Ingrid Daubechies and Je#rey C. Lagarias. Two-scale di#erence equations: I. Existence and global regularity of solutions. SIAM J. Math. Anal., 22:1388--1410, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....for innite initial sequences: If e.g. hn = 1 for all n, we immediately have [1] using (2.7) s (x) j 1, which is C whatever the regularity order of the s (x) s. 4.4. A two scale functional equation. In the dyadic case, the limit function (x) satises a itwo scale dioeerence equationj [8, 9] which was used as a starting point by Daubechies and Lagarias for deriving regularity estimates. In the rational case, this approach becomes impossible because of the lack of shift invariance mentionned in # 1. Indeed, we have the following two scale equation [1] g qk Gammaps k ( x) ....

....N times dioeerentiable functions s (x) and k=0 ( Gamma1) f s k (x) 4.11) Here is the dioeerentiation operator. The proof of the rst part (N = 1) is given in [1] The second part follows easily by induction. Note that this result generalizes the result known in the dyadic case [8, 17], where e.g. x) f(x) Gamma f(x Gamma 1) for N = 1. In words, this proposition states that the rational schemes generated by F (X) converge towards the Nth derivatives of the limit functions s (x) generated by G(X) provided we choose h k = Gamma1) Gamma N as the initial sequence ....

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I. Daubechies and J. C. Lagarias, Two-Scale Dioeerence Equations I. Existence and Global Regularity of Solutions, SIAM J.Math. Anal., Vol. 22 No. 5, pp. 13381410, September 1991.

....in the form of equation 16. The expression for in terms of the same base of is: X C 7 v v DCE A DC (19) AM C 7 v (20) which links the functions . The equation 16 has been studied more in detail by Daubechies and Lagarias [6] and other authors. The important point to note in this case is that the choice of the coefficients completely characterizes both . It is then really important to remember that the computation for the wavelet analysis and synthesis is performed by the Mallat algorithm using the ....

I. Daubechies, J.C. Lagarias. Two-Scale Difference Equations I.: Existence And Global Regularity Of Solutions. SIAM J. Math. Aml., vol. 22, no. 5, pp. 1388--1410, Sept. 1991.

....= 0:38, in the three dimensional parameter space (log p; log l; r) The two high peaks are the highest votes in the three dimensional parameter space. Dyadique renormalization equations also de ne interesting multifractals studied by Deslauriers and Dubuc [7] as well as Daubechies and Lagarias [10]. The function shown in Fig. 5 is the solution of the functional equation f(2(x ) f(2x) f(2(x ) 18) analyzed by Daubechies. When applying the voting procedure to the maxima of the wavelet transform of f(x) surprisingly there are 4 peaks that 120 40 Figure 4: Votes based on the ....

I. Daubechies, and J. Lagarias, \Two-scale dierence equations. Existence and global regularity of solutions", SIAM Journ. of Math. Anal. vol. 22, no. 5, pp. 1388-1410, Sept. 1991.

....of the cascade sequence for a refinement equation is closely related to the regularity of its solution, the conditions on h for convergence of derivatives of cascade sequences are similar to those for regularity of refinable functions. Results on regularity of refinable functions are given in [9,12] and [10] for the scalar case in one dimension, in [28] and [5] for the vector case in one dimension, in [30,21] and [19] in higher dimensions. To define the concept of fundamentality of higher order for h let m be the space of all polynomials on IR of degree m; and define V m : fv = v(j) N ....

Daubechies, I., and J. C. Lagaria, Two-scale difference equations I. Existence and global regularity of solutions, SIAM J. Math. Anal. 22 (1991), 1388--1410. Cascade Algorithms 211

....of k; and the equations in (1.1) reduce to the ordinary refinement equation OE(x) jM j X j2 Omega h(j)OE(Mx Gamma j) x 2 R s : 1.8) In this case (1.7) is equivalent to OE(0) 1: Equation (1. 8) has been extensively studied recently in connection with wavelets analysis ( 4] 6] [9] Gamma [15] 19] 20] 24] Gamma [32] 38] Gamma [41] Equation (1.8) can also be viewed as the limiting case of the nonstationary refinement equation (1.1) and we shall refer to it as the ideal refinement equation associated with (1.1) Let Gamma comprise coset representatives of Z s ....

Daubechies, I. and J. C. Lagaria, Two-scale difference equations I. Existence and global regularity of solutions, SIAM J. Math. Anal., 22(1991), 1388 - 1410.

....means of local 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 ....

