Analyse mathématique/Analyse numérique
Les polynômes orthogonaux de Bergman sur un archipel
[Bergman orthogonal polynomials on an archipelago]
Comptes Rendus. Mathématique, Volume 346 (2008) no. 9-10, pp. 499-502.

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).

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).

Received:
Accepted:
Published online:
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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Gustafsson, Björn; Putinar, Mihai; Saff, Edward B.; Stylianopoulos, Nikos. Les polynômes orthogonaux de Bergman sur un archipel. Comptes Rendus. Mathématique, Volume 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/

[1] Carleman, T. Über die Approximation analytischer Funktionen durch lineare Aggregate von vorgegebenen Potenzen, Ark. Mat., Astr. Fys., Volume 17 (1923), pp. 215-244

[2] Gaier, D. Lectures on Complex Approximation, Birkhäuser Boston Inc., Boston, MA, 1987 (Translated from the German by Renate McLaughlin)

[3] Golub, G.; Gustafsson, B.; Milanfar, P.; Putinar, M.; Varah, J. Shape reconstruction from moments: theory, algorithms, and applications (Luk, F.T., ed.), Advanced Signal Processing, Algorithms, Architecture, and Implementations X, SPIE Proceedings, vol. 4116, 2000, pp. 406-416

[4] Gustafsson, B.; He, C.; Milanfar, P.; Putinar, M. Reconstructing planar domains from their moments, Inverse Problems, Volume 16 (2000), pp. 1053-1070

[5] Levin, A.L.; Saff, E.B.; Stylianopoulos, N.S. Zero distribution of Bergman orthogonal polynomials for certain planar domains, Constr. Approx., Volume 19 (2003), pp. 411-435

[6] Miña-Díaz, E.; Saff, E.B.; Stylianopoulos, N.S. Zero distributions for polynomials orthogonal with weights over certain planar regions, Comput. Methods Funct. Theory, Volume 5 (2005), pp. 185-221

[7] Saff, E.B. Polynomials of interpolation and approximation to meromorphic functions, Trans. Amer. Math. Soc., Volume 143 (1969), pp. 509-522

[8] Saff, E.B.; Totik, V. Logarithmic Potentials with External Fields, Springer-Verlag, Berlin, 1997

[9] Stahl, H.; Totik, V. General Orthogonal Polynomials, Cambridge University Press, Cambridge, 1992

[10] Suetin, P.K. Order comparison of various norms of polynomials in a complex region, Ural. Gos. Univ. Mat. Zap., Volume 5 (1966), pp. 91-100

[11] Walsh, J.L. A sequence of rational functions with application to approximation by bounded analytic functions, Duke Math. J., Volume 30 (1963), pp. 177-189

[12] Widom, H. Extremal polynomials associated with a system of curves in the complex plane, Adv. Math., Volume 3 (1969), pp. 127-232

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