W. Dahmen and C.A. Micchelli. Biorthogonal wavelet expansion. Constr. Approx., 13:293-- 328, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....H is called a primal mask and any such mask e H in (1.3) is called a dual mask of H . H and e H are also called wavelet lters in the literature. If H and e H satisfy (1. 3) and the subdivision schemes associated with H and e H converge in the L norm, then it was proved in Dahmen and Micchelli [7] that and e satisfy (1.2) where and e are the normalized solutions to the re nement equations (1.1) with the masks H and e H , respectively. It is known that there are symmetric smooth orthogonal multiwavelet bases while there is no symmetric continuous orthogonal scalar wavelet (see [8, ....

W. Dahmen, and C. A. Micchelli, Biorthogonal wavelet expansions, Constr. Approx. 13 (1997), 293-328.

....Proposition 3.5 is: Corollary 3.6. Suppose that (1.1) has a compactly supported stable solution Phi 2 L 2 (IR ) Then, P(0) satisfies condition E(1) and lP( 0, 2 f0; 1g nf0g, for any left eigenvector l of eigenvalue 1 of P(0) Remark. A result similar to Corollary 3. 6 was obtained by [DM]. Theorem 3.8 of [S] characterizes the stability of a solution Phi of (1.1) in terms of the mask. The characterization was given under the assumption that P(0) satisfies condition E(1) and lP( 0 for 2 f0; 1g nf0g, where l is the unit left (row) eigenvector of eigenvalue 1 of P(0) ....

Dahmen, W. and C. A. Micchelli, Biorthogonal wavelet expansions, (1995), in Constr. Approx., to appear. 17

....sequence of r r complex valued matrices on Z, called the (matrix) mask for . When r = 1, the re nable function vector in (1.2) is a scalar function. The re nement equation in (1. 2) as well as various properties of its re nable function vector has been well studied in the literature, see [3, 9, 14, 15, 17, 20, 22, 23, 28, 29, 31] and references therein. A wavelet system is usually generated by some wavelet function vectors ; 1; L which are derived from a d re nable function vector as follows: c ( b ( 1; L for some appropriate matrices b ( of 2 periodic trigonometric ....

W. Dahmen and C. A. Micchelli, Biorthogonal wavelet expansions, Constr. Approx., 13 (1997), 293-328.

.... e iu ) k ) Finally, the B spline B k is factored from each of the entries of # 1 resulting in the final vector F . Thus (4.7) # F = # B 1 k DU 1 # #. We state below the corollary that summarizes those observations. In that corollary we use the following (essentially known: cf. e.g. [DM], Lemma 2.1, and the case k = 1 in Result 2.3) lemma: 21 Lemma 4.8. Let # be a refinable vector such that (i) # # is continuous on O 2#ZZ, with O some neighborhood of the origin and # #(0) #= 0; ii) the sequences # # 2#ZZ are in c 0 (2#ZZ) and are linearly independent. Let P be ....

Dahmen, W., and C. A. Micchelli, Biorthogonal wavelet expansions, Constr. Approx. 13 (1997), 293--328.

....supported mask e H such that (1.3) holds, then the mask H is called a primal mask and any such mask e H in (1.3) is call a dual mask of H . If H and e H satisfy (1. 3) and the subdivision schemes associated with H and e H converge in the L 2 norm, then it was proved in Dahmen and Micchelli [7] that OE and e OE satisfy (1.2) where OE and e OE are the normalized solutions to the refinement equations (1.1) with the masks H and e H , respectively. It is known that there are symmetric smooth orthogonal multiwavelet bases while there is no symmetric continuous orthogonal scalar wavelet (see ....

W. Dahmen and C. A. Micchelli, Biorthogonal wavelet expansions, Constr. Approx., 13 (1997), 293--328.

....dual scaling function will determine the smoothness of their derived wavelets, and the approximation orders of the scaling function and its dual scaling function will determine the vanishing moments of their derived wavelets. For more detail on (bi)orthogonal wavelets, the reader is referred to [3, 5, 6, 7, 9, 10, 12, 13, 14, 16, 24, 27, 33, 35, 37, 42, 44] and references cited there. By Omega we denote the set of the vertices of the unit cube [0; 1] s . For a positive integer k, we say that a sequence a on Z s satisfies the sum rules of order k if (1:4) X fi2Z s a(2fi )p(2fi ) X fi2Z s a(2fi)p(2fi) 8 2 Omega ; p 2 Pi ....

