A. Cohen, K. Grchenig, and L. F. Villemoes, "Regularity of multivariate refinable functions," Constr. Approx., vol. 15, pp. 241--255, 1999.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....for some k 2 IN. Let S k be the set defined by (24) and set ae 0 = maxfjj : 2 S k g. Then log m ae 0 : 26) 13 For the scalar case (r = 1) H is the space = fh 2 h(0) 0; jj 2kg; and S k is S k = foe : jfij 2kg: Theorem 3 is given in [7] 29] for r = 1; d = 1, and in [4], 9] for r = 1; d 1. Theorem 4 for r = 1; d 1 is contained in [14] For the vector case, see [10] These theorems lead to efficient ways of computing highly accurate values for the Sobolev smoothness s 2 ( Phi) by standard eigenvalue solvers. Next we discuss a result about the Holder smoothness ....

A. Cohen, K. Grochenig, L. F. Villemoes, Regularity of multivariate refinable functions, Constr. Approx. 15 (1999) 241--255.

....in this paper are focused on the study of the above problem via the transfer transition operator approach. The analysis of the regularity of refinable functions in terms of the transfer operator was developed by several authors (cf. e.g. D] DD] E] and [V] for the univariate case, RiS] [CGV], J] LMW] RS1] and [R2] for the multivariate case) In the L 2 case, the regularity estimates are in terms of a specific eigenpair of an associated transfer operator; hence, they seem to be computationally feasible. However, while the smoothness parameter of some examples was successfully ....

A. Cohen, K. Gr ochenig, and L. Villemoes, Regularity of multivariate refinable functions, Constr. Approx., 15 (1999), pp. 241--255.

....(v) and (vi) follow from the only assumptions ; e 2 L 1 c (R n ) as we illustrate in the appendix. In what concerns to (iv) the extension to higher dimensions requires some extra care. The next result is all we shall need, and it follows immediately from Theorem 2.1 and Proposition 2. 1 in [11]. Proposition 3.3 : see [11] Let L n 2 and M = diag ( 1 a 1 ; 1 an ) for some 1. Let be an admissible compactly supported M scaling function belonging to H La (R n ) Then, property (iv) holds. 3.2 The multilevel decomposition and wavelets We now recall how ....

....only assumptions ; e 2 L 1 c (R n ) as we illustrate in the appendix. In what concerns to (iv) the extension to higher dimensions requires some extra care. The next result is all we shall need, and it follows immediately from Theorem 2.1 and Proposition 2.1 in [11] Proposition 3. 3 : see [11]. Let L n 2 and M = diag ( 1 a 1 ; 1 an ) for some 1. Let be an admissible compactly supported M scaling function belonging to H La (R n ) Then, property (iv) holds. 3.2 The multilevel decomposition and wavelets We now recall how wavelet bases can be obtained from ....

A. Cohen, K. Gr ochenig and L. Villemoes, Regularity of multivariate refinable functions, Constr. Approx., 15 (1999), 241--255.

....In this section, we will study the smoothness of a refinable function in the multivariate setting. Many results on the analysis of L 2 smoothness of a refinable function both in the univariate case and in the multivariate case are obtained in the current literature. To mention only a few here, see [11, 17, 23, 29, 41, 43, 45] and references therein. For s = 1, the characterization of L p smoothness was given by Villemoes in [45] In this section, based on a result of Ditzian [19, 20] we present a simple proof to characterize the L p smoothness of a multivariate refinable function. Jia will discuss the L p smoothness ....

A. Cohen, K. Gr ochenig and L. Villemoes, Regularity of multivariate refinable functions, Constr. Approx., to appear.

....Sobolev exponent of a function as described in [22] Many results on the analysis of L 2 smoothness of a refinable function both in the univariate case and in the multivariate case are obtained in the 12 Bin Han current literature. To mention only a few here, see Cohen, Grochenig and Villemoes [16], Daubechies and Lagarias DL, Jia [42] Villemoes [60] and references therein. For s = 1, the characterization of L p smoothness was given by Villemoes in [60] Based on a result of Ditzian (see [25, 29] the following statement can be easily proved. Theorem 2.6 (see Theorem 3:2 in [29] Let a ....

A. Cohen, K. Grochenig and L. Villemoes, Regularity of multivariate refinable functions, Constr. Approx., 15 (1999), 241--255.

....step. Moreover, it is desirable to find dual functions which are as smooth as possible. We are therefore faced with the problem of estimating the regularity of a refinable function by only using the refinement mask. This problem has attracted several people in the last few years, see, e.g. [1, 9, 14, 15]. Let us briefly recall the basic ideas. We want to find ff : supfff : OE 2 C ff g: It is well known that ff sup , where sup is defined by sup : supf : Z R d (1 j j) j OE( jd 1g: 5.1) Our aim is to estimate sup from below. One typical result in this direction reads as ....

