Distribution of nodes on algebraic curves in N
[La distribution des nœuds sur les courbes algébriques de N ]
Annales de l'Institut Fourier, Tome 53 (2003) no. 5, pp. 1365-1385.

Soit A une variété algébrique de dimension 1 de N . On note m d la dimension de l’espace vectoriel complexe des restrictions à A des polynmôes holomorphes de degré d. On considère un compact non polaire K et pour chaque d=1,2,..., on choisit m d points (nœuds) {A dj } j=1,...,m d dans K. Enfin, on note Λ d la constante de Lebesgue d’ordre d associée aux noeuds {A dj } : cette constante est la norme de l’opérateur L d sur C(K), où L d (f) est le polynôme d’interpolation de Lagrange de f, de degré d, aux points {A dj }. Nous utilisons la théorie du pluripotentiel pour montrer qu’il existe une mesure m K portée par K, de masse totale égale à 1, et telle que pour n’importe quels noyaux {A dj } sur K vérifiant lim sup d Λ d 1/d 1, les mesures discrètes μ d :=1 m d j=1 m d δ A dj ,d=1,2,..., convergent faiblement vers μ K .

Given an irreducible algebraic curves A in N , let m d be the dimension of the complex vector space of all holomorphic polynomials of degree at most d restricted to A. Let K be a nonpolar compact subset of A, and for each d=1,2,..., choose m d points {A dj } j=1,...,m d in K. Finally, let Λ d be the d-th Lebesgue constant of the array {A dj }; i.e., Λ d is the operator norm of the Lagrange interpolation operator L d acting on C(K), where L d (f) is the Lagrange interpolating polynomial for f of degree d at the points {A dj } j=1,...,m d . Using techniques of pluripotential theory, we show that there is a probability measure μ K supported on K such that for any array in K satisfying lim sup d Λ d 1/d 1, the discrete measures μ d :=1 m d j=1 m d δ A dj ,d=1,2,..., converge weak-* to μ K .

DOI : 10.5802/aif.1982
Classification : 32U05, 31C10, 41A05
Keywords: algebraic curve, Lebesgue constant
Mot clés : courbe algébrique, constante de Lebesgue
Bloom, Thomas 1 ; Levenberg, Norman 2

1 University of Toronto, Department of Mathematics, Toronto, Ont. M5S 3G3 (Canada)
2 University of Auckland, Department of Mathematics, 38 Princes Street, Private Bag 92019, Auckland (New-Zealand)
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Bloom, Thomas; Levenberg, Norman. Distribution of nodes on algebraic curves in ${\mathbb {C}}^N$. Annales de l'Institut Fourier, Tome 53 (2003) no. 5, pp. 1365-1385. doi : 10.5802/aif.1982. http://archive.numdam.org/articles/10.5802/aif.1982/

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