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.

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We construct a new orthonormal basis for L 2 (R 2), whose elements are angularly integrated ridge functions — orthonormal ridgelets. The basis elements are smooth and of rapid decay in the spatial domain, and in the frequency domain are localized near angular wedges which, at radius r =2 j,haveradialextent∆r≈2 j and angular extent ∆θ ≈ 2 −j. Orthonormal ridgelet expansions expose an interesting phenomenon in nonlinear approxi-mation: they give very 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 objects are sparse: they belong to every ℓp, p>0. This implies that simple thresholding in the ridgelet orthobasis is, in a certain sense, a near-ideal nonlinear approximation scheme for such objects. Orthonormal ridgelets may be viewed as L2 substitutes for approximation by sums of ridge functions, and so can perform many of the same tasks as the ridgelets systems constructed by Candès (1997, 1998). Orthonormal ridgelets make available the machinery of orthogonal decompositions, which is not availble for ridge functions as they are not in L2 (R2). The ridgelet orthobasis is constructed as the isometric image of a special wavelet basis for

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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...

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; t-generalization of the classic Weyl dimension formula. One can expect interesting applications of this theorem since the so-called q-dimensions are undoubtedly important. It is likely that we can incorporate the Kac-Moody case as well. The necessary technique was developed in [C4]. As to the duality theorem (in its complete form), it states that the generalized trigonometric-difference zonal Fourier transform is self-dual (at least formally). We define this q; t-transform in terms of double affine Hecke algebras. The most natural way to check the self-duality is to use the connection of these algebras with the so-called elliptic braid groups (the Fourier involution will turn into the transposition of the periods of an elliptic curve). The classical trigonometric-differential 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 self-dual. The calculation of its inverse (the Plancherel theorem) is always challenging and involving. * Partially supported by NSF grant DMS--9301114 In the rational-differential setting, Charles Dunkl introduced the generalized Hankel transform which appeared to be self-dual [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 difference-analitical aspects were not touched upon). The root systems of ...

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 so-called Maass-Selberg relations from [6], according to which (cf. [9, Prop. 5.3]) the adjoint of the C-function is given by (s: \Gamma : (1.8) For the non-minimal principal series analogues of F and J have been defined and the Maass-Selberg relations have been generalized, by Carmona and Delorme (see [14], [18], [19], [15]). Using these generalized Maass-Selberg 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 Maass-Selberg relations, has been obtained independently and simultaneously by Delorme (see [20]). Later, we have found a proof of these generalized Maass-Selberg relations based on the results of the present paper. This proof will also be given in [13]

A solution to the long-standing 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 Wess-Zumino-Novikov-Witten 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 short-distance singularity of the conductance between two interior contacts to the classical conductivity σxx = 1/2 of the Chalker-Coddington network model. For this value, perturbation theory predicts a critical exponent Xt = 2/π for the typical point-contact 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 Kac-Moody. 1

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this paper. For more detailed information we refer to the surveys [Mc-Sch] 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