W. Dahmen and C. A. Micchelli, Biorthogonal wavelet expansions, Constr. Approx., 13 (1997), pp. 293--328.

....of their derived wavelets, and the accuracy orders of the scaling function and its dual scaling function will determine the vanishing moments of their derived wavelets. For more detail on (bi)orthogonal wavelets, the reader is referred Construction of Biorthogonal Wavelets by CBC Algorithm 5 to [9, 10, 11, 12, 14, 15, 17, 18, 20, 22, 31, 32, 37, 40, 44, 46, 49, 56, 59] and references cited there. The shifts of a compactly supported function OE : R s C are said to be stable if for any 2 R s , there exists a multi integer fi 2 Z s such that b OE( 2fi) 6= 0. By Omega we denote the set of the vertices of the unit cube [0; 1] s . For a positive integer ....

W. Dahmen and C. A. Micchelli, Biorthogonal wavelet expansions, Constr. Approx., 13 (1997), 293--328.

....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, 2, 4, 6, 7, 10, 11, 13, 14, 15, 18, 19, 22, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33] and references therein on discussion of various topics on multiwavelets and their applications. Let (Z) denote the linear space of all sequences on Z and 0 (Z) denote the linear space of all finitely supported sequences on Z. For any positive integer r, by Gamma (Z) Delta r Thetar we ....

.... matrix associated with a mask a as J a (0) X fi2Z a(fi) If OE 1 ; Delta Delta Delta ; OE r are functions in L 1 (R) with stable shifts and OE = OE 1 ; Delta Delta Delta ; OE r ) T satisfies the refinement equation (1) with a mask a, then it was proved in Dahmen and Micchelli [7] that J a (0) has a simple eigenvalue 2 and all other eigenvalues in modulus are less than 2: 2) Conversely, if J a (0) satisfies the condition (2) it was known that there exists a unique vector OE of compactly supported distributions such that OE satisfies (1) and J a (0) b OE(0) b ....

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W. Dahmen and C. A. Micchelli, Biorthogonal wavelet expansions, Constr. Approx., 13 (1997), 293-- 328.

....a generalization of the results of [17] to vectors of functions was provided by Hogan in [10] also see [21] and [27] but the techniques used were again inherently univariate. If OE is a vector of compactly supported functions in L 1 (IR s ) with linearly independent shifts, it is known (cf. [4], 11] 26] that the matrix M : 2 Gammas X ff2ZZ s a(ff) 1:2) has a simple eigenvalue 1 and all the other eigenvalues of M are less than 1 in modulus. In fact, this result is valid under the weaker condition that the entries of P ff2ZZ s OE( Delta Gamma ff) are linearly ....

W. Dahmen and C. A. Micchelli, Biorthogonal wavelet expansions, Constr. Approx., 13 (1997), 293--328.

....if all component functions of OE are refinable with the same coefficients. This last assumption seems to be very stringent, one could also allow matrices A k instead of the coefficients a k . This leads to matrix vector refinement equations, which were studied by Dahmen and Micchelli in [DM3]. But we will see that the above stronger condition is the right choice in our setting. However, this concept shows its full power for the case of the whole space Omega = IR n . We will have to find some appropriate approximation in the case of a bounded domain. But let us first recall those ....

W. Dahmen and C.A. Micchelli, Biorthogonal Wavelet Expansions, in preparation.

....nite dimensional, nested spaces are employed. Nevertheless, the following papers give the avor of the issues. The paper by Cohen [8] takes one subdivision rule and discusses construction via the cascade method . Cohen and Daubechies [9] discuss the issue of stability, and Dahmen and Micchelli [12] deal with existence. Our paper is not the rst to suggest that other inner products than (2.1) are possible for wavelet systems. In an earlier work [32] we used the Euclidian inner product on the data points C k 1 to construct wavelets of very small support, and Aldroubi, Eden, and Unser [2] ....

W. Dahmen and C. A. Micchelli. Biorthogonal wavelet expansions. Constructive Approximation, 13(3):293-328, 1997.

.... be studied on the basis of the refinement equation form (A) see [16, 30] The calculation of the optimal smoothness parameter t in (B) is a delicate issue, compare [30, 31, 50] Most of the papers on multiresolution analyses study the case L = 1, for the so called multiwavelet case L 1, see [28]. Assumptions (A) D) yield characterizations of Sobolev spaces H s (lR d ) 0 s s 0 , analogous to Lemma 2 (see [25] for a survey on multilevel approximations and related function spaces) E) is essentially used in the definition of the spaces on domains and for the corresponding ....

Dahmen, W. and C. Micchelli, Biorthogonal wavelet expansions, IGPM-Report Nr. 114 , RWTH Aachen, May 1995.