A. Cohen, K. Grochenig, and L. Villemoes, Regularity of multivariate refinable functions, Constr. Approx. 15 (1999), 241--255.

....in this paper are focused on the study of the above problem via the transfer transition operator approach. The analysis of the regularity of refinable functions in terms the transfer operator was developed by several authors (cf. e.g. D] DD] E] and [V] for the univariate case, RiS1] [CGV], J] LMW] RS1] and [R2] for the multivariate case) In the L 2 case, the regularity estimates are in terms of a specific eigenpair of an associated operator, hence seem to be computationally feasible. However, while the smoothness parameter of some examples was successfully computed by some ....

A. Cohen, K. Grochenig and L. Villemoes, Regularity of multivariate refinable functions, Constr. Approx. 15 (1999), 241--255.

....in the FSI case. Both [RiS] and [S] deal with the dilation matrix s = 2I . Jiang, Ji] generalized the smoothness result of [S] to a general dilation matrix s. Most recently, the transfer operator approach was employed (independently) by Jia [J1] and by Cohen, Grochenig and Villemoes, [CGV]. The results of these two papers are closely related: both show, without assuming any possible factorization, how to provide lower bounds on the smoothness of a single multivariate compactly supported refinable function, and both make fairly minimal assumptions on the dilation matrix (In [J1] ....

.... to provide lower bounds on the smoothness of a single multivariate compactly supported refinable function, and both make fairly minimal assumptions on the dilation matrix (In [J1] the matrix is assumed to be a similarity matrix, i.e. a constant multiple of a unitary one; even less is assumed in [CGV], but, alas, they have to measure smoothness, in case the matrix is not a similarity, by non standard ways) Both articles show that, if the refinable function has stable shifts, then the lower bound estimates are sharp, i.e. they characterize the smoothness class of the function. It was the ....

[Article contains additional citation context not shown here]

A. Cohen, K. Grochenig and L. Villemoes, Regularity of multivariate refinable functions, Constr. Approx. 15 (1999), 241-255.

....f ( Delta Gamma k)g k2Z n is a uniformly p stable basis of V (p) 0 , in the sense that V (p) 0 = n X k ff k ( Delta Gamma k) ff k ) 2 p (Z n ) o and k X k ff k ( Delta Gamma k)k L p (R n ) i X k jff k j p j 1=p : Remark 3. 3 In a recent article of Cohen et al. [10], it was shown that for a matrix of the form M = diag ( ff=ff 1 ; ff=ff n ) condition (iv) is always satisfied provided is smooth enough. In fact, this is the case if is in the anisotropic space H L ff ff (R n ) which occurs for instance when belongs to the isotropic ....

A. Cohen, K. Gr ochenig and L. Villemoes, Regularity of multivariate refinable functions, preprint, 1997.

....eigenvalue = 1, then an associated multiresolutionand a stable biorthogonal wavelet basis exist. This combines results relating the stability of the wavelet basis to the Sobolev regularity of the functions [12] and results relating the Sobolev regularity to the spectrum of the transfer operator [9, 25, 40]. If the filters are FIR, then this condition can be checked by computing the eigenvalues of a finite matrix, the size of which depends on the length of the filters. To actually compute the Sobolev regularity, we need to find the transfer operator T and its invariant submatrix T r . Then we ....

....size of which depends on the length of the filters. To actually compute the Sobolev regularity, we need to find the transfer operator T and its invariant submatrix T r . Then we compute the eigenvalues of T r and use the fact that an estimate of the lower bound on the Sobolev exponent is given by [9]: log ae 2 log max 6 s 6 log ae 2 log min ; 8) where max ; min are the maximum and minimum eigenvalues of the dilation matrix, respectively, and ae is the maximum nonspecial eigenvalue of T r . Special eigenvalues are eigenvalues that correspond to a polynomial left eigenvector, see ....

A. Cohen, K. Grochenig, and L. Villemoes. Regularity of multivariate refinable functions. preprint, 1996.

....where the results do not carry over to general dilations is in the context of smoothness: there, one needs to assume that the dilation is isotropic, i.e. that all the eigenvalues of s have the same modulus. Without this assumption, one can only get upper and lower bounds on the smoothness (cf. [16]) 2) The refinable OE is scalar valued (rather than vector valued) This assumption simplifies substantially the notations; however, it should be stressed that many of the arguments here can be easily carried over to that setup (specifically, the treatment of convergence and the treatment of ....

