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164
Data compression and harmonic analysis
 IEEE Trans. Inform. Theory
, 1998
"... In this paper we review some recent interactions between harmonic analysis and data compression. The story goes back of course to Shannon’s R(D) theory... ..."
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Cited by 140 (24 self)
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In this paper we review some recent interactions between harmonic analysis and data compression. The story goes back of course to Shannon’s R(D) theory...
Multilinear operators given by singular multipliers
 J. Amer. Math. Soc
"... Abstract. We prove L p estimates for a large class of multilinear operators, which includes the multilinear paraproducts studied by Coifman and Meyer [7], as well as the bilinear Hilbert transform. 1. ..."
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Cited by 65 (20 self)
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Abstract. We prove L p estimates for a large class of multilinear operators, which includes the multilinear paraproducts studied by Coifman and Meyer [7], as well as the bilinear Hilbert transform. 1.
Multilinear Calderón Zygmund theory
 ADV. IN MATH. 40
, 1996
"... A systematic treatment of multilinear CalderónZygmundoperators is presented. The theory developed includes strong type and endpoint weak type estimates, interpolation, the multilinear T1 theorem, anda variety of results regarding multilinear multiplier operators. ..."
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Cited by 46 (16 self)
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A systematic treatment of multilinear CalderónZygmundoperators is presented. The theory developed includes strong type and endpoint weak type estimates, interpolation, the multilinear T1 theorem, anda variety of results regarding multilinear multiplier operators.
Uniform bounds for the bilinear Hilbert transforms
 889–993. MR2113017 (2006e:42011), Zbl 1071.44004. Xiaochun Li
, 2004
"... Abstract. We continue the investigation initiated in [8] of uniform Lp bounds � for the family of bilinear Hilbert transforms Hα,β(f, g)(x) = p.v. f(x − αt)g(x − βt) R dt t. In this work we show that Hα,β map Lp1 (R) × Lp2 (R) into Lp (R) uniformly in the real parameters α, β satisfying  α β − 1  ..."
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Cited by 28 (15 self)
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Abstract. We continue the investigation initiated in [8] of uniform Lp bounds � for the family of bilinear Hilbert transforms Hα,β(f, g)(x) = p.v. f(x − αt)g(x − βt) R dt t. In this work we show that Hα,β map Lp1 (R) × Lp2 (R) into Lp (R) uniformly in the real parameters α, β satisfying  α β − 1  ≥ c> 0 when 1 < p1, p2 < 2 and 2 p1p2 3 < p = < ∞. p1+p2 As a corollary we obtain Lp × L ∞ → Lp uniform bounds in the range 4/3 < p < 4 for the H1,α’s when α ∈ [0, 1). 1.
Can recent innovations in harmonic analysis `explain' key findings in natural image statistics
 Network: Computation in Neural Systems
"... Recently, applied mathematicians have been pursuing the goal of sparse coding of certain mathematical models of images with edges. They have found by mathematical analysis that, instead of wavelets and Fourier methods, sparse coding leads towards new systems: ridgelets and curvelets. These new syste ..."
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Cited by 26 (1 self)
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Recently, applied mathematicians have been pursuing the goal of sparse coding of certain mathematical models of images with edges. They have found by mathematical analysis that, instead of wavelets and Fourier methods, sparse coding leads towards new systems: ridgelets and curvelets. These new systems have elements distributed across a range of scales and locations, but also orientations. In fact they have highly directionspecific elements and exhibit increasing numbers of distinct directions as we go to successively finer scales. Meanwhile, researchers in Natural Scene Statistics (NSS) have been attempting to find sparse codes for natural images. The new systems they have found by computational optimization have elements distributed across a range of scales and locations, but also orientations. The new systems are certainly unlike wavelet and gabor systems, on the one hand because of the multiorientation and on the other hand because of the multiscale nature. There is a certain degree of visual resemblance between the findings in the two fields, which suggests the hypothesis that certain important findings in the NSS literature
The Marcinkiewicz multiplier condition for bilinear operators
 Studia Math. 146 (2001), 115–156. LOUKAS GRAFAKOS
"... Abstract. This article is concerned with the question of whether Marcinkiewicz multipliers on R2n give rise to bilinear multipliers on Rn × Rn.We show that this is not always the case. Moreover we find necessary and sufficient conditions for such bilinear multipliers to be bounded. These conditions ..."