....functions or multiwavelet functions. The properties of refinable function vectors, multi scaling functions and multiwavelet functions with dilation factor m = 2 are discussed in many papers. Some of the earliest occurrences are [1] 9] 11] 12] more recent treatments include [4] [6], 14] 19] 27] 28] 30] It is straightforward to extend these results to the case of general m, following the one dimensional case which is discussed for example in [16] 29] 37] Two multiwavelets OE, OE form a biorthogonal pair if they satisfy the biorthogonality conditions Z OE ....

....based on a suggestion in [36] is presented in x5. Implementation details for both algorithms are stated in x6. Section 7 contains some examples. 2. Representations of multiwavelet masks. The results in this section are well known. Proofs or appropriate references can be found e.g. in [2] 5] [6] or [27] Throughout this paper, all calculations are based on the masks H, H alone. In terms of masks, the biorthogonality conditions (1.5) are represented as X k h ( k h ( T k mj = ffi ffi 0j I: 2.1) Here and in the remainder of this paper, I denotes an identity matrix of ....

W. Dahmen and C. A. Micchelli, Biorthogonal wavelet expansions, Constr. Approx., 13 (1997), pp. 293--328.

.... Hence, the question occurs of how the L 2 stability of B( Phi) for the solution vector Phi of (1) can be ensured, just by appropriate choice of the (two scale) symbol PP P P P P P PP ( N X l=0 PP P P P P P PP l e Gammai l : This problem has also been studied very recently in [3,4,9,14,15]. By Fourier transform of (1) we have Phi( PP P P P P P PP ( 2 ) Phi( 2 ) 2) Surface Fitting and Multiresolution Methods 293 A. Le M ehaut e, C. Rabut, and L. L. Schumaker (eds. pp. 293 300. Copyright o c 1997 by Vanderbilt University Press, Nashville, TN. ISBN ....

....MM M M M M M MM (or a linear operator) let us introduce the following Condition E. The spectral radius of MM M M M M M MM is less than or equal to 1, i.e. ae(MM M M M M M MM ) 1, and 1 is the only eigenvalue of MM M M M M M MM on the unit circle. Moreover, 1 is a simple eigenvalue. As shown in [4,10], we have: Proposition 1. Let Phi be a stable L 1 solution vector of (1) Then for the corresponding symbol PP P P P P P PP ( we have: a) PP P P P P P PP (0) satisfies Condition E. b) The solution vector Phi provides approximation order 1, i.e. we have yy y y y y y yy T 1 X l= Gamma1 ....

Dahmen, W., and C. A. Micchelli, Biorthogonal wavelet expansions, Constr. Approx., to appear.

....3.5 is: Corollary 3.6. Suppose that (1.1) has a compactly supported stable solution Phi 2 L 2 (IR s ) Then, P(0) satisfies condition E(1) and lP( 0, 2 f0; 1g s nf0g, for any left eigenvector l of eigenvalue 1 of P(0) Remark. A result similar to Corollary 3. 6 was obtained by [DM]. Theorem 3.8 of [S] characterizes the stability of a solution Phi of (1.1) in terms of the mask. The characterization was given under the assumption that P(0) satisfies condition E(1) and lP( 0 for 2 f0; 1g s nf0g, where l is the unit left (row) eigenvector of eigenvalue 1 of P(0) ....

Dahmen, W. and C. A. Micchelli, Biorthogonal wavelet expansions, (1995), in Constr. Approx., to appear.

.... Note that the shifts of OE 1 ; OE r are stable if and only if, for any 2 R, the sequences ( OE j ( 2k ) k2Z , j = 1; r, are linearly independent (see [18] If OE 1 ; OE r are functions in L 1 (R) with stable shifts, it was proved by Dahmen and Micchelli [5] that the matrix M has a simple eigenvalue 1 and all the other eigenvalues of M are less than 1 in modulus. In fact, this result is valid under a weaker condition that the sequences ( OE j (2k ) k2Z , j = 1; r, are linearly independent. Indeed, for k 2 Z, it follows from the ....

W. Dahmen and C. A. Micchelli, Biorthogonal wavelet expansions, Constr. Approx., to appear.

....that, in accordance with Lemma 3.3, one scaling function is symmetric and the other is antisymmetric. Moreover, the sum of the supports grows exactly by 1. Unfortunately, the new functions are not orthogonal and for practical applications a biorthogonal multiscaling function should be constructed [DM, SS4]. 500 G. PLONKA AND V. STRELA 0.5 1 1.5 2 1 0.5 0.5 1 0.5 1 1.5 2 0.2 0.4 0.6 0.8 1 Fig. 3.1. Symmetric multiscaling function with approximation order 3. Example 2. In the second example, we construct polynomial, symmetric multiscaling functions with two components, short support, and ....