.... v j on the centered stencil, the corresponding multiresolution transform is equivalent to the Dubuc Deslaurier interpolatory wavelet transform (see [16] and [20] For the properties of the interpolant as well as for the smoothness of the limit of this iterative process, we refer the reader to [14], 20] and [16] Example 2. Cell average multiresolution In the cell average context, IR is partitioned in disjointed dyadic cells j : f j k = k2 j ; k 1)2 j )g k2ZZ : In this context, the discrete vector v j is viewed as the average 2 j R j k v(t) dt k2 j of a locally ....

Daubechies, I. and J. Lagarias (1991)Two Scale dierences Equations: I.Existence and global regularity of solutions, SIAM J. Math. Anal.22,1388-1410.

....case of a much more general class of equations, namely two scale di#erence equations. Those are functional equations of the type f(x) N X n=0 c n f(#x # n ) x # R) 1.2) with c n # C, # n # R and # 1. They were first discussed by Ingrid Daubechies and Je#rey C. Lagarias in [DL91], who proved existence and uniqueness theorems and derived some properties of L 1 solutions. Their existence uniqueness theorem reads as follows. 4 Theorem 1. Denote # : # 1 N P n=0 c n . a) If # # 1 and # #= 1, then Equation (1.2) has no non trivial L 1 solution. b) If ....

I. Daubechies and J. C. Lagarias, Two-scale di#erence equations I. Existence and global regularity of solutions, SIAM J. Math. Anal., 22 (1991), 1388--1410.

....In particular, since f q 0 a.e. on I, from [3, Proposition 2] it follows that m e (Graph f) m(B d c ) Moreover, if f q is continuous, then from [3, Proposition 4] we get that Graph f is connected. 2) The proof of our Theorem also works in the case of general dilation equations (see [5]) If the equation (D) #(x) N X n=0 c n #(#x # n ) where # 0 . #N , c n 0, P c n = #, # 1 and #N # 0 # 1 # max # i 1 # i , has a non trivial compactly supported L 1 solution f , then f is either positive or negative almost everywhere on its support. ....

I. Daubechies, J. Lagarias, Two-scale di#erence equations I. Existence and global regularity of solutions, SIAM J. Anal. 22 (1991), 1388--1410.

....[3] and approximation of curves and surfaces in computer graphics [1] There are many interesting works concerning dilation equations; see the references for a small sample. In the context of this paper, where we are concerned with existence and uniqueness of solutions to dilation equations, [4] provides a good background. However these works concentrate either on the L 1 (R) or compactly supported L p (R) solutions of dilation equations. This ensures the continuity of the Fourier transform of the solutions, making certain uniqueness arguments possible. Mallat s multiresolution ....

....instance, if we have f 2 F (p) with f non zero on ( then it is easy to show that f is maximal. Similarly, if p is continuous at zero and f 2 F (p) is analytic then either f is identically zero or f is maximal. This shows the L 1 (R) solutions with P c n = 2 (discussed in [4]) have maximal Fourier transforms, as they are analytic and p is a trigonometric polynomial. We are now in a position to prove two quite interesting results. Theorem 7. The L 2 (R) solutions of: f(x) f(2x) f(2x 1) are in a natural one to one correspondence with the functions in L 2 ( 1; ....

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Ingrid Daubechies and Jerey C. Lagarias, Two-scale dierence equations. 1: Existence and global regularity of solutions, SIAM J. Math. Anal. 22 (1991), no. 5, 1388-1410.

....s = 1 and the choice (t) 1 Gamma jtj, t 2 ( Gamma1; 1) and zero otherwise, F k (t) is the control polygon through the control points. Stationary uniform binary subdivision schemes received a lot of attention in the literature in recent years. A general analysis of such schemes can be found in [3, 8, 9, 12, 13, 15, 19, 20, 21]. The subdivision scheme S a is termed uniformly convergent if for the initial set of control points in IR, f 0 = ffi = fffi ff;0 g ff2ZZ s , fF k (t)g converges uniformly to a non zero function OE. This property of the scheme is independent of the choice of in (1.2) 3] see also [12] and ....

Daubechies, I., and J. Lagarias, Two scale difference equations I. Existence and global regularity of solutions, SIAM J. Math. Anal. 22 (1991), 1338--1410.

...., respectively assigned to the binary points f2 Gammak jg j2ZZ . The purpose of subdivision analysis is to study the convergence of such processes and to establish the existence of a limit function on IR and its smoothness class. A general treatment of uniform subdivision can be found in [CDL] [DaLa1], DaLa2] DGL2] Dyn] MP1] MP2] Level dependent subdivision schemes, where the scheme may vary from one refinement level to the other, are discussed in [DL2] In the present work we analyze nonuniform binary subdivision schemes in which the scheme for defining the points may vary from ....

