B. Han and R. Q. Jia. Multivariate re nement equations and convergence of subdivision schemes. SIAM J. Math. Anal., 29:1177-1199, 1998.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....rules. The stability and the convergence of such re nement process, as well as the smoothness properties of its limit function if it exists, have been the subject of active research in recent years. We refer to [6] and [19] for general surveys on subdivision algorithms, and e.g. to [14] 15] [22] for more specialized results on their convergence and smoothness. An important motivation for the study of subdivision algorithms is their relation to multiresolution analysis and wavelets (see e.g. 11] or [13] In particular, the contribution of a single wavelet coecient in the representation ....

....these coecients have the form a k;l = a k 2l . The analysis of a subdivision scheme consists of establishing conditions for the convergence of the scheme, and in characterizing the smoothness as well as the order of approximation of the set of limit functions. We refer the reader to [6] 19] and [22] for a general survey on this subject, in the linear and uniform case. De nition 2 A subdivision scheme, generating recursively the data fv j : j 2 ZZ g; is called uniformly convergent if, for every set of initial control points v 0 2 1 (ZZ) there exists a continuous function f 2 ....

Han, B. and R. Jia (1998), Multivariate renement equations and convergence of subdivision schemes SIAM J. Math. Anal.29, 1177-1199.

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B. Han and R. Q. Jia, Multivariate re nement equations and convergence of subdivision schemes, SIAM J. Math. Anal. 29 (1998), 1177-1199.

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B. Han and R. Q. Jia, Multivariate re nement equations and convergence of subdivision schemes, SIAM J. Math. Anal., 29 (1998), 1177-1199.

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B. Han and R. Q. Jia, Multivariate re nement equations and convergence of subdivision schemes, SIAM J. Math. Anal., to appear.

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B. Han and R. Q. Jia, Multivariate re nement equations and convergence of subdivision schemes, SIAM J. Math. Anal., 29 (1998), 1177-1199.

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B. Han and R. Q. Jia, Multivariate re nement equations and convergence of subdivision schemes, SIAM J. Math. Anal., 29 (1998), 1177-1199.

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B. Han and R. Q. Jia, Multivariate re nement equations and convergence of subdivision schemes, SIAM J. Math. Anal., 29 (1998), 1177{ 1199.

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B. Han and R. Q. Jia, Multivariate re nement equations and convergence of subdivision schemes, SIAM J. Math. Anal. 29 (1998), 1177-1199.

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B. Han and R. Q. Jia, Multivariate re nement equations and convergence of subdivision schemes, SIAM J. Math. Anal. 29 (1998), 1177-1199.

....is called a subdivision scheme or a cascade algorithm ( 2, 18] When the sequence Q a;M 0 converges in the space L p (R ) then the limit function must be a and we say that the subdivision scheme associated with mask a and dilation matrix M converges in the L p norm. It was proved in [14] that the subdivision scheme associated with the mask a and dilation matrix M converges in the L p norm if and only if 1 (a; M; p) j det M j (By Theorem 3.1, we see that this is equivalent to p (a; M) 0) See [2, 9, 14, 18] and references therein on convergence of subdivision schemes. ....

....mask a and dilation matrix M converges in the L p norm. It was proved in [14] that the subdivision scheme associated with the mask a and dilation matrix M converges in the L p norm if and only if 1 (a; M; p) j det M j (By Theorem 3. 1, we see that this is equivalent to p (a; M) 0) See [2, 9, 14, 18] and references therein on convergence of subdivision schemes. Let be a re nable function with a nitely supported mask a and a dilation matrix M . It is known that is an interpolating re nable function if and only if the mask a is an interpolatory mask with respect to the lattice MZ and ....

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B. Han and R. Q. Jia, Multivariate re nement equations and convergence of subdivision schemes, SIAM J. Math. Anal., 29 (1998), 1177-1199.

....p = 0. If this is the case, then it is necessary that must be the normalized solution of (1.1) with the re nement mask a. For any compactly supported function f , if the series Q a f converges in the L p norm to the normalized solution 2 L p (R ) with the mask a, then it is necessary (see [4, 15, 17]) that f satis es the moment conditions of order 1, i.e. b f(0) 1 and b f(2 ) 0 for all 2 nf0g. For any function f 2 L p (R ) we say that the shifts of f are stable if there exist two positive constants C 1 and C 2 such that (1:5) C 1 k k p ( f( C ....

....a subdivision scheme. Let a be a sequence on . The subdivision operator associated with a is de ned by (1:6) S a ( a( 2 where 2 0 (Z ) Note that 0 (Z ) is a subspace of p (Z ) The p norm of an element 2 p (Z ) is denoted by k k p . It was proved in [15] that for any nitely supported re nement mask a, the subdivision scheme associated with a converges in the L p norm if and only if Moreover, it was also proved in [15] that if the subdivision scheme associated with a converges in the L p norm to the normalized solution of (1.1) with the ....