....# 4,0 and # 4,1 are symmetric, continuously di#erentiable functions. Note that # 4,0 , # 4,1 are finite element functions studied in [SS3] They are presented in Figure 3.2. Obviously, functions # 4,0 and # 4,1 are not orthogonal. For the construction of dual scaling functions and wavelets see [DM, SS4]. The procedure can be repeated as follows. Take M r 2k (#) 0 2 1 z 1 z (k # N, z = e i# ) and M r 2k 1 (#) 2k 1) 1 z) k 0 (1 z) k 2 (k 1) k # N, z = e i# ) 502 G. PLONKA AND V. STRELA 0.5 1 1.5 2 0.2 0.4 0.6 0.8 1 0.5 1 1.5 2 0.4 0.2 0.2 ....

W. Dahmen and C. A. Micchelli, Biorthogonal wavelet expansions, Const. Approx., 13 (1997), pp. 293--328.

....x and in IR s . The function f is called the Fourier transform of f . The Fourier transform is naturally extended to the space of all compactly supported distributions. If OE is a vector of compactly supported functions in L 1 (IR s ) with linearly independent shifts, it is known (cf. [2], 9] 23] that the matrix M : 2 Gammas X ff2ZZ s a(ff) 1:2) has a simple eigenvalue 1 and all the other eigenvalues of M are less than 1 in modulus. In fact, this result is valid under the weaker condition that the entries of P ff2ZZ s OE( Delta Gamma ff) are linearly independent. ....

W. Dahmen and C. A. Micchelli, Biorthogonal wavelet expansions, Constr. Approx. 13 (1997), 293--328.

....techniques from [8] As an example we construct multi scaling functions with various smoothness dual to Hermite cubics and obtain a biorthogonal pair based on Geronimo HardinMassopust (GHM) 5] orthogonal scaling functions. Other approaches to the biorthogonality of multiwavelets can be found in [1, 2, 3]. 2. Construction of Dual Symbol Let us assume that we are given a stable multi scaling function OE(t) OE 0 (t) OE r Gamma1 (t) T from L 1 (R) satisfying a matrix dilation equation OE(t) 2 X k h 0 (k)OE(2t Gamma k) 2.1) where h 0 (k) are r by r matrices. Our goal is to ....

W. Dahmen and C. Micchelli, Biorthogonal wavelet expansions, Constr. Approx. (1995).

....in [CD] The wavelets constructed are not spline type. An example of biorthogonal bivariate wavelets with dilation matrix M = 2I are constructed in [CS] where one basis is continuous piecewise linear polynomials and its dual basis is generated by compactly supported functions but not continuous. [DM] has a construction of biorthogonal wavelets in a multivariate setting. However, the dual basis constructed may only be distributions. The first example of bivariate compactly supported fundamental refinable function which is in C 1 were given in [DGL] and the smoothness estimate of the example ....

....is straightforward. see. e.g. Do] There are many papers on multivariate wavelets theory and the constructions in the literature; in particular on fast decay orthogonal wavelets and compactly supported prewavelets. We list here a few of them for the interested reader: BDR] CD] CS] CSW] [DM], JS] LLS2] M] Me] RiS1] RiS2] and references cited in these papers for the further references. For several dimensions, there are few constructions of compactly supported biorthogonal orthogonal wavelets, and the analogy of the compactly supported orthogonal wavelets constructed by ....

Dahmen, W. and C. A. Micchelli, Biorthogonal wavelet expansions, Const. Approx. to appear.

....domain. The corresponding subdivision algorithm is closely related with the cascade algorithm (see e.g. 38] Theorem 2. 1, 39] Actually, there is no reason to restrict the subdivision operator to the L 2 case, general solution vectors with components in L p can also be handled (see e.g. [12,30,31]) However, in L 2 (IR) the transition operator often provides simpler results. Since both, T and S are linear operators, their spectral properties are computable by considering their representing matrices. x2. Existence, Uniqueness, and Stability of Scaling Vectors In this section we ....

....in L 2 stable solutions. Let us introduce the following definition. A matrix (or a linear operator) AA A A A A A AA is said to satisfy Condition E if it has a simple eigenvalue 1 and the moduli of all its other eigenvalues are less then 1. First we observe some necessary conditions (see e.g. [12,25,34]) Theorem 3. Let Phi Phi Phi Phi Phi Phi Phi Phi Phi be a compactly supported, L 2 stable solution vector of (3) Then for the corresponding symbol PP P P P P P PP ( we have: a) PP P P P P P PP (0) satisfies Condition E. b) There exists a nonzero vector yy y y y y y yy 2 IR r ....

