory mask, a dual scaling function of φ with arbitrary regularity can be obtained by using the convolution method. We demonstrated in [6 that the CBC algorithm is optimal in the sense of the sum rules of the dual mask with respect to its support while the convolution method is not optimal for almost all the case. See [6, 24 for the comparison of the CBC algorithm and the convolution method. It is known that for a primal scaling function, a dual scaling function with arbitrary smoothness always exists. In applications, short support and high vanishing moments of wavelets are very important. Therefore, in applications, it is more important to construct such dual scaling function with relatively small support while achieving the optimal sum rules rather than to construct a dual scaling function with certain smoothness and a very large support. For a given scalar scaling function, even if a family of dual scaling functions with increasing smoothness can be given, but if the ratio of their smoothness (or the orders of sum rules) over their support is too small, it is still not practical in applications. See [6 for more detail. Another important observation here is that the convolution method only works for the scalar case r =. In the following, we shall study how to construct dual masks in the most general case. We shall demonstrate in a constructive way that for any given primal mask a, there always exists an dual mask ã of a such that ã satisfies the sum rules of arbitrary order. For any mask a ( l (Z s ) ) r r and for any ε ΩM, by a ε we denote the following sequence a ε (β) := a(ε + Mβ), β Z s. Under the lexicographic order, Ω M is an ordered set, say Ω M = {ε,, ε m }. It is a well known fact that a is a primal mask with a dilation matrix M if and only if Rank ( a ε (z),, a εm (z) ) = r z T s. Moreover, ã is a dual mask of a if and only if ε Ω M a ε (z) ã ε (z) T = mi r z T s. The following known fact is employed in the lifting scheme in some sense. For detail about lifting schemes, see [5, 36, 46 and references therein. Proposition 3.4. Let a in ( l (Z s ) ) r r be a primal mask with a dilation matrix M. Suppose there exist sequences ã ω, ω Ω M in ( l (Z s ) ) r r such that a ε (z) ã ω ε (z) T = mδ(ω)i r ω Ω M ε Ω M

19 HERMITE INTERPOLANTS AND BIORTHOGONAL MULTIWAVELETS 9 and the matrix ( ã ω ε (z) ) is invertible for all z T s. Then for any sequence ω,ε Ω M ã in ( l (Z s ) ) r r, ã is a dual mask of a if and only if there exists a sequence b in ( l (Z s ) ) r r with b (z) = I r such that ã ω (z) = ε Ω M b ε (z)ã ε ω(z) ω Ω M, z T s. (3.6) Proof. If ã satisfies (3.6), it is easy to verify that a ω (z) ã ω (z) T = a ω (z) ã ε ω(z) T b ε (z) T = mb (z) T = mi r. ω Ω M ω Ω M ε Ω M Thus, ã is a dual mask of a. Conversely, if ã is a dual mask of a, setting ( bε (z),, b εm (z) ) := ( ã ε (z),, ã εm (z) )( ã ω ε (z) ) ω,ε Ω M where {ε,, ε m } = Ω M, then b (z) = I r and (3.6) holds true. It is well known that the invertible matrix ( ã ω ε (z) ) is used in deriving the ω,ε Ω M associated dual wavelet masks from a and ã and such matrix can be constructed from the mask a by an algorithm proposed in [37. With the help of Theorem 3. and Proposition 3.4, we demonstrate the following (obvious?) fact: Theorem 3.5. Let a be a primal mask in ( l (Z s ) ) r r with a dilation matrix M. For any positive integer k, there exists a dual mask ã of a such that ã satisfies the sum rules of order k. Proof. Since a is a primal mask, by Quillen-Suslin Theorem, there exist sequences ã ω (ω Ω M ) in ( l (Z s ) ) r r such that the conditions in Proposition 3.4 are satisfied. Such sequences ã ω can be constructed from a by an algorithm proposed in [37. Let ã be a sequence in ( l (Z s ) ) r r given by ã ω (z) := ε Ω M b ε (z)ã ε ω(z), ω Ω M (3.7) where b ( l (Z s ) ) r r is a sequence to be determined. In order to obtain a dual mask ã (in the above form) of a such that ã satisfies the sum rules of order k, by Proposition 3.4, it suffices to demonstrate that there does exist a sequence b such that b (z) = I r and ã satisfies the following sum rule equations ( ) ν J ãω(ν) T ỹ µ ν = m µ ν ỹ ν µ Z s +, µ < k, ω Ω M, (3.8) ν µ ν = µ where the vectors ỹ µ, µ < k are computed from Theorem 3. with ỹ. Without loss of generality, we may assume ỹ = [,, T. From (3.7), for any ν Z s + and ω Ω M, we deduce that J ãω(ν) = α Z s ã ω (α) (M ω + α) ν ν! = b ε (β) (M ε + β)η ã ε η! ω(α β) (M (ω ε) + α β)ν η (ν η)! ε Ω M α Z s β Z s α Z s = Jε(η)F b ω(ν ε η), ε Ω M η ν