....refinement equation OE(x) n X i= Gammam p i OE(2x Gamma i) 1:10) Therefore, subdivision analysis is almost equivalent to the analysis of compact solutions of corresponding refinement equations. This direction, which is also motivated by wavelet analysis, is pursued in many works, e.g. [DaLa1], DaLa2] The main tool for this analysis being of course Fourier analysis. 2. Analysis of non uniform binary subdivision schemes The analysis tools for uniform binary subdivision schemes in [DGL2] Dyn] and [MP2] or the Fourier analysis approach in [DaLa1] do not seem to be appropriate for ....

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Daubechies, I. and J. Lagarias, Two scale difference equations I. Existence and global regularity of solutions, SIAM J. Math. Anal. 22 (1991), 1338--1410.

....triangularizability. AMS subject classifications. 26A15, 26A18, 41A05 PII. S0895479894262327 1. Introduction. Infinite products of matrices occur in a wide variety of fields. They may be used to study subdivision algorithms [2, 13, 14] Markov chains [3] lattice two scale di#erence equations [8, 9], and orthonormal bases of compactly supported wavelets [6] Our interest is in characterizing certain classes of smooth compactly supported N dimensional prewavelets or scaling functions using infinite products of matrices. These functions are the solutions of the N dimensional dilation ....

....1 #(2x 1) c m#(2x m) 2.1) where # : R # R and c i , i = 0, m, are given real coe#cients. The regularity properties of the solutions of dilation equations have been extensively studied. In particular, nontrivial L 1 solutions having compact support are characterized in [8] and shown to have their support in [0, m] Moreover, it is shown that if # is r times continuously di#erentiable, then r m 1. The Holder exponent and fractal structure of # are determined in [4, 5, 9] Continuous solutions are characterized in terms of the general and joint spectral radii of a ....

I. Daubechies and J. Lagarias, Two-scale di#erence equations, I. Existence and global regularity of solutions, SIAM J. Math. Anal., 22 (1991), pp. 1388--1410.

....matsunqy leonis.nus.edu.sg 1 Here the Fourier transform f of an integrable function f is defined by f( Z IR f(x)e Gammaix dx; and H(z) the symbol of the refinement equation (1.1) is defined by H(z) N X j=0 c j z j : 1. 3) For the symbol H, Daubechies and Lagarias proved in [7] that H(1) M i(H) 1.4) for some positive integer i(H) when M = 2. Their proof is easy generalized to the case M 2. Refinable distribution arises in many contexts, in dyadic interpolation as well as in the construction of various wavelets and multiresolutions. There are a lot of papers on ....

I. Daubechies and J. L. Lagarias, Two-scale difference equations I. existences and global regularity of solutions, SIAM J. Math. Anal., 22(1991), 13881410.

....to non standard limits. For example see de Rahm [21] Catmull and Clark [4] Doo and Sabin [12] Dubuc [13] Deslauriers and Dubuc [11] Dyn, Gregory and Levin [15,16,17] The convergence analysis of uniform BSS has been developed considerably in the last few years in a series of works [5] [9,10], 14] 13] 15,17] 18] 19,20] The analysis of BSS for surface design is well developed for the case of a square grid topology of control points [5] 18] 14] The case of general Curves and Surfaces II 0 P. J. Laurent, A. Le M ehaut e, and L. L. Schumaker (eds. pp. 0 0. Copyright o ....

Daubechies, I. and J. Lagarias, Two scale difference equations I. Existence and global regularity of solutions, SIAM J. Math. Anal. 22 (1991), 1338-- 1410. 8 N. Dyn, J.A. Gregory and D. Levin

....of the re nement equation (1.1) if it satis es (1.1) and R R f(x) dx = 1. Not every re nement equation has an associated re nable function, since the requirment f(x) 2 L 1 (R) can not be met in general. When it does, the associated re nable function is unique, and is compactly supported, see [DL1]. We shall study (1.1) primarily in conjunction with subdivision schemes. A comprehensive discussion of subdivision schemes can be found in [CDM] The subdivision scheme relates to the re nement equation (1.1) as follows: Start with a set of vectors fv 0 n : n 2 Zg with each v 0 n 2 R m , ....

I. Daubechies and J. C. Lagarias, Two-scale dierence equations I. Existence and global regularity of solutions, SIAM J. Math. Analysis 22 (1991), 1388-1410.

....compactly supported orthogonal wavelets have this amount of regularity. Many symmetric biorthogonal wavelets (the WVS can be defined for biorthogonal wavelets with a slight modification; see [21] and [32] with relatively small support do not satisfy condition (2) either. See, e.g. 6] [12], and [13] There are several disadvantages in using smoother wavelets to approximate a function that is much less smooth than the given wavelets. First, as long as (7) and (8) hold, there is no gain in approximation order with smoother wavelets. Second, since smoother wavelets have wider support, ....