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Dahmen, W., and C. A. Micchelli, Biorthogonal wavelet expansions, Constr. Approx. 13 (1997), 293--328.

.... that these assumptions hold with respect to the L 2 norm (or H s norms when applicable) using information about the refinement equations (10) and (17) i.e. about the masks and symbols (a ff ) m( and (a ff ) m ( are well known and can be formulated in various terms (see [34, 23, 16]) A1. The scaling function OE 2 L 2 (R d ) is of compact support, and generates a MRA fV j g. Its mask (a ff ) defined by (10) is finite, the symbol m( is a trigonometric polynomial. A2. There are 2 d Gamma 1 wavelets 2 V 1 , 2 0 , of compact support such that the system Phi 0 [ ....

W. Dahmen, C. A. Micchelli, Biorthogonal wavelet expansions, Constr. Approx. 13, 1997, 293--328.

.... that the shifts of OE 1 ; OE r are stable if and only if, for any 2 IR, the sequences ( OE j ( 2k ) k2ZZ , j = 1; r, are linearly independent (see [18] If OE 1 ; OE r are functions in L 1 (IR) with stable shifts, it was proved by Dahmen and Micchelli [5] that the matrix M has a simple eigenvalue 1 and all the other eigenvalues of M are less than 1 in modulus. In fact, this result is valid under a weaker condition that the sequences ( OE j (2k ) k2ZZ , j = 1; r, are linearly independent. Indeed, for k 2 ZZ, it follows from the ....

W. Dahmen and C. A. Micchelli, Biorthogonal wavelet expansions, Constr. Approx., to appear.

....e OE b 2 (ec Gamma x) e OE b 1 (x) e b 2 ( e d Gamma x) e b 1 (x) 4. 6) where Phi b = OE b 1 ; OE b 2 ) T , Psi b = b 1 ; b 2 ) T , e Phi b = e OE b 1 ; e OE b 2 ) T , e Psi b = e b 1 ; e b 2 ) T (see [14] It is known (see [3], 16] that if H; G; e H; e G form a PR FIR multifilter bank which generates symmetric antisymmetric biorthogonal multiwavelets, then H(0) p 2; e H(0) p 2 satisfy Condition E, and H= p 2; e H= p 2 satisfy the vanishing moment conditions of order at least one: 8 : 1; 0)H(0) p ....

W. Dahman and C. A. Micchelli, Biorthogonal wavelet expansions, Constr. Approx. 13 (1997), 293--328.

....time domain. The corresponding subdivision algorithm is closely related to the cascade algorithm (see e.g. 38] Theorem 2. 1, 39] Actually, there is no reason to restrict the subdivision operator to the L 2 case, general solution vectors with components in L p can also be handled (see e.g. [12,30,31]) However, in L 2 (IR) the transition operator often provides simpler results. Since both, T and S are linear operators, their spectral properties are computable by considering their representing matrices. 2. Existence, Uniqueness, and Stability of Scaling Vectors In this section we ....

....in L 2 stable solutions. Let us introduce the following definition. A matrix (or a linear operator) AA A A A A A AA is said to satisfy Condition E if it has a simple eigenvalue 1 and the moduli of all its other eigenvalues are less then 1. First we observe some necessary conditions (see e.g. [12,25,34]) Theorem 3. Let ## # # # # # ## be a compactly supported, L 2 stable solution vector of (3) Then for the corresponding symbol PP P P P P P PP (#) we have (a) PP P P P P P PP (0) satisfies Condition E. b) There exists a nonzero vector yy y y y y y yy # IR r such that yy y y y y y yy ....

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Dahmen, W., and C. A. Micchelli, Biorthogonal wavelet expansions, Constr. Approx. 13 (1997), 293--328.

....j;j 0 ffi ff;ff 0 ; 3.13) i.e. the system f ae g ae2Rnf0g ; f ae 0 g ae 0 2Rnf0g forms a biorthogonal wavelet basis. For further information concerning biorthogonal bases and suitable extensions the reader is referred e.g. to Cohen and Daubechies [8] and to Dahmen and Micchelli [12]. In general, the extension problem is nontrivial. However, in our special case, the solution is quite simple and can always be given explicitly. Unless otherwise stated, all symbols and subsymbols are assumed to be defined with respect to the scaling matrix M from Theorem 2.1. Theorem 3.1 ....

