M. Frazier and B. Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J. 34 (1985), pp. 777--799. 92 References  Home/Search   Document Not in Database   Summary   Related Articles   Check

....dense , discrete subset of the plane. Moreover, the theory is not restricted to L (IR) but the size of the samples actually characterizes smoothness and decay properties of functions. See [10, 11] For other aspects of sampling of wavelet and short time Fourier transforms we refer to [1, 6, 9]. In this note we show how recent quantitative results on irregular sampling of band limited functions also yield irregular sampling theorems for wavelet and short time Fourier transforms. This extends the program of [3] and [4] to irregular sampling. The novelty lies (a) in the explicit ....

M. Frazier, B. Jawerth. Decomposition of Besov spaces. Indiana Univ. Math. J. 34 (1985), 777--799.

....compression 19 Solution of linear systems 20 Appendices A Fourier transforms B A collection of formulas 1History Some ideas related to wavelets already existed at the beginning of the century, but the real development came only in the mid eighties. Besides a paper by Frazier and Jawerth (1985) [12], wavelets were in the initial stage developed in France, the so called French school lead by J. Morlet, A. Grossmann and Y. Meyer. Wavelets, or Ondelettes as they are called in French were used at the beginning of the eighties by J. Morlet, a geophysicist, as a tool for signal analysis for ....

M. Frazier and B. Jawerth. Decomposition of Besov spaces. Indiana University Mathematics Journal, 34(4):777--799, 1985.

....and go back to the context of subband filters, or more precisely quadrature mirror filters [36, 37, 41, 51, 52, 53, 54, 58, 56, 59] In mathematical analysis, wavelets were defined as translates and dilates of one fixed function and were used to both analyze and represent general functions. [15, 20, 25, 35, 24]. In the late eighties the introduction of multiresolution analysis and the fast wavelet transform by Mallat and Meyer provided the connection between subband filters and wavelets [33, 34, 35] this led to the first construction of smooth, orthogonal, and compactly supported wavelets in 1987 [18] ....

M. Frazier and B. Jawerth. Decomposition of Besov spaces. Indiana Univ. Math. J., 34(4):777--799, 1985.

....equivalent norm kf j C s (R n )k = X jffj[s] kD ff f j L1 (R n )k X jffj= s] sup jD ff f(x) Gamma D ff f(y)j jx Gamma yj fsg ; 5) where the supremum is taken over all x 2 R n and y 2 R n with x 6= y. 2. 2 Atoms We adapt the atoms introduced by Frazier and Jawerth in [12], 13] and [14] to our later purposes. Let again N be the natural numbers, N 0 = f0g [ N, and Z n be the lattice of all points in R n with integer valued components. Let b 0 be given, 2 N 0 and k 2 Z n . Then Q k denotes a cube in R n with sides parallel to the axes, centered at x ;k ....

....value of L) and 2 f pq . Furthermore, inf k j f pq k; 14) where the infimum is taken over all admissible representations (12) is an equivalent quasi norm in F s pq (R n ) Remark 3. As we said this theorem is at least in principle covered by the work of Frazier and Jawerth, see [12], 13] and [14] Our formulation is different and, as we hope, more handsome, even on R n , and we switched from requirements like (2.2 11) to their counterparts (2.2 6) This latter modification prepares the atomic approach to spaces on domains. Starting from (2.2 11) one has to replace 0 oe = ....

Frazier,M., Jawerth,B.: Decomposition of Besov spaces. Indiana Univ. Math. J. 34, 777-- 799 (1985)

.... ; 2 J ; 2.7) 2.6) implies the dual relation kD 1 dk 2 (J ) kd T kH 0 : 2.8) Similar relations are also known to hold for Sobolev spaces in L p for p 6= 2. Moreover, interpolation between such spaces provides norm equivalences for a whole range of Besov spaces B q (L p ) [24, 37, 39, 44]. In the present context we will have to make use of the following special case kdk (J ) kd T kB (L ) 2.9) where the smoothness index and the integrability index are related by 1 = d 1 2 : 2.10) 2.3. Cancellation Property. The second main requirement on the ....

M. Frazier and B. Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J., 34 (1985), pp. 777-799. 32 A. BARINKA, T. BARSCH, P. CHARTON, A. COHEN, S. DAHLKE, W. DAHMEN, AND K. URBAN

....B ff p . These spaces measure oscillatory properties of functions both at large and small scales. In particular for p = 2, these spaces coincide with the (homogeneous) Sobolev spaces. It is true that ( B ff p ) B Gammaff p 0 , for 1 p 1. Based on the work of Frazier and Jawerth [5] the function OE in (12) can be chosen to generate an almost orthogonal wavelet (OE transform) decomposition of the Besov spaces. That is, every f 2 B ff p can be written in the form f = X ;k hf; OE k iOE k ; 13) and kfk B ff p X ;k Gamma jhf; OE k ij2 (ff n=2 Gamman=p) ....

