Orthogonal functions

Researchers

and formerly ...

Description

Orthogonal rational functions are studied on the complex unit circle and the real line. They generalize orthogonal polynomials.

Two cases are considered: either the poles of these rational functions are off the support of the measure (outside the open unit disk or in the lower half plane) or the poles are exactly on the boundary (i.e. on the circle or the real line).

Many topics from classical polynomials are generalized: recurrence relations, reproducing kernels, interpolation properties, quadrature formula, moment problems, Favard theorems, convergence behaviour, etc.

Among the applications one can find Pisarenko frequency modeling, inverse scattering, network synthesis, H-infinity control, prediction of time series and stochastic processes, rational wavelets, ...

Current research topics are quadrature formulas and application in identification.

Orthogonal rational functions

Much of this work was done in the framework of a HCM project ROLLS, in a collaboration between

K.U.Leuven (A. Bultheel),
U. La Laguna, Tenerife (P. González-Vera)
U. van Amsterdam (E. Hendriksen)
NTNU Trondheim (O. Njåstad).

The main result was the compilation of a monograph on the topic.

A vizualization applet is available on the net. It was prepared by K. Müller from the U. Leipzig in the framework of the ROLLS project.

The collaboration existed before and continues after the project.

More recent collaboration is with the mathematical department of the Universidad Carlos III de Madrid (F. Marcellan).

More general orthogonal systems

Previous research is related to a project Orthogonal Systems and applications that is set up in collaboration with the department of mathematics ( research group of applied mathematics) at the K.U.Leuven ( W. Van Assche) and the department of mathematics and computer science ( research group Computer Arithmetic and Numerical Techniques) at the University of Antwerp (A. Cuyt).

The current research project is related to multiorthogonality and orthogonal rational functions. See the project CORFU which is in collaboration with the department of mathematics ( research group of applied mathematics) at the K.U.Leuven ( W. Van Assche).

The purpose is to exploit all the properties of orthogonal basis functions, formal or informal, in pure and numerical analysis and many applications.

They do show up in the theory of classical and less classical orthogonal polynomials and rational functions, both on the unit crircle and the real line, wavelet analysis, recurrence relations, iterative methods in linear algebra, continued fractions, and many other places.