Distribution of nodes on algebraic curves in ${\mathbb {C}}^N$

Bloom, Thomas; Levenberg, Norman

doi:10.5802/aif.1982

Distribution of nodes on algebraic curves in

ℂ^{N}

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)

Annales de l'Institut Fourier, Volume 53 (2003) no. 5, pp. 1365-1385.

Abstract
Résumé

Given an irreducible algebraic curves $A$ in $ℂ^{N}$ , 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 ${A_{d j}}_{j = 1, . . ., m_{d}}$ in $K$ . Finally, let $Λ_{d}$ be the $d$ -th Lebesgue constant of the array ${A_{d j}}$ ; 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 ${A_{d j}}_{j = 1, . . ., m_{d}}$ . Using techniques of pluripotential theory, we show that there is a probability measure $μ_{K}$ supported on $\partial K$ such that for any array in $K$ satisfying ${\lim \sup}_{d \to \infty} Λ_{d}^{1 / d} \leq 1$ , the discrete measures $μ_{d} : = \frac{1}{m_{d}} \sum_{j = 1}^{m_{d}} δ_{A_{d j}}, d = 1, 2, . . .,$ converge weak- $*$ to $μ_{K}$ .

Soit $A$ une variété algébrique de dimension 1 de $ℂ^{N}$ . On note $m_{d}$ la dimension de l’espace vectoriel complexe des restrictions à $A$ des polynmôes holomorphes de degré $\leq d$ . On considère un compact non polaire $K$ et pour chaque $d = 1, 2, . . .,$ on choisit $m_{d}$ points (nœuds) ${A_{d j}}_{j = 1, . . ., m_{d}}$ dans $K$ . Enfin, on note $Λ_{d}$ la constante de Lebesgue d’ordre $d$ associée aux noeuds ${A_{d j}}$ : 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 ${A_{d j}}$ . Nous utilisons la théorie du pluripotentiel pour montrer qu’il existe une mesure $m_{K}$ portée par $\partial K$ , de masse totale égale à 1, et telle que pour n’importe quels noyaux ${A_{d j}}$ sur $K$ vérifiant ${\lim \sup}_{d \to \infty} Λ_{d}^{1 / d} \leq 1$ , les mesures discrètes $μ_{d} : = \frac{1}{m_{d}} \sum_{j = 1}^{m_{d}} δ_{A_{d j}}, d = 1, 2, . . .,$ convergent faiblement vers $μ_{K}$ .

MR: 2032937 | Zbl: 1044.32026

DOI: 10.5802/aif.1982

Classification: 32U05, 31C10, 41A05
Keywords: algebraic curve, Lebesgue constant

Author's affiliations:

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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     author = {Bloom, Thomas and Levenberg, Norman},
     title = {Distribution of nodes on algebraic curves in ${\mathbb {C}}^N$},
     journal = {Annales de l'Institut Fourier},
     pages = {1365--1385},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {53},
     number = {5},
     year = {2003},
     doi = {10.5802/aif.1982},
     zbl = {1044.32026},
     mrnumber = {2032937},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1982/}
}

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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. https://aif.centre-mersenne.org/articles/10.5802/aif.1982/

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