Results 1  10
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27
Near Optimal Signal Recovery From Random Projections: Universal Encoding Strategies?
, 2004
"... Suppose we are given a vector f in RN. How many linear measurements do we need to make about f to be able to recover f to within precision ɛ in the Euclidean (ℓ2) metric? Or more exactly, suppose we are interested in a class F of such objects— discrete digital signals, images, etc; how many linear m ..."
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Cited by 1513 (20 self)
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Suppose we are given a vector f in RN. How many linear measurements do we need to make about f to be able to recover f to within precision ɛ in the Euclidean (ℓ2) metric? Or more exactly, suppose we are interested in a class F of such objects— discrete digital signals, images, etc; how many linear measurements do we need to recover objects from this class to within accuracy ɛ? This paper shows that if the objects of interest are sparse or compressible in the sense that the reordered entries of a signal f ∈ F decay like a powerlaw (or if the coefficient sequence of f in a fixed basis decays like a powerlaw), then it is possible to reconstruct f to within very high accuracy from a small number of random measurements. typical result is as follows: we rearrange the entries of f (or its coefficients in a fixed basis) in decreasing order of magnitude f  (1) ≥ f  (2) ≥... ≥ f  (N), and define the weakℓp ball as the class F of those elements whose entries obey the power decay law f  (n) ≤ C · n −1/p. We take measurements 〈f, Xk〉, k = 1,..., K, where the Xk are Ndimensional Gaussian
Compressive Sensing and Structured Random Matrices
 RADON SERIES COMP. APPL. MATH XX, 1–95 © DE GRUYTER 20YY
, 2011
"... These notes give a mathematical introduction to compressive sensing focusing on recovery using ℓ1minimization and structured random matrices. An emphasis is put on techniques for proving probabilistic estimates for condition numbers of structured random matrices. Estimates of this type are key to ..."
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Cited by 157 (18 self)
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These notes give a mathematical introduction to compressive sensing focusing on recovery using ℓ1minimization and structured random matrices. An emphasis is put on techniques for proving probabilistic estimates for condition numbers of structured random matrices. Estimates of this type are key to providing conditions that ensure exact or approximate recovery of sparse vectors using ℓ1minimization.
Restriction and Kakeya phenomena for finite fields
 DUKE MATH. J
, 2004
"... The restriction and Kakeya problems in Euclidean space have received much attention in the last few decades, and they are related to many problems in harmonic analysis, partial differential equations (PDEs), and number theory. In this paper we initiate the study of these problems on finite fields. I ..."
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Cited by 41 (0 self)
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The restriction and Kakeya problems in Euclidean space have received much attention in the last few decades, and they are related to many problems in harmonic analysis, partial differential equations (PDEs), and number theory. In this paper we initiate the study of these problems on finite fields. In many cases the Euclidean arguments carry over easily to the finite setting (and are, in fact, somewhat cleaner), but there
Fourier analysis, Schur multipliers on Sp and noncommutative
 Λ(p)sets,’ Studia Math
, 1999
"... This work deals with various questions concerning Fourier multipliers on Lp, Schur multipliers on the Schatten class Sp as well as their completely bounded versions when Lp and Sp are viewed as operator spaces. We use for this aim subsets of Z enjoying the noncommutative Λ(p)property which is a ne ..."
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Cited by 18 (2 self)
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This work deals with various questions concerning Fourier multipliers on Lp, Schur multipliers on the Schatten class Sp as well as their completely bounded versions when Lp and Sp are viewed as operator spaces. We use for this aim subsets of Z enjoying the noncommutative Λ(p)property which is a new analytic property much stronger than the classical Λ(p)property. We start by studying the notion of noncommutative Λ(p)sets in the general case of an arbitrary discrete group before
Subspaces and orthogonal decompositions generated by bounded orthogonal systems
"... We investigate properties of subspaces of L2 spanned by subsets of a finite orthonormal system bounded in the L ∞ norm. We first prove that there exists an arbitrarily large subset of this orthonormal system on which the L1 and the L2 norms are close, up to a logarithmic factor. Considering for exa ..."
