Given an irreducible algebraic curves in , let be the dimension of the complex vector space of all holomorphic polynomials of degree at most restricted to . Let be a nonpolar compact subset of , and for each choose points in . Finally, let be the -th Lebesgue constant of the array ; i.e., is the operator norm of the Lagrange interpolation operator acting on , where is the Lagrange interpolating polynomial for of degree at the points . Using techniques of pluripotential theory, we show that there is a probability measure supported on such that for any array in satisfying , the discrete measures converge weak- to .
Soit une variété algébrique de dimension 1 de . On note la dimension de l’espace vectoriel complexe des restrictions à des polynmôes holomorphes de degré . On considère un compact non polaire et pour chaque on choisit points (nœuds) dans . Enfin, on note la constante de Lebesgue d’ordre associée aux noeuds : cette constante est la norme de l’opérateur sur , où est le polynôme d’interpolation de Lagrange de , de degré , aux points . Nous utilisons la théorie du pluripotentiel pour montrer qu’il existe une mesure portée par , de masse totale égale à 1, et telle que pour n’importe quels noyaux sur vérifiant , les mesures discrètes convergent faiblement vers .
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}, pages = {1365--1385}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {53}, number = {5}, year = {2003}, doi = {10.5802/aif.1982}, mrnumber = {2032937}, zbl = {1044.32026}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1982/} }
TY - JOUR TI - Distribution of nodes on algebraic curves in ${\mathbb {C}}^N$ JO - Annales de l'Institut Fourier PY - 2003 DA - 2003/// SP - 1365 EP - 1385 VL - 53 IS - 5 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.1982/ UR - https://www.ams.org/mathscinet-getitem?mr=2032937 UR - https://zbmath.org/?q=an%3A1044.32026 UR - https://doi.org/10.5802/aif.1982 DO - 10.5802/aif.1982 LA - en ID - AIF_2003__53_5_1365_0 ER -
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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