....for all sufficiently large k, in the domain H OE of T OE . Hence, by the weak E condition, T k Phi ) k is bounded in H OE (say, in the L 1 (TT d ) norm) Thus k b OEk L2 (IR d ) 1. The next result was proved first in [51] under the assumption that the mask is polynomial; see also [16]. We remind the reader that, for a compactly supported OE 2 L 2 (IR d ) the stability of E(OE) is characterized by the positivity everywhere of the function Phi from Lemma 3.1.4 (cf. 77] 24] 47] 48] 5] Theorem 3.1.11. 65] Let OE be a compactly supported refinable distribution. ....

[Article contains additional citation context not shown here]

Cohen, A., K. Grochenig, and L. Villemoes, Regularity of multivariate refinable functions, 1996, preprint.

....that the starting point needs to be unrealistically high. However, in practice things are better since the smoothness can be assessed much more accurately. Here we summarize the approach to be used in our examples at the end. By now it has received treatment at many levels, by several people [CGV], CD] E] H] J] RiS1] RS4] and [V] The criterion to be used to bound sup from below is contained in the following statement: For an integer r, let V r : Phi v 2 0 (Z s ) X ff2Z s p(ff)v(ff) 0; 8 p 2 Pi r Psi ; where Pi r denotes the polynomials of total degree ....

....1 ;N1 ] s ;fi2[ GammaN 1 ;N1 ] s : Let ae IHj V r be the spectral radius of IHj V r . Then the exponent sup satisfies (3:11) sup Gamma log Gamma ae IHj V r Delta log Gamma j max j Delta : The proof of this statement can be obtained by modifying the proof in [RiS1] or from [CGV], J] and [RS4] The choice N 1 can also be gleamed from [CGV] and [J] as the smallest N 1 such that (3:12) i M Gamma1 : M Gammaj M Gammaj j [ GammaN; N ] s [ GammaN 1 ; N 1 ] s ; 8 j 1; where [ GammaN; N ] s contains the support of a. 4. Dual Functions and ....

[Article contains additional citation context not shown here]

A. Cohen, K. Grochenig and L. Villemoes, Regularity of multivariate refinable functions, preprint, (1996).

....that the starting point needs to be unrealistically high. However, in practice things are better since the smoothness can be assessed much more accurately. Here we summarize the approach to be used in our examples at the end. By now it has received treatment at many levels, by several people [CGV], CD] E] H] J] RiS1] RS4] and [V] The criterion to be used to bound sup from below is contained in the following statement: For an integer r, let V r : Phi v 2 0 (Z s ) X ff2Z s p(ff)v(ff) 0; 8 p 2 Pi r Psi ; where Pi r denotes the polynomials of total degree r. ....

....ff Gamma fi) i ff2 Omega ;fi2 Omega : Let ae IHj Vr be the spectral radius of IHj Vr . Then the exponent sup satisfies (3:11) sup Gamma log Gamma ae IHj Vr Delta log Gamma j max j Delta : The proof of this statement can be obtained by modifying the proof in [RiS1] or from [CGV], J] and [RS4] The invariant set Omega was defined in [LLS1] and [LLS2] and [HJ1] An explicit formula for the invariant set was given in [HJ1, Theorem 4.2] 3:12) Omega : 1 X j=1 M Gammaj supp a: 4. Dual Functions and Biorthogonal Wavelets In this section we combine the construction ....

A. Cohen, K. Grochenig and L. Villemoes, Regularity of multivariate refinable functions, preprint, (1996).

....detail here. Similarly we can establish the corresponding results of Theorems 2.3 and 2. 5 to the generalized nonhomogeneous refinement equation (19) 3 Regularity The analysis of the smoothness of compactly supported solution of refinement equations is widely studied (for instance see [CDP] CGV] J] Ji] MaS] MS] RS] and references therein) In this section, we divide two subsections to discuss the regularity in Bessel potential spaces and estimate of Sobolev exponent of the solution of the nonhomogeneous refinement equation (1) These estimates can used directly to the ....

A. Cohen, K. Grochenig and L. Villemoes, Regularity of multivariate refinable functions, Preprint 1996.

....FSI case. Both [RiS] and [S] deal with the dilation matrix s = 2I , however, it is easy to carry out of the proof of [S] for a general dilation matrix s (see e.g. Ji] Most recently, the transfer operator approach was employed (independently) by Jia [J1] and by Cohen, Grochenig and Villemoes, [CGV]. The results of these two papers are closely related: both show, without assuming any possible factorization, how to provide lower bounds on the smoothness of a single multivariate compactly supported refinable function, and both make fairly minimal assumptions on the dilation matrix (In [J1] ....

