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Polynômes orthogonaux simultanés et systèmes dynamiques infinis

Abstract : First, I define vector polynomials orthogonal with respect to a matrix of measures. I recall some usual properties like the r+s+1 terms recurrence relation, the Shohat-Favard theorem or a Christoffel-Darboux formula. Then, using Padé approximants, I describe the resolvent set of the operator associated to the recurrence relations. This result was given by Duren but I give here a new proof. I then define Moebius transforms on the set of matrix of size r x s. I find again all cases of continued fractions : scalar, vector or matrix cases. These transforms help us to prove a theorem of convergence acceleration of matrix continued fractions. This generalizes a theorem on generalized continued fractions given by de Bruin et Jacobsen. Vector polynomials are then used to calculate recurrence coefficients of other polynomials with the help of the vector modified Chebsyhev algorithm, which is a generalization of the scalar case. In this last case we give some criterion of stability. Finally, modified Chebsyhev algorithm is used to study the Toda-Langmuir semi-infinite dynamical system. In this system, the particles are on the real semi-axis and act with each other with respect to a decreasing exponential law. To solve this problem, I use, again, a new approach. In fact, since I study only the first n particles, I am interested in the error done if we cut the system to N>>n quantities. We, then, work with a finite system. I present the theoretical study of the error in which we use our results on the stability of modified Chebyshev algorithm. I give also some numerical examples.
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Contributor : Emmanuel Bourreau <>
Submitted on : Tuesday, March 9, 2004 - 6:29:08 PM
Last modification on : Friday, April 19, 2019 - 1:21:00 AM
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  • HAL Id : tel-00005227, version 1



Emmanuel Bourreau. Polynômes orthogonaux simultanés et systèmes dynamiques infinis. Mathématiques [math]. Université des Sciences et Technologie de Lille - Lille I, 2002. Français. ⟨tel-00005227⟩



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