Dahmen, W. and Micchelli, C.A., Biorthogonal wavelet expansions, IGPM-Bericht Nr. 114, RWTH Aachen, 1995.

....) k ) Finally, the B spline B k is factored from each of the entries of Phi 1 resulting in the final vector F . Thus (4:7) b F = b B Gamma1 k DU 1 b Phi: We state below the corollary that summarizes those observations. In that corollary we use the following (essentially known: cf. e.g. [DM], Lemma 2.1, and the case k = 1 in Result 2.3) lemma: Lemma 4.8. Let Phi be a refinable vector such that (i) b Phi is continuous on O 2 ZZ, with O some neighborhood of the origin and b Phi(0) 6= 0; ii) the sequences b Phi j2 ZZ are in c 0 (2 ZZ) and are linearly independent. Let P be the ....

Dahmen, W., and C. A. Micchelli, Biorthogonal wavelet expansions, Constr. Approx. 13 (1997), 293--328.

....vector refinement equation: OE = X k2ZZ a(k)OE(2 Delta Gamma k) 1) where a : ZZ IR r Thetar is a finitely supported sequence of r Theta r matrices called the (matrix refinement) mask. Vector refinement equations and refinable function vectors have been well studied recently (cf. e.g. [1, 2, 3, 4, 5, 8, 9, 11]) We investigate how perturbation of a finitely supported mask affects the corresponding refinable function vector and associated pyramid scheme. As in [11] we assume throughout that the mask a satisfies the following condition: X k2ZZ a(k) 2 0 0 with lim n 1 ( 2) n = 0: 2) ....

....h(2r Delta Gamma 2(j Gamma 1) for j = 2; r. Then OE 0 satisfies the moment conditions in (4) If there exists OE 2 (L p (IR) r such that lim n 1 kQ n a OE 0 Gamma OEk p = 0, we say that the pyramid scheme associated with a converges to OE in the L p norm. It is well known (cf. [4, 8, 11]) that if a satisfies condition (2) and the pyramid scheme associated with a converges in L p norm for some 1 p 1, then e T 1 X k2ZZ a(2k) e T 1 X k2ZZ a(1 2k) e T 1 : 6) Also, the limit function vector OE is the normalized solution of equation (1) The Effect of ....

Dahmen, W. and C. A. Micchelli, Biorthogonal wavelet expansions, Constr. Approx. 13 (1997), 293--328.

.... of the refinable vectors was approached in [62] Using TST and the general form of the factorization of P ( we created a handy algorithm for the construction of multi scaling functions with given approximation order [59] Biorthogonal multiwavelet bases wre discussed in several recent papers [1, 17, 40, 52], but no explicit algorithms for the construction of the corresponding scaling functions and wavelets were given. We filled this gap and suggested the cofactor method for the construction of biorthogonal multiwavelets [75] Some results on stability and linear independece of multiwavelets and ....

....An effective algorithm for the construction of multi scaling functions with symmetry and approximation was described in Section 3.6. Now we are going to present an algorithm for the construction of dual multi scaling functions. Biorthogonality of multiwavelets was already approached in [1, 17, 40, 52] but no systematic procedure for the construction of duals were given. The advantage of our method is that it gives a transparent way to get the dual basis with the same symmetry and approximation properties as the direct one. In this section we slightly change the notation and denote by H 0 (z) ....

W. Dahmen and C. Micchelli, Biorthogonal wavelet expansions, preprint (1995).

....can be ensured if the transfer operator T associated with PP P P( satisfies special spectral conditions. This method can also be applied in the multivariate setting (see [21] In the meantime, it turned out that the basic conditions assumed by Shen [21] are necessary for stability of Phi (see [4,11,14]) In order to handle the transfer operator T in practice, one has to use its representing matrix, which in fact can be given explicitly in terms of Kronecker products of coefficient matrices PP P P n (see [15,20] The resulting conditions, which are spectral conditions to the representing ....

....that the stability of scaling vectors is closely related with the convergence of corresponding subdivision schemes and cascade algorithms. In fact, the convergence of the stationary subdivision scheme can be taken as a criteria for stability of Phi in L p (IR) This subject is addressed in [3,4,12]. In particular, relations between spectral conditions of the transfer operator and the convergence of the cascade algorithm are considered in [21] We are especially interested in the first two methods. The purpose of this paper is to study the relation between the spectral properties of the ....

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Dahmen, W., and Micchelli, C. A., Biorthogonal wavelet expansions, Constr. Approx., to appear.