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Cited by 25 (7 self)
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Abstract. This article is concerned with the question of whether Marcinkiewicz multipliers on R2n give rise to bilinear multipliers on Rn × Rn.We show that this is not always the case. Moreover we find necessary and sufficient conditions for such bilinear multipliers to be bounded. These conditions in particular imply that a slight logarithmic modification of the Marcinkiewicz condition gives multipliers for which the corresponding bilinear operators are bounded on products of Lebesgue and Hardy spaces. 1.
Euclidean algorithms are Gaussian
, 2003
"... Abstract. We prove a Central Limit Theorem for a general class of costparameters associated to the three standard Euclidean algorithms, with optimal speed of convergence, and error terms for the mean and variance. For the most basic parameter of the algorithms, the number of steps, we go further an ..."
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Cited by 22 (10 self)
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Abstract. We prove a Central Limit Theorem for a general class of costparameters associated to the three standard Euclidean algorithms, with optimal speed of convergence, and error terms for the mean and variance. For the most basic parameter of the algorithms, the number of steps, we go further and prove a Local Limit Theorem (LLT), with speed of convergence O((log N) −1/4+ǫ). This extends and improves the LLT obtained by Hensley [27] in the case of the standard Euclidean algorithm. We use a “dynamical analysis ” methodology, viewing an algorithm as a dynamical system (restricted to rational inputs), and combining tools imported from dynamics, such as the crucial transfer operators, with various other techniques: Dirichlet series, Perron’s formula, quasipowers theorems, the saddle point method. Dynamical analysis had previously been used to perform averagecase analysis of algorithms. For the present (dynamical) analysis in distribution, we require precise estimates on the transfer operators, when a parameter varies along vertical lines in the complex plane. Such estimates build on results obtained only recently by Dolgopyat in the context of continuoustime dynamics [20]. 1.
Regression in Random Design and Warped Wavelets
 BERNOULLI,10
, 2004
"... We consider the problem of estimating an unknown function f in a regression setting with random design. Instead of expanding the function on a regular wavelet basis, we expand it on the basis jk (G), j, k} warped with the design. This allows to perform a very stable and computable thresholding alg ..."
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Cited by 19 (0 self)
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We consider the problem of estimating an unknown function f in a regression setting with random design. Instead of expanding the function on a regular wavelet basis, we expand it on the basis jk (G), j, k} warped with the design. This allows to perform a very stable and computable thresholding algorithm. We investigate the properties of this new basis. In particular, we prove that if the design has a property of Muckenhoupt type, this new basis has a behavior quite similar to a regular wavelet basis. This enables us to prove that the associated thresholding procedure achieves rates of convergence which have been proved to be minimax in the uniform design case.
The dichotomy between structure and randomness, arithmetic progressions, and the primes
"... Abstract. A famous theorem of Szemerédi asserts that all subsets of the integers with positive upper density will contain arbitrarily long arithmetic progressions. There are many different proofs of this deep theorem, but they are all based on a fundamental dichotomy between structure and randomness ..."
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Cited by 19 (1 self)
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Abstract. A famous theorem of Szemerédi asserts that all subsets of the integers with positive upper density will contain arbitrarily long arithmetic progressions. There are many different proofs of this deep theorem, but they are all based on a fundamental dichotomy between structure and randomness, which in turn leads (roughly speaking) to a decomposition of any object into a structured (lowcomplexity) component and a random (discorrelated) component. Important examples of these types of decompositions include the Furstenberg structure theorem and the Szemerédi regularity lemma. One recent application of this dichotomy is the result of Green and Tao establishing that the prime numbers contain arbitrarily long arithmetic progressions (despite having density zero in the integers). The power of this dichotomy is evidenced by the fact that the GreenTao theorem requires surprisingly little technology from analytic number theory, relying instead almost exclusively on manifestations of this dichotomy such as Szemerédi’s theorem. In this paper we survey various manifestations of this dichotomy in combinatorics, harmonic analysis, ergodic theory, and number theory. As we hope to emphasize here, the underlying themes in these arguments are remarkably similar even though the contexts are radically different. 1.