I. Daubechies and J. C. Lagarias, Two-scale di#erence equations I. Existence and global regularity of solutions,SIAMJ.Math. Anal., 22 (1991), pp. 1388--1410.

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Ingrid Daubechies and Jerey C. Lagarias, Two-scale dierence equations. 1: Existence and global regularity of solutions, SIAM Journal of Mathematical Analysis 22 (1991), no. 5, 1388-1410.

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I. Daubechies and J. C. Lagarias, Two-scale di#erence equations I. Existence and global regularity of solutions, SIAM J. Math. Analysis 22 (1991), 1388-1410.

....the study of dilation equations, is how it interacts with translation and dilation: f(#) f(#x) ## # 1 f(# 1 #) e i#k f(#) The normalisation of the Fourier transform is irksomely nonstandard. For example [40] define it with an extra factor of 2# inside the exponential and [11] doesn t bother with the minus sign. # The convolution of two functions f and g is given by (f g) x) f(t)g(x t) dt. Applying this to: we get: c k e i . Letting p(#) c k e i#k we can rewrite this as: The trigonometric polynomial p is referred ....

....by many authors and will be used frequently in Chapters 2 and 3. Most authors are concerned with the case where g is integrable, which ensures the continuity of g, allowing the iteration of the transformed equation until it becomes an infinite product. By estimating the decay of this product, [11] shows that if the function is compactly supported and the equation has N coe#cients, then the support of the function will be of length N 1. 1.6 Matrix methods Another technique commonly applied to dilation equations involves rewriting the dilation equation in matrix form. The most obvious ....

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Ingrid Daubechies and Je#rey C. Lagarias, Two-scale di#erence equations. 1: Existence and global regularity of solutions, SIAM Journal of Mathematical Analysis 22 (1991), no. 5, 1388--1410.

.... Theta r (complex) matrix. In this paper, by a dilation matrix we mean an integer matrix whose eigenvalues lie outside the closed unit disk. When g = 0, 1. 1) becomes the homogeneous refinement equation OE(x) X ff2ZZ s a(ff)OE(Mx Gamma ff) x 2 IR s : The readers are referred to [1] and [4] for some basic properties of homogeneous refinement equations. For vector homogeneous refinement equations, see [9] 2] 22] 19] and [27] Nonhomogeneous refinement equations generalized from their homogeneous counterpart are motivated by constructions of multiwavelets to obtain ....

I. Daubechies and J. C. Lagarias, Two-scale difference equations: I. Existence and global regularity of solutions, SIAM J. Math. Anal. 22 (1991), 1388--1410.

....equation can be written as OE(x) X ff2Z s a(ff)OE(Mx Gamma ff) x 2 R s ; 1.3) where the refinement mask a is finitely supported. When the dilation matrix M is two times the s Theta s identity matrix I s , existence and uniqueness of the solutions of (1. 3) were studied in [3] and [7]. In particular, for the univariate case s = 1, it was proved in [7] that (1.3) has a nontrivial L 1 solution with compact support only if P ff2Z s a(ff) 2 n for some positive integer n. For the vector case (i.e. r 1) the coefficients a(ff) ff 2 Z s in (1.3) become r Theta r ....

....ff) x 2 R s ; 1.3) where the refinement mask a is finitely supported. When the dilation matrix M is two times the s Theta s identity matrix I s , existence and uniqueness of the solutions of (1. 3) were studied in [3] and [7] In particular, for the univariate case s = 1, it was proved in [7] that (1.3) has a nontrivial L 1 solution with compact support only if P ff2Z s a(ff) 2 n for some positive integer n. For the vector case (i.e. r 1) the coefficients a(ff) ff 2 Z s in (1.3) become r Theta r complex matrices. The existence of compactly supported distributional ....

I. Daubechies and J. Lagarias, Two-scale difference equations: I. Existence and global regularity of solutions, SIAM J. Math. Anal. 22 (1991), 1388--1410.

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Daubechies, I. and Lagarias, J. C., Two--scale difference equations I. Existence and global regularity of solutions, SIAM J. Math. Anal. 22 (1991), 1388--1410.

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Ingrid Daubechies and Je#rey C. Lagarias. Two-scale di#erence equations: I. Existence and global regularity of solutions. SIAM J. Math. Anal., 22:1388--1410, 1991.

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I. Daubechies and J. C. Lagarias. Two-scale dierence equations. i. existence and global regularity of solutions. SIAM J. Math. Anal., 22:1388-1410, 1992.

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I. Daubechies and J. C. Lagarias. Two-scale difference equations I. Existence and global regularity of solutions. SIAM J. Math. Anal., 22(5):1388--1410, 1991.

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