.... to provide lower bounds on the smoothness of a single multivariate compactly supported refinable function, and both make fairly minimal assumptions on the dilation matrix (In [J1] the matrix is assumed to be a similarity matrix, i.e. a constant multiple of a unitary one; even less is assumed in [CGV], but, alas, they have to measure smoothness, in case the matrix is not a similarity, by non standard ways) Both articles show that, if the refinable function has stable shifts, then the lower bound estimates are sharp, i.e. they characterize the smoothness class of the function. It was the ....

[Article contains additional citation context not shown here]

A. Cohen, K. Grochenig and L. Villemoes, Regularity of multivariate refinable functions, preprint, 1996.

....3 (z Gamma1 )Q n;n;n (z Gamma1 ) n = 2; 8; that will lead to suitable functions. In this way, we will provide an array of examples in which it is interesting to see how the smoothness changes. The quantities to be used to measure smoothness are the ones used in [RiS3] J] [CGV] and [RS5] A function 2 C ff for n ff n 1, provided that 2 C n but 62 C n 1 and, for some constant independent of x, jD fl (x t) Gamma D fl (x)j constjtj ff Gamman ; for all jflj = n and jtj 1: The number ff is related to weighted L p critical exponents p defined ....

A. Cohen, K. Grochenig and L. Villemoes, Regularity of multivariate refinable functions, preprint, (1996).

....eigenvalue = 1, then an associated multiresolution and a stable biorthogonal wavelet basis exist. This combines results relating the stability of the wavelet basis to the Sobolev regularity of the functions [12] and results relating the Sobolev regularity to the spectrum of the transfer operator [9, 24, 38]. If the filters are FIR, then this condition can be checked by computing the eigenvalues of a finite matrix, the size of which depends on the length of the filters. To actually compute the Sobolev regularity, we need to find the transfer operator T and its invariant submatrix T r . Then we ....

....size of which depends on the length of the filters. To actually compute the Sobolev regularity, we need to find the transfer operator T and its invariant submatrix T r . Then we compute the eigenvalues of T r and use the fact that an estimate of the upper bound on the Sobolev exponent is given by [9]: log ae 2 log max 6 s 6 log ae 2 log min ; 11) where max ; min are the maximum and minimum eigenvalues of the dilation matrix, respectively, and ae is the maximum nonspecial eigenvalue of T r . Special eigenvalues are eigenvalues that correspond to a polynomial left eigenvector, see ....

A. Cohen, K. Grochenig, and L. Villemoes. Regularity of multivariate refinable functions. preprint, 1996.

....supported functions derived from a multiresolution analysis does form such a Riesz basis. This involves determining the smoothness of the dual system. The elements of the dual system typically consist of non compactly supported functions, whose smoothness can be treated by extending the results of [7, 22, 9]. We show how to determine the exact range of Sobolev exponents in the multivariate case, both theoretically and numerically, from spectral properties of transfer operators. This technique is applied to several bases deriving from linear finite elements which have been proposed in the literature. ....

....of this study, the exposition has changed substantially. We have corrected a number of misprints and inconsistencies, improved our numerical experiments and incorporated recent results on wavelet regularity. We would like to thank A. Cohen and R. J. Jia for making available to us the manuscripts [9, 22], and the referees for their constructive criticism. 2 Theory: Riesz bases in H s (R d ) 2.1 General definitions and assumptions By f , we denote the Fourier transform of f 2 L 2 (R d ) f( Z R d f(x)e Gammaix dx ; and by kfkH s the Sobolev norm of f 2 H s (R d ) kfk ....

[Article contains additional citation context not shown here]

A. Cohen, K. Grochenig, L. Villemoes, Regularity of multivariate refinable functions, Constr. Approx., 1998 (to appear).

No context found.

A. Cohen, K. Grochenig, and L. Villemoes, Regularity of multivariate refinable functions, to appear in: Constr. Approx.

....case such a factorization is a necessary condition for OE 2 H s ; s L. In the multivariate case, the relations between Strang Fix conditions and regularity are not so easy, but (C5) always serves as an indicator for regularity. For a further discussion of this topic, the reader is referred to [8]. In this paper we derive an algorithm for the construction of symbols satisfying (C1) C5) To understand the underlying idea, we briefly recall the univariate situation. There one has to find a non negative trigonometric polynomial m( satisfying m( m( 1=2) 1; 2.8) m (l) 1=2) ....

A. Cohen, K. Grochenig, and L. Villemoes, Regularity of multivariate refinable functions, to appear in: Constr. Approx.

No context found.

A. Cohen, K. Grchenig, and L. F. Villemoes, "Regularity of multivariate refinable functions," Constr. Approx., vol. 15, pp. 241--255, 1999.

No context found.

A. Cohen, K. Grochenig, and L. Villemoes, Regularity of multivariate refinable functions, Constr. Approx. 15(1999), 241--255.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC