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

Bloom, Thomas; Levenberg, Norman

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

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 $\le d$. On considère un compact non polaire $K$ et pour chaque $d=1,2,...,$ on choisit $m_d$ points (nœuds) ${\left\{A_{dj}\right\}}_{j=1,...,m_d}$ dans $K$. Enfin, on note ${\Lambda }_d$ la constante de Lebesgue d’ordre $d$ associée aux noeuds $\left\{A_{dj}\right\}$ : cette constante est la norme de l’opérateur $L_d$ sur $C\left(K\right)$, où $L_d\left(f\right)$ est le polynôme d’interpolation de Lagrange de $f$, de degré $d$, aux points $\left\{A_{dj}\right\}$. 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 $\left\{A_{dj}\right\}$ sur $K$ vérifiant ${\lim \sup }_{d\to \infty }{\Lambda }_d^{1/d}\le 1$, les mesures discrètes ${\mu }_d:=\frac{1}{m_d}{\sum }_{j=1}^{m_d}{\delta }_{A_{dj}},d=1,2,...,$ convergent faiblement vers ${\mu }_K$.

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 ${\left\{A_{dj}\right\}}_{j=1,...,m_d}$ in $K$. Finally, let ${\Lambda }_d$ be the $d$-th Lebesgue constant of the array $\left\{A_{dj}\right\}$; i.e., ${\Lambda }_d$ is the operator norm of the Lagrange interpolation operator $L_d$ acting on $C\left(K\right)$, where $L_d\left(f\right)$ is the Lagrange interpolating polynomial for $f$ of degree $d$ at the points ${\left\{A_{dj}\right\}}_{j=1,...,m_d}$. Using techniques of pluripotential theory, we show that there is a probability measure ${\mu }_K$ supported on $\partial K$ such that for any array in $K$ satisfying ${\lim \sup }_{d\to \infty }{\Lambda }_d^{1/d}\le 1$, the discrete measures ${\mu }_d:=\frac{1}{m_d}{\sum }_{j=1}^{m_d}{\delta }_{A_{dj}},d=1,2,...,$ converge weak-$*$ to ${\mu }_K$.

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

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