Objectif
This INTAS project brings together the expert researchers in Western Europe and the school around A. A. Gonchar in the New Independent States. The backbone of the project is the development of the theory of rational approximation of analytic functions, in particular the extension to simultaneous rational approximation to a system of functions. 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 project.
One of the main objectives is to work out applications of rational approximation in four areas:
Spectral theory of difference operators. There is a close connection between polynomials satisfying a second order recurrence relation (difference equation), orthogonal polynomials, and denominators of Padé approximation. One of the objectives is to find a similar connection between polynomials satisfying a higher order recurrence relation, multiple orthogonal polynomials, and common denominators of Hermite-Padé approximation. This extension deals with non-symmetric finite order difference equations which have not been considered in the literature but which occur naturally in some situations. 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.
Non-linear dynamical systems. Certain discrete integrable dynamical systems (Toda lattice, Langmuir 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 transformations between discrete dynamical systems of higher order, 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 (as in the continuum limit of the Toda lattice).
Special functions. The multiple orthogonal polynomials arising from Hermite-Padé approximation for very specific systems of functions can be expressed in terms of special functions with various important structure properties. These special functions extend the classical orthogonal polynomials. The objective is to make a full characterisation of the classical multiple orthogonal polynomials, with the relevant properties involving the differential operators, the difference operator, and the q-difference operator, and to investigate their asymptotic behaviour using the Riemann-Hilbert approach. We expect to work out explicit formulas for all these classical polynomials and to list all the relevant quantities characterising these special functions.
Number theory. Historically Hermite-Padé approximation was introduced in the nineteenth century to prove the transcendence of the remarkable constant e (the basis of the natural logarithm). Our objective is to use Hermite-Padé 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.
Programme(s)
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Programmes de financement pluriannuels qui définissent les priorités de l’UE en matière de recherche et d’innovation.
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Les appels à propositions sont divisés en thèmes. Un thème définit un sujet ou un domaine spécifique dans le cadre duquel les candidats peuvent soumettre des propositions. La description d’un thème comprend sa portée spécifique et l’impact attendu du projet financé.
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Procédure par laquelle les candidats sont invités à soumettre des propositions de projet en vue de bénéficier d’un financement de l’UE.
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Régime de financement (ou «type d’action») à l’intérieur d’un programme présentant des caractéristiques communes. Le régime de financement précise le champ d’application de ce qui est financé, le taux de remboursement, les critères d’évaluation spécifiques pour bénéficier du financement et les formes simplifiées de couverture des coûts, telles que les montants forfaitaires.
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Coordinateur
3001 Leuven
Belgique
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