CORFU research project

CORFU: Constructive study of Orthogonal Functions (01-01-2002 - 31-12-2005)

Two fields of applications

In this project, we want to concentrate on two points: first the constructive aspects of orthogonal rational functions with prescribed poles and their applications and on the other hand also on the constructive aspects of multi-orthogonality with applications in associated dynamical systems (Toda grid and higher order analogs). Orthogonal rational functions (ORF) generalize the theory of polynomials orthogonal with respect to a positive measure on the real line or the complex unit circle. Multi-orthogonality is a property of the common denominators of Hermite-Padé approximation of a vector of functions (simultaneous rational approximation) and goes back to Hermite who studied this kind of approximation for his proof that e is a rational number. During the last decade there has been an important stimulus from Eastern Europe to study Hemite-Padé approximations and multi-orthogonal polynomials from an analytical (and not an algebraic) point of view. Riemann surfaces, operator theory and non-symmetric operators and vector potential theory turn out to be important tools.

One of the applications of ORF is numerical quadrature. It concerns in particular quadrature based on orthogonal rational functions instead of the classical Gauss or Szegö quadrature, which is based on orthogonal polynomials. Of course, this is important for the integration of functions with singularities. Quadrature for a rational integrand has already been studied by Van Assche-Vanherwegen and Gautschi. There remain however many open problems: there is of course the problem of an optimal choice of the poles to be chosen, but also quadrature for a finite or half infinite interval requires an adaptation of the theory of orthogonal rational functions, practical computational techniques for nodes and weights, an appropriate convergence analysis, etc. Also simultaneous quadrature has to be studied. Here the multi-orthogonal polynomials show up and the classical Gauss quadrature has to be adapted to the simultaneous situation. Borges has given a first impulse for this research, with as an application the integration of a light beam over three color filters.

A second field of applications of orthogonal rational functions is to be found in linear systems theory and in some nonlinear systems. Classically, one tries to construct a rational transfer function of a linear system as the ratio of two polynomials that are chosen in an optimal way with respect to some criterion. Very recently one started to construct a rational function as a linear combination of orthogonal rational basis functions. Hereby orthogonality was considered with respect to the Lebesgue measure (weight equal to 1) and only one real or a complex conjugate pair of repeated poles is used. The general theory of ORF however, can deal with an arbitrary distribution of poles and an arbitrary weight function. For nonlinear systems, our attention goes especially to The Toda grid and related systems (relativistic Toda grid, Bogoyavlenskii grid). Moser showed in 1975 that these non-linear systems are solvable via an isospectral deformations, in particular via the solution of a direct and an inverse problem for orthogonal polynomials on the real line. The generalizations that we have in mind are for orthogonal Laurent polynomials (relativistic Toda grid) and multi-orthogonal polynomials (Bogoyavlenskii grid).

The project:

It is the objective of the project to elaborate further in a constructive way on the applications mentioned: numerical quadrature and linear and non-linear systems, both from the viewpoint of orthogonal rational functions and multi-orthogonality. At first sight, these applications seem to be quite different, but, although they have each their own specific problems, the fundamental properties of the orthogonal functions involved that have to be investigated are the same: an optimal choice of the poles, properties of the zeros, convergence theory. Moreover, orthogonal rational functions and multi-orthogonal polynomials have in common that they are both related to orthogonal polynomials with respect to varying weights: for orthogonal rational functions, the varying weight contains the information of the poles, for multi-orthogonal functions, the varying weights on one interval contain the information about the zeros of the multi-orthogonal polynomials on the other intervals.

Numerical quadrature:

Many of the constructive aspects of this theory are yet to be discovered. The basic idea for orthogonal rational functions is that by choosing a set of poles, a set of rational functions is fixed. Then the integrand is interpolated for certain set of interpolation points by a function from the set of rational functions. The integral of this interpolant is an interpolating quadrature formula. If the interpolation points are chosen as the zeros of an orthogonal basis function (or of a related function to ensure that the zeros are in the interval of integration) then a larger class of rational functions will be integrated exactly. This is the same principle as for Gauss quadrature formulas. In this theory, the properties adapted to finite intervals or the half line are still lacking. The following questions need to find an answer in this project:

• What is the right way to compute orthogonal (and quasi-orthogonal) functions in the case of real poles or complex conjugate poles? Here it is important to derive theoretical properties concerning the zeros of these functions. Research on this topic is just starting.
• How can the zeros of these functions be computed numerically? In the polynomial case, this is an eigenvalue problem for a tridiagonal matrix (Jacobi matrix). This can be generalized to a generalized or a quadratic eigenvalue problem, but an appropriate stable and efficient way for its numerical solution is not yet known. This is related to an operator theoretical approach to orthogonal rational functions. Concerning this matter, there are intensive contacts with E. Hendriksen (Univ. Amsterdam).
• What conditions are needed for the poles that are chosen to get convergence for the largest possible set of functions. This is related to potential theoretical properties and convergence in capacity. For this expertise is available in the participating teams and there are contacts between the two promoters and the University of La Laguna (P. Gonzalez-Vera and R. Orive).
• Are the proposed methods competitive with existing techniques of numerical quadrature based on rational functions as they are proposed by Gautschi?
• Are these methods direct generalizations of techniques proposed for integrals over a half line? In this respect, there are only results related to the situation where only poles at 0 and infinity are allowed so that one has very special cases of orthogonal rational functions, namely Laurent polynomials. These results are by Jones-Njåstad-Thron. The two promoters have close contacts with these authors. See also the work by G. Min for an interval.

