PDM research project

ORF04: Orthogonal rational functions: analysis and applications (1-10-2004 to 30-09-2005)

In linear system theory systems are usually modelled using rational transfer functions. In some way or another, the abovementioned problems all appear here. However, only recently there has been a growing interest in regarding these rational functions no longer as a ratio of two polynomials, but instead as a linear combination of orthogonal rational functions. These are obtained orthogonalizing a base of rational functions of increasing degree with a set of predetermined poles. If the poles are properly chosen, using an orthogonal rational basis will lead to better approximations and faster convergence. Of course, this is not only true for problems in linear system theory, but is generally valid for rational approximation problems.

From a mathematical point of view, orthogonal rational functions form an extension of orthogonal polynomials. Many properties and problems of orthogonal polynomials can therefore be generalized to the rational case, like moment problems, quadrature formulas, asymptotic behaviour, etc. The aim of this project is to study several of these aspects from a numerical point of view, keeping in mind the efficient computation and applications in system theory.

One of the first objectives is to study the asymptotic behaviour of orthogonal rational functions on the complex unit circle and the real interval on the one hand, and on a halfline or line on the other hand. The convergence of orthogonal polynomials as the degree tends to infinity, has become a very broad field with diverse results. The rational case is often analogous, but in most cases the presence of poles complicates the theory and yields more interesting theorems. Several results are known about ratio asymptotics, strong and weak asymptotics and zero asymptotics, but many questions are still unanswered. Most results were obtained for poles not converging to the support of the measure and especially the case of a halfline or line has not been studied well. Also convergence on the support of the measure has not been studied yet. Many of these asymptotic results are useful in estimating the errors and precision in numerical applications, because of the relatively simple limit functions, as opposed to the more complex computation of the orthogonal functions themselves.

Moment problems are the next topic we wish to study. In classical moment problems one tries to find a positive measure, given the moments of this measure. Necessary and sufficient conditions for existence and uniqueness are studied. For the classical (weak) moment problems from the polynomial case, such as Hamburger, Hausdorff or Stieltjes moment problems on (a subset of) the real line and trigonometric moment problems on the unit circle, the moments are given in 0. In strong moment problems one considers two points (0 and infinity) and orthogonal Laurent polynomials are used. A natural generalization to the rational case are multipoint moment problems. Extended Hamburger moment problems (on the real line) have been studied in the literature, but for the case of extended Stieltjes (halfline) and especially Hausdorff (interval) moment problems, a lot of work has to be done and the numerical aspects have hardly been studied.

Quadrature formulas of gaussian type are closely related to moment problems. Given the number of nodes, these formulas will integrate exactly a maximal space of polynomials (rational functions). The connection with moment problems is that the discrete measure defined by the nodes and weights of the quadrature formula is an approximation to the measure solving the moment problem. Convergence of these quadrature formulas is therefore closely related to the determinacy of the moment problems. In the polynomial case these quadrature formulas have been extensively studied, as well from a theoretical point of view as from a numerical one. However, for some integrals the integrand may not be suited to be approximated by a polynomial, e.g. because of the presence of poles. In this case quadrature formulas based on orthogonal rational functions will be more interesting. Quadrature formulas with poles close to the poles of the integrand will of course yield better results than polynomial based quadrature formulas. For different supports of the measure (circle, line, halfline and interval) the theoretical foundations for these formulas have been set up and in some cases convergence results have been obtained, but a numerical approach to the problem has hardly been taken. Efficiently constructing and computing these formulas is a weak point in the competitivity with existing quadrature formulas. The known algorithms to compute nodes and weights require a lot of computational effort, which nullifies the gain in accuracy completely. One of the main objectives in this project is to look for quadrature formulas of this type which can easily be constructed for an arbitrary degree. The connection with orthogonal Laurent polynomials (which can be viewed as a special case of orthogonal rational functions) may prove to be interesting here. Next we also would like to study how the location of the poles influences the precision and speed of convergence, for certain classes of integrands (continuous functions, analytic functions, ...).

A final objective of this project is to apply the foregoing in system theory. Representing the transfer function in linear system theory as a linear combination of (orthogonal) rational basis functions has only recently gained interest. Often basis functions are used which are orthogonal with respect to the Lebesgue measure. This has the advantage that explicit expressions for the basis functions are known, but the measure may not always be the most appropriate one, such that this can lead to illconditioning of the systems of equations to be solved. By using a basis which better fits the needs of the problem, these numerical problems can be controlled more, but the analysis and the study of convergence become much more complicated. We wish to study the numerical aspects (optimal location of the poles, condition analysis of the problem, efficient and accurate computation of the basis) as well as the theoretical aspects (stability of the system, convergence, error estimates). If time allows, also time varying systems and systems with multiple input and output (MIMO) can be studied.