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Cited by 15 (4 self)
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We investigate properties of subspaces of L2 spanned by subsets of a finite orthonormal system bounded in the L ∞ norm. We first prove that there exists an arbitrarily large subset of this orthonormal system on which the L1 and the L2 norms are close, up to a logarithmic factor. Considering for example the Walsh system, we deduce the existence of two orthogonal subspaces of Ln 2, complementary to each other and each of dimension roughly n/2, spanned by ±1 vectors (i.e. Kashin’s splitting) and in logarithmic distance to the Euclidean space. The same method applies for p> 2, and, in connection with the Λp problem (solved by Bourgain), we study large subsets of this orthonormal system on which the L2 and the Lp norms are close (again, up to a logarithmic factor).
CHARACTERIZATIONS OF HANKEL MULTIPLIERS
"... Abstract. We give characterizations of radial Fourier multipliers as acting on radial L p functions, 1 < p < 2d/(d + 1), in terms of Lebesgue space norms for Fourier localized pieces of the convolution kernel. This is a special case of corresponding results for general Hankel multipliers. Besi ..."
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Cited by 7 (4 self)
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Abstract. We give characterizations of radial Fourier multipliers as acting on radial L p functions, 1 < p < 2d/(d + 1), in terms of Lebesgue space norms for Fourier localized pieces of the convolution kernel. This is a special case of corresponding results for general Hankel multipliers. Besides L p − L q bounds we also characterize weak type inequalities and intermediate inequalities involving Lorentz spaces. Applications include results on interpolation of multiplier spaces. 1.
Necessary conditions for vectorvalued operator inequalities in harmonic analysis
 Proc. London Math. Soc
"... Abstract. Via a random construction we establish necessary conditions for L p (ℓ q) inequalities for certain families of operators arising in harmonic analysis. In particular we consider dilates of a convolution kernel with compactly supported Fourier transform, vector maximal functions acting on cl ..."
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Cited by 6 (2 self)
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Abstract. Via a random construction we establish necessary conditions for L p (ℓ q) inequalities for certain families of operators arising in harmonic analysis. In particular we consider dilates of a convolution kernel with compactly supported Fourier transform, vector maximal functions acting on classes of entire functions of exponential type, and a characterization of Sobolev spaces by square functions and pointwise moduli of smoothness. 1.
Lacunary matrices
 Indiana Univ. Math. J
"... We study unconditional subsequences of the canonical basis (erc) of elementary matrices in the Schatten class Sp. They form the matrix counterpart to Rudin’s Λ(p) sets of integers in Fourier analysis. In the case of p an even integer, we find a sufficient condition in terms of trails on a bipartite ..."
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Cited by 3 (1 self)
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We study unconditional subsequences of the canonical basis (erc) of elementary matrices in the Schatten class Sp. They form the matrix counterpart to Rudin’s Λ(p) sets of integers in Fourier analysis. In the case of p an even integer, we find a sufficient condition in terms of trails on a bipartite graph. We also establish an optimal density condition and present a random construction of bipartite graphs. As a byproduct, we get a new proof for a theorem of Erdős on circuits in graphs. 1
Compressive Sensing for Sparse Approximations: Constructions, Algorithms, and Analysis
, 2010
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New examples of noncommutative Λ(p) sets
 Illinois J. Math
"... In this paper, we introduce a certain combinatorial property Z?(k), which is defined for every integer k ≥ 2, and show that every set E ⊂ Z with the property Z?(k) is necessarily a noncommutative Λ(2k) set. In particular, using number theoretic results about the number of solutions to socalled “Su ..."
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Cited by 2 (0 self)
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In this paper, we introduce a certain combinatorial property Z?(k), which is defined for every integer k ≥ 2, and show that every set E ⊂ Z with the property Z?(k) is necessarily a noncommutative Λ(2k) set. In particular, using number theoretic results about the number of solutions to socalled “Sunit equations, ” we show that for any finite set Q of prime numbers, EQ is noncommutative Λ(p) for every real number 2 < p < ∞, where EQ is the set of natural numbers whose prime divisors all lie in the set Q. 1 1