Objective
This INTAS network brings together expert researchers in Western Europe and in the New Independent States. The backbone of the project is the development of the theory of rational approximation, in particular simultaneous rational approximation to a system of functions (Hermite-Pade approximation). Important tools for rational approximation are Riemann-Hilbert problems, the theory of orthogonal polynomials, logarithmic potential theory, and operator theory for difference operators. Most European experts for these tools are members of the various teams in the network. One of the main objectives is to work out applications of rational approximation in four areas: 1) Inverse problems in rational approximation. The inverse problem can be stated as follows. Given the rational approximants to a function f, how can one obtain useful information of the function f. The most relevant information is the location of the singularities of the function f. The inverse problem amounts to the question: how can the behaviour of the poles of the rational approximants be translated into information about the singularities of the function f. 2) Spectral theory of difference operators. One of the objectives is to find a close connection between polynomials satisfying a higher order recurrence relation, multiple orthogonal polynomials, and common denominators of Hermite-Pade approximation. We expect to obtain necessary and/or sufficient conditions on the coefficients in the recurrence relations implying multiple orthogonality on the real line, to set up perturbation theory of certain standard recurrence relations (constant coefficients), and to work out the scattering theory for such operators using the asymptotic behaviour of the polynomials and the corresponding Riemann-Hilbert problem. 3) Non-linear dynamical systems. Certain discrete integrable dynamical systems (Toda lattice) can be solved using tridiagonal operators. Our objective is to study discrete dynamical systems corresponding to banded Hessenberg operators, appearing in the spectral theory of multiple orthogonal polynomials. We expect to find an explicit method for obtaining the solution of these dynamical systems, and we hope to find useful results for the continuous dynamical systems obtained as continuum limits of the discrete systems. 4) Number theory. Our objective is to use Hermite-Pade approximation to prove properties of other remarkable constants and we hope to be able to prove irrationality and transcendence of certain constants that are not yet known to be irrational, such as Catalan's constant and Euler's constant. We expect to find explicit rational numbers, expressed in terms of the special function found higher, approximating these constants.
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B-3001 Leuven
Belgium