....almost everywhere on IR s . x3. Vector Subdivision Let A = AA A A A A A AA ff : ff 2 ZZ s ) be a finitely supported bi infinite vector of N Theta N matrices; hence, the set fff 2 ZZ s : AA A A A A A AA ff 6= 0g is finite. Also, let M 2 ZZ s Thetas be an expanding scaling matrix, cf. [3], i.e. all eigenvalues of M are greater than 1. These two quantities determine the subdivision operator SA;M which is a bounded mapping from the sequence space N p (ZZ s ) Gamma cc c c c c c cc ff 2 IR N : ff 2 ZZ s Delta into itself defined by setting, for c 2 N p (ZZ s ) ....

....= Gamma ff f f f f f ff (M Gammar ff) ff 2 ZZ s Delta : Note that fl fl r p (ff f f f f f ff ) fl fl p m r=p kff f f f f f ff k p ; r 2 IN; 4) so that r p is a bounded linear operator for any r 2 IN. With this information at hand, we can formulate the following result from [3]. Proposition 2. Let A be a finitely supported bi infinite vector of N Theta N matrices and let g be a p test function. Then the following statements are equivalent: 1) A admits a W p;N (IR) convergent subdivision scheme. 2) for any c 2 N p (ZZ s ) there exists a function ff f f f f f ....

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Dahmen, W. and C. A. Micchelli, Biorthogonal wavelet expansions, 1995, IGPM report 114, to appear in Constr. Approx..

....N matrices. We say that A admits a W p;N (R) convergent vector subdivision scheme if for any c 2 N p (R) there exists a function f c 2 W p;N (R) such that (2:8) lim r 1 kf c Gamma [ S r A c] 2 r Delta)k p = 0; and for some c 2 N p (Z) we have f c 6= 0. It has been shown in [DM] that equation (2.8) is equivalent to (2:9) lim r 1 2 Gammar=p fl fl S r A c Gamma r p (f c ) fl fl p = 0: where (2:10) r p (f) 2 r Z (j 1) 2 r j=2 r f (t)dt : j 2 Z ; 1 p 1; and (2:11) r 1 (f) Gamma f Gamma j=2 r Delta : j 2 Z Delta : Moreover, ....

....even holds in wider generality. Let g 2 W p (R)be any p test function, i.e. g is a stable and compactly supported function whose integer translates form a partition of unity. Then, for any f 2 W N p (R) and any bi infinite vector c 2 N p (Z) it can be proved by the methods presented in [DM] that there exists a constant C depending on g only such that for any r 2 N (2:20) 2 Gammar=p fl fl c Gamma Gamma r p (f) Delta fl fl p C i kf Gamma [g; c] 2 r Delta)k p p Gamma f; 2 Gammar Delta j and (2:21) kf Gamma [g; c] 2 r Delta)k p C i 2 Gammar=p ....

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Dahmen, W., Micchelli, C. A.: Biorthogonal wavelet expansions, Constr. Approx. 13, 294--328 (1997)

....Let us look into this matter in the next section. 2 Subdivision Subdivision is an iterative process for generating refinable vector fields. The introduction suggests to us two methods to construct refinable vector fields by subdivision. The first approach is to iterate on vectors on R N , [6], and the second is to work on R Z and iterate with (2N; N ) periodic two slanted bi infinite matrices, 1] Let us establish some suitable notation to formalize these two notions and review important facts about them. For 1 p 1 we let p (Z) and L p (R) be the usual spaces of real valued ....

....used in [22] see also Remark 2.3) the result follows. We are now ready to give alternative criteria for convergence of both vector and scalar subdivision. For this purpose, we recall that u is the bi infinite vector all of whose components are one and recall the following definition from [6]. Definition 2.7 (Test functions) Any function g 2 H p (R) which is stable, of compact support and has the property that [g; u] 1; is called a test function. Let us denote this class by T p (R) A refinable test function also has the property that there exists a bi infinite vector v = v i : ....

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W. Dahmen and C. A. Micchelli, Biorthogonal wavelet expansions, Constr. Approx., to appear.

....Here one can take I j as the set of vertices in T j and j k as the unique continuous piecewise linear functions relative to T j satisfying j k (m) 2 j ffi m;k ; m; k 2 I j : It is easy to see that the Phi j are uniformly stable in the sense of (2.14) and that (2. 15) is valid (see e.g. [12]) Moreover, one readily checks that j k = X m2I j 1 2 Gammaj Gamma1 j k (m) j 1 m ; i.e. a j m;k : 2 Gammaj Gamma1 j k (m) Example 2.3. Let Omega = IR s and let M be some fixed s Theta s matrix with integer entries whose eigenvalues are all strictly larger than ....