M. Frazier, B. Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J. 34 (1985), 777799.

....; 2 J ; 2.9) 2.8) implies the dual relation kD 1 dk 2 (J ) kd T kH 0 : 2.10) Similar relations are also known to hold for Sobolev spaces in L p for p 6= 2. Moreover, interpolation between such spaces provides norm equivalences for a whole range of Besov spaces B q (L p ) [26, 38, 40, 46]. In the present context we will have to make use of the following special case kdk (J ) kd T kB (L ) 2.11) where the smoothness index and the integrability index are related by 1 = d 1 2 : 2.12) 2.3 Cancellation Property The second main requirement on the ....

M. Frazier and B. Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J. 34 (1985), 777-799.

....directly from (2.1) for all x; y 2 R n one has j; f(x) j; f(y) 1 2 j jx Gamma yj) 2.4) The next assertion supplies a very useful sufficient condition for the convergence of the series in B s pq and F s pq . It is contained in well known results of Frazier and Jawerth [8], 9] and Netrusov [15] on atomic characterizations of these spaces. Let Q j;k for j 2 N 0 j N [ f0g and k 2 Z n denote the cube given by Q j;k = 2 Gammaj k 1 ; 2 Gammaj (k 1 1) Theta : Theta [2 Gammaj kn ; 2 Gammaj (kn 1) dyadic cubes) If Q ae R n is a cube ....

....s; p; q; n. Remark 2.1. The role of the assumptions s ( n p Gamma n) for B s pq ; s ( n min(p;q) Gamma n) for F s pq : 2. 6) is that if s does not satisfy them, then one has to impose moment conditions on the atoms in the relevant atomic characterizations of B s pq and F s pq (see [8], 9] 15] In other words, the convergence of the series P 1 j=0 f j becomes then determined not only by smoothness and size conditions, but also by certain cancellations. Dealing with cancellation phenomena would require methods different from those developed in this paper, and we do not ....

M. Frazier and B. Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J. 34 (1985), 777-799.

....an isomorphic map from F s pq (u) onto F s pq . In particular, kuf j F s pq k is an equivalent norm on F s pq (u) Theorem 3 has been proved in [10] As a basic tool we used an atomic decomposition of F s pq (u) Concerning atomic decompositions of (unweighted) F s pq spaces, we refer to [5], 6] 7] and [14] 272 THOMAS SCHOTT Let 1 p 1, k 2 N 0 , and u 2 W e . Then the weighted Sobolev space W k p (u) is the set of all f 2 D 0 such that D ff f 2 L p (u) jffj k. The norm on W k p (u) is given by kf j W k p (u)k = i X jffjk kD ff f j L p (u)k p j 1=p : The ....

M. Frazier and B. Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J. 34 (1985), 777--799.

....with important topics outside of signal processing. CHAPTER 3. BASIS PURSUIT 39 3.4. 1 Atomic Decomposition The terminology atomic decomposition arose in the mathematical discipline of harmonic analysis [13] Decompositions based on optimizations like BP have been used extensively in that field [36, 37, 47]; we give two examples. Yves Meyer has described in his book on wavelets [64] the so called Bump Algebra B, which is defined by a BP type optimization. Begin with a dictionary DBump of Gaussian Bumps : D = OE fl : fl = t; s) where OE fl is the normalized Gaussian expf Gamma(x Gamma t) ....

M. Frazier and B. Jawerth. Decomposition of Besov spaces. Indiana University Mathematics Journal, pp. 777--799, 1985.

....over the last ten years. We refer to the work of (in alphabetical order) Aldroubi and Unser [2, 3, 108, 107] Battle and Lemarie [13, 78] Chui and Wang [19, 25, 24, 23] Cohen and Daubechies [28] Cohen, Daubechies, and Feauveau [29] Daubechies [47, 49, 48] Donoho [57, 56] Frazier and Jawerth [65, 67, 66], Herley and Vetterli [73, 110] Kovacevic and Vetterli [77, 111] Mallat [85, 84, 86] Meyer [87] and many more. Except for Donoho, they all rely on the Fourier transform as a basic construction tool. The reason is that translation and dilation become algebraic operations in the Fourier domain. ....

