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

Bloom, Thomas, and Levenberg, Norman. "Distribution of nodes on algebraic curves in ${\mathbb {C}}^N$." Annales de l’institut Fourier 53.5 (2003): 1365-1385. <http://eudml.org/doc/116075>.

@article{Bloom2003,
abstract = {Given an irreducible algebraic curves $A$ in $\{\mathbb \{C\}\}^N$, 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 $\lbrace A_\{dj\}\rbrace _\{j=1,...,m_d\}$ in $K$. Finally, let $\Lambda _d$ be the $d$-th Lebesgue constant of the array $\lbrace A_\{dj\}\rbrace $; i.e., $\Lambda _d$ 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 $\lbrace A_\{dj\}\rbrace _\{j=1,...,m_d\}$. 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 $\{\rm lim\,sup\}_\{d\rightarrow \infty \}\Lambda ^\{1/d\}_d\le 1$, the discrete measures $\mu _d:=\{1\over m_d\}\sum ^\{\{m_d\}\}_\{j=1\}\delta _\{A_\{dj\}\},\; d=1,2,...,$ converge weak-$*$ to $\mu _K$.},
affiliation = {University of Toronto, Department of Mathematics, Toronto, Ont. M5S 3G3 (Canada); University of Auckland, Department of Mathematics, 38 Princes Street, Private Bag 92019, Auckland (New-Zealand)},
author = {Bloom, Thomas, Levenberg, Norman},
journal = {Annales de l’institut Fourier},
keywords = {algebraic curve; Lebesgue constant},
language = {eng},
number = {5},
pages = {1365-1385},
publisher = {Association des Annales de l'Institut Fourier},
title = {Distribution of nodes on algebraic curves in $\{\mathbb \{C\}\}^N$},
url = {http://eudml.org/doc/116075},
volume = {53},
year = {2003},
}

TY - JOUR
AU - Bloom, Thomas
AU - Levenberg, Norman
TI - Distribution of nodes on algebraic curves in ${\mathbb {C}}^N$
JO - Annales de l’institut Fourier
PY - 2003
PB - Association des Annales de l'Institut Fourier
VL - 53
IS - 5
SP - 1365
EP - 1385
AB - Given an irreducible algebraic curves $A$ in ${\mathbb {C}}^N$, 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 $\lbrace A_{dj}\rbrace _{j=1,...,m_d}$ in $K$. Finally, let $\Lambda _d$ be the $d$-th Lebesgue constant of the array $\lbrace A_{dj}\rbrace $; i.e., $\Lambda _d$ 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 $\lbrace A_{dj}\rbrace _{j=1,...,m_d}$. 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 ${\rm lim\,sup}_{d\rightarrow \infty }\Lambda ^{1/d}_d\le 1$, the discrete measures $\mu _d:={1\over m_d}\sum ^{{m_d}}_{j=1}\delta _{A_{dj}},\; d=1,2,...,$ converge weak-$*$ to $\mu _K$.
LA - eng
KW - algebraic curve; Lebesgue constant
UR - http://eudml.org/doc/116075
ER -