no. 9-10
Analyse mathématique/Analyse numérique
Les polynômes orthogonaux de Bergman sur un archipel

Présenté par : Philippe G. Ciarlet

Gustafsson, Björn1 ; Putinar, Mihai2 ; Saff, Edward B.3 ; Stylianopoulos, Nikos4
1 Department of Mathematics, The Royal Institute of Technology, S-10044, Stockholm, Suède
2 Department of Mathematics, University of California at Santa Barbara, Santa Barbara, CA 93106-3080, États-Unis
3 Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, TN 37240, États-Unis
4 Department of Mathematics and Statistics, University of Cyprus, P.O. Box 20537, 1678 Nicosia, Chypre
Comptes Rendus. Mathématique, Tome 346 (2008) no. 9-10, pp. 499-502.

Des estimations de croissance pour la suite de polynômes orthogonaux se rapportant à la mesure d'aire sur une réunion finie de domaines de Jordan sont obtenues par une étude detaillée de la fonction de Green du complément et de la reflection de Schwarz dans les portions analytiques de la frontière. Deux applications en découlent : la distribution limite des zéros de la suite des polynômes orthogonaux de Bergman et un algorithme de reconstruction robuste de l'ouvert original à partir de données incomplètes (tomographiques par exemple).

Growth estimates for orthogonal polynomials with respect to area measure (Bergman polynomials) over the union of finitely many Jordan regions with piecewise smooth boundary are obtained by a careful investigation of the Green function of the complement, and of Schwarz reflection in analytic arcs of the boundary. As applications we obtain a detailed picture of the limiting zero distribution of Bergman's orthogonal polynomials, and also we propose a robust reconstruction algorithm of the original open set, starting from incomplete data (such as obtained by geometric tomography).

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2008.03.001
Gustafsson, Björn 1 ; Putinar, Mihai 2 ; Saff, Edward B. 3 ; Stylianopoulos, Nikos 4

1 Department of Mathematics, The Royal Institute of Technology, S-10044, Stockholm, Suède
2 Department of Mathematics, University of California at Santa Barbara, Santa Barbara, CA 93106-3080, États-Unis
3 Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, TN 37240, États-Unis
4 Department of Mathematics and Statistics, University of Cyprus, P.O. Box 20537, 1678 Nicosia, Chypre 
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     title = {Les polyn\^omes orthogonaux de {Bergman} sur un archipel},
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Gustafsson, Björn; Putinar, Mihai; Saff, Edward B.; Stylianopoulos, Nikos. Les polynômes orthogonaux de Bergman sur un archipel. Comptes Rendus. Mathématique, Tome 346 (2008) no. 9-10, pp. 499-502. doi : 10.1016/j.crma.2008.03.001. http://www.numdam.org/articles/10.1016/j.crma.2008.03.001/

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