....domain Sigma j k ae Omega with diam ( Sigma j k ) c h j (4:7) such that Omega Gamma oe j;l k ) Sigma j k for all k 2 I j ; l 2 IN 0 : 4:8) These properties are easily verified for the stationary matrices A j;0 = A 0 from Example 2.3. when the mask is finitely supported (see e.g. [12]) Now, viewing Phi j as a vector with components j k , 2.16) may be rewritten as Phi j = A T j;0 Phi j 1 , and for Phi = f Phi j g j2IN0 and S as above it will be convenient to abbreviate X k2I j c k j k = Phi T j c: Thus refinability of Phi with respect to (A) fA j;0 g ....

Dahmen, W., and C. A. Micchelli, Biorthogonal Wavelet Expansions, in preparation.

.... More generally, one could consider scaling by powers of some matrix M whose eigenvalues are all larger than one or vector valued versions of the form Phi(x) X ff2ZZ a ff Phi(M x Gamma ff) where now Phi(x) 1 (x) N (x) T and the a ff are N Theta N matrices (see e.g. [DM2]) For our purposes it will suffice to stick with (5.1) for the special univariate case n = 1. In this case the concept of biorthogonal wavelets is well developed. To describe this, we call two refinable functions , a dual pair if D ; Delta Gamma k) E IR : Z IR (x) x ....

W. Dahmen and C.A. Micchelli, Biorthogonal wavelet expansions, IGPM Report #114, RWTH Aachen, May 1995.

....in (IC n f0g) s . The above assumptions may be weakened. For instance, in the first part of the assertion it would suffice to assume 0 2 L 2 . The idea of the proof is the same as in the case M = 2I (see e.g. 25] The result may also be viewed as a special case of a corresponding result in [18]. The proof of Theorem 4.2 relies now on the following consequence of Lemma 4.1. Proposition 4.1 Let 0 satisfy the hypotheses of Lemma 4.1 with finitely supported a 0 . Then there exist additional finitely supported masks a e , e 2 E such that the matrix A(z) a e 0 e (z) e;e 0 2E ....

W. Dahmen, C.A. Micchelli, Biorthogonal wavelet expansions, IGPM Report # 114, RWTH Aachen, May 1995.

No context found.

W. Dahmen, C.A. Micchelli, Biorthogonal wavelet expansions, in preparation.

....(5. 1) More generally, one could consider scaling by powers of some matrix M whose eigenvalues are all larger than one or vector valued versions of the form Phi(x) X ff2ZZ a ff Phi(Mx Gamma ff) where now Phi(x) 1 (x) N (x) T and the a ff are N Theta N matrices (see e.g. [DM2]) For our purposes it will suffice to stick with (5.1) for the special univariate case n = 1. In this case the concept of biorthogonal wavelets is well developed. To describe this, we call two refinable functions , a dual pair if D ; Delta Gamma k) E IR : Z IR (x) ....

W. Dahmen and C.A. Micchelli, Biorthogonal wavelet expansions, IGPM Report #114, RWTH Aachen, May 1995.

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W. Dahmen and C.A. Micchelli. Biorthogonal wavelet expansion. Constr. Approx., 13:293-- 328, 1997.

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W. Dahmen and C. Micchelli. Biorthogonal wavelet expansion. Constr. Approx., 13:294-328, 1997.

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W. Dahmen and C. A. Micchelli, Biorthogonal wavelet expansions, Constr. Approx. 13 (1997), 293-328.

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W. Dahmen and C. A. Micchelli, Biorthogonal wavelet expansions, Constr. Approx., 13 (1997), 293-328.

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W. Dahmen and C. A. Micchelli, Biorthogonal wavelet expansions, Constr. Approx., 13 (1997), 293-328.

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W. Dahmen and C. A. Micchelli, Biorthogonal wavelet expansions, Constr. Approx. 13 (1997), 293-328.

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W. Dahmen and C. A. Micchelli, Biorthogonal wavelet expansions, Constr. Approx., 13 (1997), pp. 293--328.

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W. Dahmen and C. A. Micchelli, Biorthogonal wavelet expansions, Constr. Approx. 13 (1997), 293--328.

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W. Dahmen and C. A. Micchelli, Biorthogonal wavelet expansions, Constr. Approx., 13 (1997), 293-328.

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W. Dahmen and C. A. Micchelli, Biorthogonal wavelet expansions, preprint, (1995).

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W. Dahmen and C. A. Micchelli, Biorthogonal wavelet expansions, Const. Approx., to appear.

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