For simultaneous quadrature we want to compute several integrals of the same function simultaneously. This situation shows up for example when a light signal f is filtered with three color filters (RGB: a red filter, a green filter and a blue filter). For simultaneous quadrature one uses zeros of a multi-orthogonal polynomial. This kind of quadrature is already described by Borges but it is obvious that the author was not aware of the already existing theory of multi-orthogonal polynomials. In the project the following aspects will be considered:

• The eigenvalue problem is in this case a band Hessenberg matrix. Stable and efficient algorithms for this type of matrices have to be designed.
• The computation of the weights of the quadrature formula for tridiagonal matrices is done via the first component of the normalized eigenvectors. For band Hessenberg matrices it is to be found out how the weights are related to the first components of the eigenvectors.
• Band Hessenberg matrices are not symmetric, so that it is not directly clear that all the eigenvalues are real. The matrices turn out to have supplementary properties which imply that certain eigenvalues are real. A promising class of matrices that has to be taken into account is the class of oscillation matrices and totally non-negative matrices. For the project we shall investigate this type of matrices, their properties and their relation with multi-orthogonality.

Linear and non-linear systems:

For the applications in linear systems, the idea is to find for a given input u and output y some transfer function H that describes the system ``in the best possible way''. There are numerous techniques to solve this problem. Usually this problem is solved in a linearized form after which an iterative technique is used to improve this estimate. For an approximation in the least squares sense, sometimes use is made of orthogonal polynomials to solve the linear problem. For a given set of poles, one may however also write H as a linear combination of orthogonal rational functions and solve the problem is this form. Because the date are usually in digital form, orthogonality with respect to summation is appropriate. Some promising experiments have been done in this respect. About this subject there are contacts with B. De Moor and S. Van Huffel (ESAT/K.U.Leuven), with P. Van Dooren and Y. Genin (UCL), and J. Schoukens and R. Pintelon (VUB). Also here there remain a number of questions that we want to answer in this project:

• What are the most efficient ways to compute orthogonal rational functions with respect to a discrete inner product? It is important to find out how good a discrete approximation will approximate a continuous approximant. We have to derive convergence properties: on one hand there is the duality between discrete (sampled) and continuous systems, which corresponds to the duality between orthogonality on the unit circle or orthogonality on the real line, but on the other hand, there is the approximation of an inner product on the unit circle with a continuous measure by s discrete measure (viz.\ orthogonality with respect to summation).
• The previous problem is linear and can be solved in a fast and recursive way, and could for example be used in an interactive program to find the poles. But how can an automatic algorithm be designed to make an optimal choice of the poles? We can try a classical method, but these are usually based on the analysis of the spectrum of the system, which is exactly what we want to approximate here. Another way could be to consider the non-linear optimization problem where the poles (denominator) are considered to be the only non-linear parameters. This non-linear least squares problem is however very ill conditioned and does not determine the poles very well. Therefore certain regularization techniques have to be used to obtain acceptable results.

For applications in non-linear systems, our attention goes in the first place to the Toda grid and related systems like the relativistic Toda grid and the Bogoyavlenskii grid. It is known that for the Toda grid the solution can be found via orthogonal polynomials: the initial conditions define a sequence of recursion coefficients and a Jacobi matrix for orthogonal polynomials. If we multiply the orthogonality measure by the exponential e^xt, we get new polynomials and recursion coefficients (Jacobi matrix) that depend on the time t and that are solutions of the non linear differential equations of the Toda grid. For the project we will concentrate on the following generalizations:

• The relativistic Toda grid is not related with a Jacobi matrix, but with orthogonal Laurent polynomials. There is again the possibility to do an isospectral deformation. A concrete elaboration of this solution method is one of the research aspects of this project.
• The Bogoyavlenskii grid is a higher order analogue of the Toda grid. The corresponding matrix is a band Hessenberg matrix with only two diagonals that contain non zero elements. Again the possibility exists to do an isospectral deformation but now of the orthogonality measures for the corresponding multi-orthogonal polynomials. This will be worked out in practice.
• Continuum limits of the relativistic Toda grids and the Bogoyavlenskii grid will be studied with methods like described by Deift and McLaughlin.