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Minimax Estimation via Wavelet Shrinkage
, 1992
"... We attempt to recover an unknown function from noisy, sampled data. Using orthonormal bases of compactly supported wavelets we develop a nonlinear method which works in the wavelet domain by simple nonlinear shrinkage of the empirical wavelet coe cients. The shrinkage can be tuned to be nearly minim ..."
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Cited by 246 (32 self)
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We attempt to recover an unknown function from noisy, sampled data. Using orthonormal bases of compactly supported wavelets we develop a nonlinear method which works in the wavelet domain by simple nonlinear shrinkage of the empirical wavelet coe cients. The shrinkage can be tuned to be nearly minimax over any member of a wide range of Triebel and Besovtype smoothness constraints, and asymptotically minimax over Besov bodies with p q. Linear estimates cannot achieve even the minimax rates over Triebel and Besov classes with p <2, so our method can signi cantly outperform every linear method (kernel, smoothing spline, sieve,:::) in a minimax sense. Variants of our method based on simple threshold nonlinearities are nearly minimax. Our method possesses the interpretation of spatial adaptivity: it reconstructs using a kernel which mayvary in shape and bandwidth from point to point, depending on the data. Least favorable distributions for certain of the Triebel and Besov scales generate objects with sparse wavelet transforms. Many real objects have similarly sparse transforms, which suggests that these minimax results are relevant for practical problems. Sequels to this paper discuss practical implementation, spatial adaptation properties and applications to inverse problems.
Unconditional bases are optimal bases for data compression and for statistical estimation
 Applied and Computational Harmonic Analysis
, 1993
"... An orthogonal basis of L 2 which is also an unconditional basis of a functional space F is a kind of optimal basis for compressing, estimating, and recovering functions in F. Simple thresholding operations, applied in the unconditional basis, work essentially better for compressing, estimating, and ..."
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Cited by 142 (23 self)
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An orthogonal basis of L 2 which is also an unconditional basis of a functional space F is a kind of optimal basis for compressing, estimating, and recovering functions in F. Simple thresholding operations, applied in the unconditional basis, work essentially better for compressing, estimating, and recovering than they do in any other orthogonal basis. In fact, simple thresholding in an unconditional basis works essentially better for recovery and estimation than other methods, period. (Performance is measured in an asymptotic minimax sense.) As an application, we formalize and prove Mallat's Heuristic, which says that wavelet bases are optimal for representing functions containing singularities, when there may be an arbitrary number of singularities, arbitrarily distributed.
On the Design of Maximally Sparse Beamforming Arrays
 IEEE Trans. on Antennas and Prop
, 1991
"... AbstractThe problem of designing nonuniformly spaced arrays is formulated as one of constrained optimization in which the cost function is chosen to select the array with the minimum number of elements. The response of the array is controlled by a set of inequality point response constraints. It is ..."
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Cited by 18 (1 self)
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AbstractThe problem of designing nonuniformly spaced arrays is formulated as one of constrained optimization in which the cost function is chosen to select the array with the minimum number of elements. The response of the array is controlled by a set of inequality point response constraints. It is shown that a suitable cost function for this problem is the I,, quasinorm for 0 < p < 1 and that there exists a number pl such that for all 0 < p < p1 the resulting array is maximally sparse. Furthermore it is shown that a solution to the problem lies at an extreme point of the simplex formed by the point constraints. A simplex search algorithm is described which will converge to a local minimum of the cost function on this simplex. The algorithm is illustrated in application to the design of sparse linear and planar arrays. I.
Noncommutative approximation: inverseclosed subalgebras and offdiagonal decay of matrices
 Constr. Approx
"... Abstract. We investigate two systematic constructions of inverseclosed subalgebras of a given Banach algebra or operator algebra A, both of which are inspired by classical principles of approximation theory. The first construction requires a closed derivation or a commutative automorphism group on ..."
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Cited by 7 (0 self)
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Abstract. We investigate two systematic constructions of inverseclosed subalgebras of a given Banach algebra or operator algebra A, both of which are inspired by classical principles of approximation theory. The first construction requires a closed derivation or a commutative automorphism group on A and yields a family of smooth inverseclosed subalgebras of A that resemble the usual HölderZygmund spaces. The second construction starts with a graded sequence of subspaces of A and yields a class of inverseclosed subalgebras that resemble the classical approximation spaces. We prove a theorem of JacksonBernstein type to show that in certain cases both constructions are equivalent. These results about abstract Banach algebras are applied to algebras of infinite matrices with offdiagonal decay. In particular, we obtain new and unexpected conditions of offdiagonal decay that are preserved under matrix inversion. 1.
Interpolation on sparse grids and Nikol'skijBesov spaces of dominating mixed smoothness
, 1998
"... We investigate the order of convergence of periodic interpolation on sparse grids (blending interpolation) in the framework of Nikol'skijBesov spaces of dominating mixed smoothness. The main ingredients are a unified approach to error estimates in univariate Nikol'skijBesov spaces and th ..."
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Cited by 4 (1 self)
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We investigate the order of convergence of periodic interpolation on sparse grids (blending interpolation) in the framework of Nikol'skijBesov spaces of dominating mixed smoothness. The main ingredients are a unified approach to error estimates in univariate Nikol'skijBesov spaces and the tensor product characterization of the bivariate Nikol'skijBesov spaces of dominating mixed smoothness.
Some Error Estimates for Periodic Interpolation of Functions from Besov Spaces
, 1998
"... Using periodic StrangFix conditions, we can give an approach to error estimates for periodic interpolation on equidistant and sparse grids for functions from certain Besov spaces. 1 Introduction We investigate the L 2 error of interpolation on equidistant and sparse grids for periodic functions ..."
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Cited by 1 (0 self)
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Using periodic StrangFix conditions, we can give an approach to error estimates for periodic interpolation on equidistant and sparse grids for functions from certain Besov spaces. 1 Introduction We investigate the L 2 error of interpolation on equidistant and sparse grids for periodic functions from isotropic L 2 Besov spaces and L 2 Besov spaces of functions with dominating mixed smoothness properties. The interpolation of periodic functions by translates of a given function and the corresponding error estimates have been analyzed by several authors (e.g. [3, 8, 14]) in the univariate as well as in the multivariate case. The periodic StrangFix conditions were introduced in [2, 14]. There, they were used to find L 2 error estimates for functions from isotropic L 2 Sobolev spaces. The approximation of functions on sparse grids and the related field of hyperbolic approximation have a fairly long tradition (e.g. [4, 5, 27]) as well. For bivariate functions, the number of i...
We study the Cauchy singular integral operator SwI on (−1, 1), where w  is
"... Dedicated to Professor P. Junghanns on the occasion of his 50th birthday ..."