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302
Ridgelets: A key to higherdimensional intermittency?
, 1999
"... In dimensions two and higher, wavelets can efficiently represent only a small range of the full diversity of interesting behavior. In effect, wavelets are welladapted for pointlike phenomena, whereas in dimensions greater than one, interesting phenomena can be organized along lines, hyperplanes, and ..."
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Cited by 112 (10 self)
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In dimensions two and higher, wavelets can efficiently represent only a small range of the full diversity of interesting behavior. In effect, wavelets are welladapted for pointlike phenomena, whereas in dimensions greater than one, interesting phenomena can be organized along lines, hyperplanes, and other nonpointlike structures, for which wavelets are poorly adapted. We discuss in this paper a new subject, ridgelet analysis, which can effectively deal with linelike phenomena in dimension 2, planelike phenomena in dimension 3 and so on. It encompasses a collection of tools which all begin from the idea of analysis by ridge functions ψ(u1x1+...+unxn) whose ridge profiles ψ are wavelets, or alternatively from performing a wavelet analysis in the Radon domain. The paper reviews recent work on the continuous ridgelet transform (CRT), ridgelet frames, ridgelet orthonormal bases, ridgelets and edges and describes a new notion of smoothness naturally attached to this new representation.
Orthonormal Ridgelets and Linear Singularities
, 1998
"... We construct a new orthonormal basis for L2 (R2), whose elements are angularly integrated ridge functions — orthonormal ridgelets. The new basis functions are in L2 (R2) and so are to be distinguished from the ridge function approximation system called ridgelets by Candès (1997, 1998), as ridge func ..."
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Cited by 56 (16 self)
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We construct a new orthonormal basis for L2 (R2), whose elements are angularly integrated ridge functions — orthonormal ridgelets. The new basis functions are in L2 (R2) and so are to be distinguished from the ridge function approximation system called ridgelets by Candès (1997, 1998), as ridge functions are not in L2 (R2). Orthonormal ridgelet expansions have an interesting application in nonlinear approximation: the problem of efficient approximations to objects such as 1 {x1 cos θ+x2 sin θ>a} e−x2 1−x2 2 which are smooth away from a discontinuity along a line. The orthonormal ridgelet coefficients of such an object are sparse: they belong to every ℓp, p>0. This implies that simple thresholding in the ridgelet orthobasis is, in a certain sense, a nearideal nonlinear approximation scheme. The ridgelet orthobasis is the isometric image of a special wavelet basis for Radon space; as a consequence, ridgelet analysis is equivalent to a special wavelet analysis in the Radon domain. This means that questions of ridgelet analysis of linear singularities can be answered by wavelet analysis of point singularities. At the heart of our nonlinear approximation result is the study of a certain tempered distribution on R2 defined formally by S(u, v) =v  −1/2σ(u/v) with σ a certain smooth bounded function; this is singular at (u, v) =(0,0) and C ∞ elsewhere. The key point is that the analysis of this point singularity by tensor Meyer wavelets yields sparse coefficients at high frequencies; this is reflected in the sparsity of the ridgelet coefficients and the good nonlinear approximation properties of the ridgelet basis.
A CUBIC DIRAC OPERATOR AND THE EMERGENCE OF EULER NUMBER MULTIPLETS OF REPRESENTATIONS FOR EQUAL RANK SUBGROUPS
 VOL. 100, NO. 3 DUKE MATHEMATICAL JOURNAL
, 1999
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Quantum graphs: II. Some spectral properties of quantum and combinatorial graphs
 J. Phys. A: Math. Gen
"... The paper deals with some spectral properties of (mostly infinite) quantum and combinatorial graphs. Quantum graphs have been intensively studied lately due to their numerous applications to mesoscopic physics, nanotechnology, optics, and other areas. A Schnol type theorem is proven that allows one ..."
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Cited by 45 (5 self)
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The paper deals with some spectral properties of (mostly infinite) quantum and combinatorial graphs. Quantum graphs have been intensively studied lately due to their numerous applications to mesoscopic physics, nanotechnology, optics, and other areas. A Schnol type theorem is proven that allows one to detect that a point λ belongs to the spectrum when a generalized eigenfunction with an subexponential growth integral estimate is available. A theorem on spectral gap opening for “decorated ” quantum graphs is established (its analog is known for the combinatorial case). It is also shown that if a periodic combinatorial or quantum graph has a point spectrum, it is generated by compactly supported eigenfunctions (“scars”). 1
Fast Discrete Polynomial Transforms with Applications to Data Analysis for Distance Transitive Graphs
, 1997
"... . Let P = fP 0 ; : : : ; Pn\Gamma1 g denote a set of polynomials with complex coefficients. Let Z = fz 0 ; : : : ; z n\Gamma1 g ae C denote any set of sample points. For any f = (f 0 ; : : : ; fn\Gamma1 ) 2 C n the discrete polynomial transform of f (with respect to P and Z) is defined as the col ..."
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Cited by 36 (7 self)
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. Let P = fP 0 ; : : : ; Pn\Gamma1 g denote a set of polynomials with complex coefficients. Let Z = fz 0 ; : : : ; z n\Gamma1 g ae C denote any set of sample points. For any f = (f 0 ; : : : ; fn\Gamma1 ) 2 C n the discrete polynomial transform of f (with respect to P and Z) is defined as the collection of sums, f b f(P 0 ); : : : ; b f(Pn\Gamma1 )g, where f(P j ) = hf; P j i = P n\Gamma1 i=0 f i P j (z i )w(i) for some associated weight function w. These sorts of transforms find important applications in areas such as medical imaging and signal processing. In this paper we present fast algorithms for computing discrete orthogonal polynomial transforms. For a system of N orthogonal polynomials of degree at most N \Gamma 1 we give an O(N log 2 N) algorithm for computing a discrete polynomial transform at an arbitrary set of points instead of the N 2 operations required by direct evaluation. Our algorithm depends only on the fact that orthogonal polynomial sets satisfy a thre...
