Adhemar Bultheel

Orthogonal rational functions

List of symbols Introduction 1. Preliminaries 2. The fundamental spaces 3. The kernel functions 4. Recurrence and second kind functions 5. Para-orthogonality and quadrature 6. Interpolation 7. Density of the rational functions 8. Favard theorems 9. Convergence 10. Moment problems 11. The boundary case 12. Some applications Conclusion Bibliography Index.

Pade´ techniques for model reduction in linear system theory: a survey

Abstract Techniques of Pade approximation and continued fractions have been used often in model reduction problems. An extensive bibliography on this topic is given. The ideas are explained for the simple situation of a scalar function where no singular blocks appear in the Pade table. Extensions to the matrix case and the multivariable case are not explained in detail.

Empirical Bayes approach to improve wavelet thresholding for image noise reduction

Wavelet threshold algorithms replace small magnitude wavelet coefficients with zero and keep or shrink the other coefficients. This is basically a local procedure, because wavelet coefficients characterize the local regularity of a function. Although a wavelet transform has decorrelating properties, structures in images, like edges, are never decorrelated completely, and these structures appear in the wavelet coefficients: a classification based on a local criterion-like coefficient magnitude is not the perfect method to distinguish important, uncorrupted coefficients from coefficients dominated by noise. We therefore introduce a geometrical prior model for configurations of important wavelet coefficients and combine this with local characterization of a classical threshold procedure into a Bayesian framework. The local characterization is incorporated into the conditional model, whereas the prior model describes only configurations, not coefficient values. More precisely, local characterization favors configurations with clusters of important coefficients. In this way, we can compute, for each coefficient, the posterior probability of being “sufficiently clean.” This article proposes and motivates the particular and original choice of the conditional model. Instead of introducing this Bayesian framework, we could also apply heuristic image processing techniques to find clustered configurations of large coefficients. This article also explains the benefits of the Bayesian approach compared to these simple techniques. The parameter of the prior model is estimated on an empirical basis using a pseudolikelihood criterion.

A general module theoretic framework for vector M-Padé and matrix rational interpolation

A general module theoretic framework is used to solve several classical interpolation problems and generalizations thereof in a unified way. These problems are divided into two main families. The first family contains the classical linearized Padé, Padé-Hermite and M-Padé problems and the generalization to the vector M-Padé problem. The second family consists of the Padé problem, the scalar, vector and matrix rational interpolation problems. The solution method is straightforward, recursive and efficient. It can follow any “path” in the “solution table” even if this “solution table” is nonnormal (nonperfect). Reordering of the interpolation data is not required.

Stabilised wavelet transforms for non-equispaced data smoothing

This paper discusses wavelet thresholding in smoothing from non-equispaced, noisy data in one dimension. To deal with the irregularity of the grid we use the so-called second generation wavelets, based on the lifting scheme. The lifting scheme itself leads to a grid-adaptive wavelet transform. We explain that a good numerical condition is an absolute requisite for successful thresholding. If this condition is not satisfied the output signal can show an arbitrary bias. We examine the nature and origin of stability problems in second generation wavelet transforms. The investigation concentrates on lifting with interpolating prediction, but the conclusions are extendible. The stability problem is a cumulated effect of the three successive steps in a lifting scheme: split, predict and update. The paper proposes three ways to stabilise the second generation wavelet transform. The first is a change in update and reduces the influence of the previous steps. The second is a change in prediction and operates on the interval boundaries. The third is a change in splitting procedure and concentrates on the irregularity of the data points. Illustrations show that reconstruction from thresholded coefficients with this stabilised second generation wavelet transform leads to smooth and close fits.

Quadrature on the half-line and two-point Pade´ approximants to Stieltjes functions. Part I: algebraic aspects

Abstract Let f(z) be a Stieltjes function with asymptotic expansions L0 and L∞ at z = 0 and z = ∞, respectively. Let [ k n ] denote the two-point Pade approximant of type (m,n) which matches k of the coefficients of the series L0 and which takes its remaining interpolation conditions from L∞. We discuss in this paper the algebraic aspects of this problem. We shall emphasize the relation between quadrature formulas and two-point Pade approximants and derive expressions for the error. In a subsequent paper we shall consider the convergence aspects of these approximants. For example, the positivity of the error, which is obtained here, will result in the monotonic convergence of certain sequences of two-point Pade approximants, a property which is well known in the case of classical Pade approximants.

The computation of orthogonal rational functions and their interpolating properties

We shall consider nested spacesln,n = 0, 1, 21... of rational functions withn prescribed poles outside the unit disk of the complex plane. We study orthogonal basis functions of these spaces for a general positive measure on the unit circle. In the special case where all poles are placed at infinity,ln =∏n, the polynomials of degree at mostn. Thus the present paper is a study of orthogonal rational functions, which generalize the orthogonal Szegö polynomials. In this paper we shall concentrate on the functions of the second kind which are natural generalizations of the corresponding polynomials.

On computing rational Gauss-Chebyshev quadrature formulas

We provide an algorithm to compute the nodes and weights for Gauss-Chebyshev quadrature formulas integrating exactly in spaces of rational functions with arbitrary real poles outside [-1,1]. Contrary to existing rational quadrature formulas, the computational effort is very low, even for extremely high degrees, and under certain conditions on the poles it can be shown that the complexity is of order O(n). This method is based on the derivation of explicit expressions for Chebyshev orthogonal rational functions, which are (thus far) the only examples of explicitly known orthogonal rational functions on [-1,1] with arbitrary real poles outside this interval.

A connection between quadrature formulas on the unit circle and the interval [ - 1,1]

Abstract We establish a relation between Gauss quadrature formulas on the interval [−1,1] that approximate integrals of the form I σ (F)= ∫ −1 +1 F(x)σ(x) d x and Szegő quadrature formulas on the unit circle of the complex plane that approximate integrals of the form I ω (f)= ∫ − π π f( e i θ )ω(θ) d θ . The weight σ(x) is positive on [−1,1] while the weight ω(θ) is positive on [−π,π]. It is shown that if ω(θ)=σ( cos θ)| sin θ| , then there is an intimate relation between the Gauss and Szegő quadrature formulas. Moreover, as a side result we also obtain an easy derivation for relations between orthogonal polynomials with respect to σ(x) and orthogonal Szegő polynomials with respect to ω(θ). Inclusion of Gauss–Lobatto and Gauss–Radau formulas is natural.

Rational approximation in linear systems and control

Abstract In this paper we want to describe some examples of the active interaction that takes place at the border of rational approximation theory and linear system theory. These examples are mainly taken from the period 1950–1999 and are described only at a skindeep level in the simplest possible (scalar) case. We give comments on generalizations of these problems and how they opened up new ranges of research that after a while lived their own lives. We also describe some open problems and future work that will probably continue for some years after 2000.