M. Frazier and B. Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J., 34 (1995), pp. 777--799.

....[0; 1] Such a basis has elements which are in C R and have, at high resolutions, D vanishing moments. It acts as an unconditional basis for a very wide range of smoothness spaces: all the Besov classes B oe p;q [0; 1] and Triebel classes F oe p;q [0; 1] in a certain range 0 oe min(R; D) [25, 29, 18, 16, 17]. Each of these classes has a norm k Delta kB oe p;q or k Delta k F oe p;q which measures smoothness. Special cases include the traditional Holder ( Zygmund) classes C oe = B oe 1;1 and Sobolev Classes W oe p = F oe p;2 . Definition. S is the scale of all spaces B oe p;q and all ....

Frazier, M. and Jawerth, B. (1985). Decomposition of Besov spaces. Indiana Univ. Math. J., 777--799.

....( 2 N) in the sense of Definition 3 (i) and (ii) respectively, with (32) and = f k g 2N 0 ;k2Z n 2 b p;q . Furthermore, inf k j b p;q k; where the infimum is taken over all admissible representations (33) is an equivalent quasi norm in B s p;q (R n ) As for the proof we refer to [3]. Local means. A function K 2 S(R n ) is an admissible kernel if it can be represented as K = Delta N K 0 , where N 2 N 0 is sufficiently large, Delta is the Laplacian, K 0 2 S(R n ) has a compact support and c K 0 (0) 6= 0. If f 2 S 0 (R n ) then the local means are defined ....

M. Frazier and B. Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J. 34 (1985), 777-- 799.

....or generalized basis, for each subspace, we show that the union of the local atoms can generate a global frame for the Hilbert space. Corresponding duals can be calculated in a exible way by means of the systems of quasi projectors. Decomposition methods were introduced by Frazier and Jawerth [8] in order to construct wavelet type bases for Besov spaces. A general presentation of this kind of methods was proposed by Gr obner and Feichtinger [4, 6, 9] showing how many di erent classical and widely used Banach spaces (L p spaces, Besov and Triebel spaces, Modulation spaces, Wiener ....

M. Frazier and B. Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J. 34 (1985), 777-799.

....be used both to give a unified view of many of the results in harmonic analysis and also, at least potentially, could be effective substitutes for Fourier series in numerical applications. The first named author of this paper came to this understanding through the joint work with Mike Frazier [57, 58, 59]. As the emphasis shifted more towards the representations themselves, and the building blocks involved, the name also shifted: Yves Meyer and Jean Morlet suggested the word wavelet for the building blocks, and what earlier had been referred to as Littlewood Paley theory now started to be called ....

....and this was done successfully e.g. in geophysics. The transform is often graphically represented as two two dimensional images with color or grey value corresponding to the modulus and phase of W(a; b) The continuous wavelet transform is also used in singularity detection and characterization [57, 82]. A typical result in this direction is that if a function f is Holder (Lipschitz) continuous of order 0 ff 1, so that jf(x h) Gamma f(x)j = O(h ff ) then the continuous wavelet transform has an asymptotic behavior like W(a; b) O(a ff 1=2 ) for a 0: In fact, the converse is true ....

M. Frazier and B. Jawerth. Decomposition of Besov spaces. Indiana Univ. Math. J., 34(4):777--799, 1985.

No context found.

M. Frazier and B. Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J. 34 (1985), pp. 777--799. 92 References

No context found.

M. Frazier, B. Jawerth, Decomposition of Besov Spaces, Indiana Univ. Math. J. 34 (1985) 777-799.

No context found.

M. Frazier, B. Jawerth. Decomposition of Besov spaces. Indiana Univ. Math. J. 34 (1985), 777-799.

No context found.

Frazier, M. and Jawerth, B., Decomposition of Besov spaces, Indiana Univ. Math. J. 34 (1985), 777-799.

No context found.

M. Frazier and B. Jawerth. Decomposition of Besov spaces. Indiana University Mathematics Journal, pp. 777--799, 1985.

No context found.

Frazier, M., and Jawerth, B. (1985). Decomposition of Besov Spaces. Indiana Univ. Math J. 34 777-799.

No context found.

Frazier, M. and Jawerth, B. (1985). Decomposition of Besov Spaces. Indiana Univ. Math J. 34 (1985) 777-799.

No context found.

Frazier, M. and Jawerth, B. (1985). Decomposition of Besov Spaces. Indiana Univ. Math J. 34 (1985) 777-799.

No context found.

Frazier, M., and Jawerth, B. (1985). Decomposition of Besov Spaces. Indiana Univ. Math J. 34 777-799.

No context found.

M. Frazier and B. Jawerth, Decomposition of Besov-spaces, Indiana Univ. Math. J. 34 (1985), 777--799.

First 50 documents

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