Fourier Inversion On A Reductive Symmetric Space
, 1999
"... this paper is an important step towards the Plancherel formula for X. What remains for the Plancherel formula is essentially to identify the contributions TF f in terms of generalized principal series representations. For example, T \Delta f should be identified as being in the discrete series for X ..."
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Cited by 34 (13 self)
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this paper is an important step towards the Plancherel formula for X. What remains for the Plancherel formula is essentially to identify the contributions TF f in terms of generalized principal series representations. For example, T \Delta f should be identified as being in the discrete series for X. These identifications will be given in a sequel [13] to this paper, but since it is an important application we outline the argument here. For F = ; the identification is inherent already in the definition of F and J by means of the minimal principal series  an important ingredient is the regularity (from [8]) of the normalized Eisenstein integrals on ia q . This regularity is, in turn, based on the socalled MaassSelberg relations from [6], according to which (cf. [9, Prop. 5.3]) the adjoint of the Cfunction is given by (s: \Gamma : (1.8) For the nonminimal principal series analogues of F and J have been defined and the MaassSelberg relations have been generalized, by Carmona and Delorme (see [14], [18], [19], [15]). Using these generalized MaassSelberg relations we obtain the necessary identifications of TF f for F 6= \Delta. In particular, these functions are tempered. As a consequence of Theorem 1.2 it follows then that T \Delta f is in the discrete series, and the Plancherel formula is established. A different proof of the Plancherel formula, also based on the generalized MaassSelberg relations, has been obtained independently and simultaneously by Delorme (see [20]). Later, we have found a proof of these generalized MaassSelberg relations based on the results of the present paper. This proof will also be given in [13]
Continuous Rotation Invariant Valuations on Convex Sets
, 1999
"... this paper. For more detailed information we refer to the surveys [McSch] and [Mc3]. Throughout this paper all the valuations are assumed to be continuous with respect to the Hausdor# metric. Note that the theory of valuations which are invariant or covariant with respect to translations belongs to ..."
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Cited by 32 (2 self)
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this paper. For more detailed information we refer to the surveys [McSch] and [Mc3]. Throughout this paper all the valuations are assumed to be continuous with respect to the Hausdor# metric. Note that the theory of valuations which are invariant or covariant with respect to translations belongs to the classical part of convex geometry. There exists an explicit description of translation invariant continuous valuations on
Macdonald's Evaluation Conjectures and Difference Fourier Transform
, 1994
"... This paper contains the proof of the remaining two (the duality and evaluation conjectures). The evaluation conjecture (now a theorem) is in fact a q; tgeneralization of the classic Weyl dimension formula. One can expect interesting applications of this theorem since the socalled qdimensions are ..."
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Cited by 32 (1 self)
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This paper contains the proof of the remaining two (the duality and evaluation conjectures). The evaluation conjecture (now a theorem) is in fact a q; tgeneralization of the classic Weyl dimension formula. One can expect interesting applications of this theorem since the socalled qdimensions are undoubtedly important. It is likely that we can incorporate the KacMoody case as well. The necessary technique was developed in [C4]. As to the duality theorem (in its complete form), it states that the generalized trigonometricdifference zonal Fourier transform is selfdual (at least formally). We define this q; ttransform in terms of double affine Hecke algebras. The most natural way to check the selfduality is to use the connection of these algebras with the socalled elliptic braid groups (the Fourier involution will turn into the transposition of the periods of an elliptic curve). The classical trigonometricdifferential Fourier transform (corresponding to the limit q = t as t ! 1 for certain special k) plays one of the main roles in the harmonic analysis on symmetric spaces. It sends symmetric trigonometric polynomials to the corresponding radial parts of Laplace operators (HarishChandra, Helgason) and is not selfdual. The calculation of its inverse (the Plancherel theorem) is always challenging and involving. * Partially supported by NSF grant DMS9301114 In the rationaldifferential setting, Charles Dunkl introduced the generalized Hankel transform which appeared to be selfdual [D,J]. We demonstrate in this paper that one can save this very important property if trigonometric polynomials come together with difference operators. At the moment, it is mostly an algebraic observation (the differenceanalitical aspects were not touched upon). The root systems of ...
Conformal field theory of the integer quantum Hall plateau transition
"... A solution to the longstanding problem of identifying the conformal field theory governing the transition between quantized Hall plateaus of a disordered noninteracting 2d electron gas, is proposed. The theory is a nonlinear sigma model with a WessZuminoNovikovWitten term, and fields taking valu ..."
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Cited by 31 (1 self)
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A solution to the longstanding problem of identifying the conformal field theory governing the transition between quantized Hall plateaus of a disordered noninteracting 2d electron gas, is proposed. The theory is a nonlinear sigma model with a WessZuminoNovikovWitten term, and fields taking values in a Riemannian symmetric superspace based on H 3 × S 3. Essentially the same conformal field theory appeared in very recent work on string propagation in AdS3 backgrounds. We explain how the proposed theory manages to obey a number of tight constraints, two of which are constancy of the partition function and noncriticality of the local density of states. An unexpected feature is the existence of a truly marginal deformation, restricting the extent to which universality can hold in critical quantum Hall systems. The marginal coupling is fixed by matching the shortdistance singularity of the conductance between two interior contacts to the classical conductivity σxx = 1/2 of the ChalkerCoddington network model. For this value, perturbation theory predicts a critical exponent Xt = 2/π for the typical pointcontact conductance, in agreement with numerical simulations. The irrational exponent is tolerated by the fact that the symmetry algebra of the field theory is Virasoro but not KacMoody. 1