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}

[La distribution des nœuds sur les courbes algébriques de

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, Tome 53 (2003) no. 5, pp. 1365-1385.

Résumé
Abstract

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 $μ_{K}$ .

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 $μ_{K}$ .

MR Zbl

DOI : 10.5802/aif.1982

Classification : 32U05, 31C10, 41A05
Keywords: algebraic curve, Lebesgue constant
Mot clés : courbe algébrique, constante de Lebesgue

Affiliations des auteurs :

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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     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, Tome 53 (2003) no. 5, pp. 1365-1385. doi : 10.5802/aif.1982. https://aif.centre-mersenne.org/articles/10.5802/aif.1982/

Bibliographie
Cité par

[BBCL] T. Bloom; L. Bos; C. Christensen; N. Levenberg Polynomial interpolation of holomorphic functions in $ℂ$ and $ℂ^{n}$ , Rocky Mtn. J. Math, Volume 22 (1992) no. 2, pp. 441-470 | DOI | MR | Zbl

[Be] E. Bedford The operator ${(d d^{c})}^{n}$ on complex spaces, Séminaire Pierre Lelong-Henri Skoda 1980-1981 et Colloque de Wimereux, Mai 1981 (Lecture Notes in Math.), Volume 919 (1982), pp. 294-323 | Zbl

[BLMT] L. Bos; N. Levenberg; P. Milman; B. A. Taylor Tangential Markov Inequalities Characterize Algebraic Submanifolds of $B b b R^{n}$ , Indiana J. Math, Volume 44 (1995) no. 1, pp. 115-138 | MR | Zbl

[D] J.-P. Demailly Mesures de Monge-Ampère et caractérisation géométrique des variétés algébriques affines, Bull. de la Soc. Math. de France, Volume 113, Fasc. 2 (1985) no. 19, pp. 1-125 | Numdam | MR | Zbl

[DG] F. Docquier; H. Grauert Levisches problem und rungescher Satz für teilgebiete Steinscher Mannigfaltigkeiten (German), Math. Ann, Volume 140 (1960), pp. 94-123 | DOI | MR | Zbl

[GMS] M. Götz; V. Maymeskul; and E. B. Saff Asymptotic distribution of nodes for near-optimal polynomial interpolation on certain curves in $ℝ^{2}$ , Constructive Approximation, Volume 18 (2002) no. 2, pp. 255-284 | DOI | MR | Zbl

[H] L. Hörmander Notions of Convexity, Birkhäuser, Boston, 1994 | MR | Zbl

[K] M. Klimek Pluripotential Theory, Clarendon Press, Oxford, 1991 | MR | Zbl

[Kr] S. Krantz Function Theory of Several Complex Variables, Wiley, New York, 1982 | MR | Zbl

[Ru] W. Rudin A geometric criterion for algebraic varieties, J. Math. Mech, Volume 20 (1968) no. 7, pp. 671-683 | MR | Zbl

[Sa1] A. Sadullaev An estimate for polynomials on analytic sets, Math. USSR Izvestiya, Volume 20 (1983) no. 3, pp. 493-502 | DOI | MR | Zbl

[Sa2] A. Sadullaev Extension of plurisubharmonic functions from a submanifold, (Russian), Dokl. Akad. Nauk UzSSR, Volume 5 (1982), pp. 3-4 | MR | Zbl

[T] B. A. Taylor An estimate for an extremal plurisubharmonic function on $ℂ^{n}$ , Séminaire P. Lelong, P. Dolbeault, H. Skoda, Année 1982/1983 (Lecture Notes in Math.), Volume 1028 (1983), pp. 318-328 | Zbl

[Ze] A. Zeriahi Fonction de Green pluricomplex à pôle à l'infini sur un espace de Stein parabolique et applications, Math. Scand, Volume 69 (1991), pp. 89-126 | MR | Zbl

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