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58
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...
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 140 (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.
The absolutely continuous spectrum of onedimensional Schrödinger operators with decaying potentials
, 2008
"... This paper deals with general structural properties of onedimensional Schrödinger operators with some absolutely continuous spectrum. The basic result says that the ω limit points of the potential under the shift map are reflectionless on the support of the absolutely continuous part of the spectr ..."
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Cited by 53 (7 self)
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This paper deals with general structural properties of onedimensional Schrödinger operators with some absolutely continuous spectrum. The basic result says that the ω limit points of the potential under the shift map are reflectionless on the support of the absolutely continuous part of the spectral measure. This implies an Oracle Theorem for such potentials and DenisovRakhmanov type theorems. In the discrete case, for Jacobi operators, these issues were discussed in my recent paper [19]. The treatment of the continuous case in the present paper depends on the same basic ideas.
Localized tight frames on spheres
 SIAM J. Math. Anal
, 2006
"... Abstract. In this paper we wish to present a new class of tight frames on the sphere. These frames have excellent pointwise localization and approximation properties. These properties are based on pointwise localization of kernels arising in the spectral calculus for certain selfadjoint operators, ..."
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Cited by 36 (7 self)
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Abstract. In this paper we wish to present a new class of tight frames on the sphere. These frames have excellent pointwise localization and approximation properties. These properties are based on pointwise localization of kernels arising in the spectral calculus for certain selfadjoint operators, and on a positiveweight quadrature formula for the sphere that the authors have recently developed. Improved bounds on the weights in this formula are another byproduct of our analysis.
Mutual Information Approach to Blind Separation of Stationary Sources
 IEEE Transactions on Information Theory
, 1999
"... This paper presents an unified approach to the problem of separation of sources, based on the consideration of mutual information. The basic setup is that the sources are independent stationary random processes which are mixed either instantaneously or through a convolution, to produce the observed ..."
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Cited by 34 (8 self)
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This paper presents an unified approach to the problem of separation of sources, based on the consideration of mutual information. The basic setup is that the sources are independent stationary random processes which are mixed either instantaneously or through a convolution, to produce the observed records. We define the entropy of stationary processes and then the mutual information between them as a measure of their independence. This provides us with a contrast for the separation of source problem. For practical implementation, we introduce several degraded forms of this contrast, which can be computed from a finite dimensional distribution of the reconstructed source processes only. From them, we derive several sets of estimating equations generalising those considered earlier. 1 Introduction Blind separation of sources is a topic which have received much attention recently, as it has many important applications (speech analysis, radar, sonar, : : : ). Basically, one observes seve...
On a high order numerical method for functions with singularities
 Math. Comp
, 1998
"... Abstract. By splitting a given singular function into a relatively smooth part and a specially structured singular part, it is shown how the traditional Fourier method can be modified to give numerical methods of high order for calculating derivatives and integrals. Singular functions with various t ..."
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Cited by 24 (0 self)
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Abstract. By splitting a given singular function into a relatively smooth part and a specially structured singular part, it is shown how the traditional Fourier method can be modified to give numerical methods of high order for calculating derivatives and integrals. Singular functions with various types of singularities of importance in applications are considered. Relations between the discrete and the continuous Fourier series for the singular functions are established. Of particular interest are piecewise smooth functions, for which various important applications are indicated, and for which numerous numerical results are presented. 1.
R.: Halfline Schrödinger operators with no bound states
"... Abstract. We consider Schödinger operators on the halfline, both discrete and continuous, and show that the absence of bound states implies the absence of embedded singular spectrum. More precisely, in the discrete case we prove that if ∆+V has no spectrum outside of the interval [−2,2], then it ha ..."
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Cited by 19 (7 self)
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Abstract. We consider Schödinger operators on the halfline, both discrete and continuous, and show that the absence of bound states implies the absence of embedded singular spectrum. More precisely, in the discrete case we prove that if ∆+V has no spectrum outside of the interval [−2,2], then it has purely absolutely continuous spectrum. In the continuum case we show that if both −∆+V and −∆−V have no spectrum outside [0, ∞), then both operators are purely absolutely continuous. These results extend to operators with finitely many bound states. 1.
Remeztype inequalities and their applications
, 1993
"... The Remez inequality gives a sharp uniform bound on [1, 1] for real algebraic polynomials p of degree at most n if the Lebesgue measure of the subset of [1, 1], where I pl is at most 1, is known. Remeztype inequalities give bounds for classes of functions on a line segment, on a curve or on a reg ..."
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Cited by 13 (12 self)
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The Remez inequality gives a sharp uniform bound on [1, 1] for real algebraic polynomials p of degree at most n if the Lebesgue measure of the subset of [1, 1], where I pl is at most 1, is known. Remeztype inequalities give bounds for classes of functions on a line segment, on a curve or on a region of the complex plane, given that the modulus of the functions is bounded by 1 on some subset of prescribed measure. This paper offers a survey of the extensive recent research on Remeztype inequalities for polynomials, generalized nonnegative polynomials, exponentials of logarithmic potentials and Miintz polynomials. Remeztype inequalities play a central role in proving other important inequalities for the above classes. The paper illustrates the power of Remeztype inequalities by giving a number of applications.
From the semidiscrete to fully discrete: Stability of RungeKutta schemes by the energy method
 SIAM REV
, 1998
"... The integration of semidiscrete approximations for timedependent problems is encountered in a variety of applications. The Runge–Kutta (RK) methods are widely used to integrate the ODE systems which arise in this context, resulting in large ODE systems called methods of lines. These methods of line ..."
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Cited by 11 (4 self)
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The integration of semidiscrete approximations for timedependent problems is encountered in a variety of applications. The Runge–Kutta (RK) methods are widely used to integrate the ODE systems which arise in this context, resulting in large ODE systems called methods of lines. These methods of lines are governed by possibly illconditioned systems with a growing dimension; consequently, the naive spectral stability analysis based on scalar eigenvalues arguments may be misleading. Instead, we present here a stability analysis of RK methods for wellposed semidiscrete approximations, based on a general energy method. We review the stability question for such RK approximations, and highlight its intricate dependence on the growing dimension of the problem. In particular, we prove the strong stability of general fully discrete RK methods governed by coercive approximations. We conclude with two nontrivial examples which demonstrate the versatility of our approach in the context of general systems of convectiondiffusion equations with variable coefficients. A straightforward implementation of our results verify the strong stability of RK methods for local finitedifference schemes as well as global spectral approximations. Since our approach is based on the energy method (which is carried in the physical space), and
The Periodic Schrödinger Operators with Potentials in the C. FeffermanPhong Class
, 1999
"... We consider the periodic Schrödinger operator − ∆ + V (x) in R d, d ≥ 3 with potential V in the C. FeffermanPhong class. Let Ω be a periodic cell for V. We show that, for p ∈ ((d − 1)/2, d/2], there exists a positive constant ε depending only on the shape of Ω, p and d such that, if lim sup r→0 su ..."
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Cited by 10 (1 self)
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We consider the periodic Schrödinger operator − ∆ + V (x) in R d, d ≥ 3 with potential V in the C. FeffermanPhong class. Let Ω be a periodic cell for V. We show that, for p ∈ ((d − 1)/2, d/2], there exists a positive constant ε depending only on the shape of Ω, p and d such that, if lim sup r→0 sup r x∈Ω 2