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.

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...

Abstract. This paper deals with general structural properties of one-dimensional 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 Denisov-Rakhmanov 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. 1.

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...

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.

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. Remez-type 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 Remez-type inequalities for polynomials, generalized nonnegative polynomials, exponentials of logarithmic potentials and Miintz polynomials. Remez-type inequali-ties play a central role in proving other important inequalities for the above classes. The paper illustrates the power of Remez-type inequalities by giving a number of applications.

We consider circulant preconditioners for Hermitian Toeplitz systems from the view point of function theory. We show that some well-known circulant preconditioners can be derived from convoluting the generating function f of the Toeplitz matrix with famous kernels like the Dirichlet and the Fej'er kernels. Several circulant preconditioners are then constructed using this approach. Finally, we prove that if the convolution product converges to f uniformly, then the circulant preconditioned Toeplitz systems will have clustered spectrum.

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The integration of semidiscrete approximations for time-dependent 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 ill-conditioned 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 well-posed 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 convection-diffusion equations with variable coefficients. A straightforward implementation of our results verify the strong stability of RK methods for local finite-difference schemes as well as global spectral approximations. Since our approach is based on the energy method (which is carried in the physical space), and

this paper we look at algorithms for generating random variates with a density f on the real line in just these situations. The generation problem when only the characteristic function can exactly be computed is dealt with in a series of papers by the present author [7,]0,11,12]. When a sequence of moments is known, i.e., for each n, the nth moment is available at unit computational cost, a solution is developed in [13] for integer-valued random variables. Unfortunately, this method cannot be generalized to the continuous case. So, we consider densities that are specified either through the sequence of Fourier coefficients or through the sequence of moments. It is known that under some conditions the density can be written as an infinite series involving these Fourier coefficients or moments, for example, by using Fourier series or smoothed Fourier series in the former case and Hermite (Gram-Charlier, Edgeworth), Laguerre, Jacobi or Legendre series in the latter case. Unfortunately, the evaluation of these series takes an infinite amount of computational effort when we assume that each computation of a Fourier coefficient or moment takes one unit of time. Hence, we cannot compute the density at all in the algorithms. Instead, we resort to algorithms that compute a random but finite number of coefficients or moments per generated random variate. The fact that we can do this and still insure theoretical exactness shows once again that random variate generation is 'easier' than computing quantities related to a distribution. Most methods discussed here boil down to the acceptance-rejection method combined with the series method developed in [8,11, pp. 151-171]. There is one interesting exception related to the Fourier coefficient problem. It is known that any Fourier cosin...