20 2 BIN HAN where Fω(µ) ε := β Z ã ε ω(β) ( M (ω ε) + β ) µ /µ! for any µ Z s s +. Thus, the linear system (3.8) can be rewritten as: for all µ < k and ω Ω M, ( ) ν Fω(ν ε η) T Jε(η) b T ỹ µ ν = m µ ν ỹ ν. (3.9) ε Ω M ν µ η ν ν = µ Therefore, after changing the order of summation in the left side of (3.9), we get ( ) ν+η Fω(ν) ε T Jε(η) b T ỹ µ η ν = m µ ν ỹ ν. ε Ω M η µ ν µ η ν = µ In other words, (3.8) can be rewritten in the following form: Fω() ε T ( ) µ Jε(µ) b T ỹ + ( ) ν+η Fω(ν) ε T Jε(η) b T ỹ µ η ν ε Ω M η<µ ν µ η ε Ω M = m µ ν ỹ ν, µ Z s +, µ < k, ω Ω M ν = µ with a sequence b to be determined. Note that each b ε (β) is an r r matrix. By b ε (β)[i, j we denote the (i, j) entry of the matrix b ε (β). Regard all b ε (β)[i, j as parameters for all ε Ω M, β Z s, 2 i r and j r. Let c in ( l (Z s ) ) r ( ) T be given by c(β) = b(β)[,,, b(β)[, r. Then it is easily seen that Jω(µ) c = Jω(µ) b T ỹ since ỹ = [,,, T. Therefore, we end up with the following equations: Fω() ε T ( ) µ Jε c (µ) + G ε ω,νjε c (ν) = gµ ω, ω Ω M, µ < k, ε Ω M η<µ ε Ω M (3.) where G ε ω,ν are constant matrices, and gµ ω are vectors with parameters {b ε (β)[i, j : ε Ω M, β Z s, 2 i r, j r}. Since Fω() ε = ã ε ω() and ( ã ε ω(z) ) is ε,ω Ω M invertible, the matrix ( Fω() ) ε has full rank which implies that the equations ε,ω Ω M (3.) can be uniquely solved as J c ε (µ) = f ε µ ε Ω M, µ Z s +, µ < k, (3.) where f ε µ are vectors with parameters {b ε (β)[i, j : ε Ω M, β Z s, 2 i r, j r}. We now prove that the following linear system J c (µ) = f µ µ Z s +, µ < k (3.2) has a solution. Note that (3.7) and the biorthogonal condition imply that mb (z) = a ε (z) ã ε (z) T z T s ε Ω M which can be rewritten as mb(mα) = ε Ω M β Z s a(ε + Mβ) ã ε (ε + Mβ + Mα) T, α Z s. (3.3)

21 HERMITE INTERPOLANTS AND BIORTHOGONAL MULTIWAVELETS 2 From (3.3), by the same argument as in Theorem 3., we have J b (ν) = ε Ω M η ν ( ) η J a ε (η) J ãε (ν η) T, ν Z s + which is similar to (3.). Since ã must satisfy (3.8), by a similar argument as in Theorem 3., we have m ν µ ( ) ν J b (ν) ỹ µ ν = η µ ν = η Since ỹ µ, µ < k satisfy (3.2), we deduce that ν µ m η ν J a (µ η) ỹ ν µ Z s +, µ < k. (3.4) ( ) ν J b (ν) ỹ µ ν = ỹ µ µ Z s +, µ < k. (3.5) By the uniqueness of f ε µ, ε Ω M and µ < k, we conclude that the linear system (3.2) must be equivalent to the linear system (3.5) since J c (µ), µ < k can also be uniquely solved from (3.5). It is evident that b (z) = I r is a solution to (3.5). Therefore, b (z) = I r is a solution to (3.2). Set b (Mβ) = δ(β)i r and b ε (β)[i, j := (or any other numbers as long as b ε in ( l (Z s ) ) r r ) for all ε ΩM \{}, β Z s, 2 i r and j r. Then for each ε Ω M \{}, f ε µ in (3.) are constants. To demonstrate that for each ε Ω M \{}, (3.) has at least one solution, it suffices to prove that for each ε Ω M \{} and j =,, r, the following linear system β E ε C(β)(M ε + β) µ /µ! = h ε µ µ Z s +, µ < k (3.6) has a solution for C(β), β E ε where E ε is a subset of Z s, C(β) = b ε (β)[, j and h ε µ := f ε µ[j. It is evident that (3.6) can be rewritten as β E ε C(β)β µ = h ε µ µ Z s +, µ < k (3.7) for some constants h ε µ. The above linear system always has a solution for {C(β) : β E ε } when E ε is large enough. For example, if E ε := {β Z s + : β < k}, then (3.7) has a unique solution for {C(β) : β E ε }. For the solvability of (3.7), the reader is referred to [39 on multivariate polynomial interpolation. The proof of Theorem 3.5 illustrates the general procedure to construct all dual masks of any given primal mask such that the dual masks satisfy the sum rules of any arbitrary order. Though a general CBC algorithm for the general primal mask is possible but the exposition of such algorithm is much more complicated and technical. For the sake of simplicity, we shall not discuss the details here. In general, there are three major steps in constructing a dual mask with arbitrary order of sum rules. First, solve the linear system given in the discrete biorthogonal relation (.9) by Proposition 3.4 or by any other method. Secondly, for any fixed positive integer k, compute the vectors ỹ µ, µ < k as in Theorem 3.. Finally, solve the linear system given in the sum rule equations (.6). The existence of a solution