Distribution of nodes on algebraic curves in N
Annales de l'Institut Fourier, Volume 53 (2003) no. 5, p. 1365-1385

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 .

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 .

DOI : https://doi.org/10.5802/aif.1982
Classification:  32U05,  31C10,  41A05
Keywords: algebraic curve, Lebesgue constant
@article{AIF_2003__53_5_1365_0,
     author = {Bloom, Thomas and Levenberg, Norman},
     title = {Distribution of nodes on algebraic curves in ${\mathbb {C}}^N$},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {53},
     number = {5},
     year = {2003},
     pages = {1365-1385},
     doi = {10.5802/aif.1982},
     zbl = {1044.32026},
     mrnumber = {2032937},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2003__53_5_1365_0}
}
Bloom, Thomas; Levenberg, Norman. Distribution of nodes on algebraic curves in ${\mathbb {C}}^N$. Annales de l'Institut Fourier, Volume 53 (2003) no. 5, pp. 1365-1385. doi : 10.5802/aif.1982. http://www.numdam.org/item/AIF_2003__53_5_1365_0/

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