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B. Han and R. Q. Jia, Multivariate re nement equations and convergence of subdivision schemes, SIAM J. Math. Anal. 29 (1998), 1177-1199.

....p;k (a) maxf lim p : j j = k; 2 g; k 2 [ f0g: 1.6) Such quantity p;k (a) plays an important rule in the study of convergence of subdivision schemes and smoothness analysis of re nable functions. In general, it is hard to compute p;k (a) Even though, when p = 2, it is known [5, 15, 20, 25, 26] that 2;k (a) can be obtained by computing the spectral radius of a nite matrix, in higher dimensions, the size of such matrix is too large to practically compute its spectral radius. Therefore, it is worth studying how to estimate p;k (a) and how to compute 2;k (a) eciently. We shall ....

....radius of a nite matrix, in higher dimensions, the size of such matrix is too large to practically compute its spectral radius. Therefore, it is worth studying how to estimate p;k (a) and how to compute 2;k (a) eciently. We shall discuss these topics in Sections 3 and 4. It was proved in [15] that the subdivision scheme associated with a converges in the L p norm if and only if p;1 (a) 2 . If a is an interpolatory mask, then it is easily seen that each Q a 0 in the subdivision scheme is a fundamental function. Thus, if the subdivision scheme associated with an interpolatory ....

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B. Han and R. Q. Jia, Multivariate re nement equations and convergence of subdivision schemes, SIAM J. Math. Anal., 29 (1998), 1177-1199.

.... W k p (R d ) When M is an isotropic dilation matrix, the subdivision scheme associated with the mask a and dilation M converges in the Sobolev space W k p (R d ) if and only if k 1 (a; M; p) j det M j 1=p k=d (we shall see in Section 3 that this is equivalent to p (a; M) k) See [10] for the case k = 0 and [17] for the case p = 2 on the characterization of the convergence of a subdivision scheme. The L p smoothness of f 2 L p (R d ) is measured by its L p smoothness exponent: p (f) supf 0 : kr n y fk p 6 Ckyk 8 y 2 R d for some constant C and for large ....

....on stability. Given a mask a and a dilation matrix M , it is known that M a is an interpolating re nable function if and only if the mask a is an interpolatory mask and the subdivision scheme associated with mask a and dilation M converges in the L1 norm (equivalently, 1 (a; M;1) 1, see [10]) However, in general, it is dicult to directly check the condition 1 (a; M;1) 1. The following result is useful to indirectly check such a condition. Corollary 3.2 Let a be a nitely supported mask on Z d and M be a dilation matrix. Suppose that b is a dual mask of a with respect to the ....

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B. Han and R. Q. Jia, Multivariate renement equations and convergence of subdivision schemes, SIAM J. Math. Anal., 29 (1998), 1177-1199.

....x 2 R. Then we consider an iteration scheme Q n a 0 ; n = 0; 1; 2; where Q a is the linear operator on L p (R s ) 1 p 1) given by Q a f : X k2Z s a k f(2 k) f 2 L p (R s ) The iteration scheme is called a subdivision scheme or cascade algorithm in the literature (see [1, 3]) If there exists a function f 2 L p (R s ) such that lim n 1 kQ n a 0 fk p = 0, then we say that the subdivision scheme associated with a converges in the L p norm. If this is the case, then f must be the unique distributional solution a to the re nement equation (1.1) A characterization ....

....lim n 1 kQ n a 0 fk p = 0, then we say that the subdivision scheme associated with a converges in the L p norm. If this is the case, then f must be the unique distributional solution a to the re nement equation (1. 1) A characterization of L p convergence of a subdivision scheme was given in [3]. Let be an orthonormal scaling function in L 2 (R s ) with a nitely supported mask a. Then its mask a must satisfy the following discrete orthogonal condition X k2Z s a k 2j a k = 2 s 0j 8 j 2 Z s : 2.1) Let a be a nitely supported mask on Z s and a be its associated re nable ....

B. Han and R. Q. Jia, Multivariate renement equations and convergence of subdivision schemes, SIAM J. Math. Anal., 29 (1998), 1177-1199.

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B. Han and R. Q. Jia. Multivariate re nement equations and convergence of subdivision schemes. SIAM J. Math. Anal., 29:1177-1199, 1998.

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B. Han and R.-Q. Jia. Multivariate renement equations and convergence of subdivision schemes. #### ## ##### #####, 29:1177-1199, 1998.

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