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Zbl 1044.32026
Bloom, Thomas; Levenberg, Norman
Distribution of nodes on algebraic curves in ${\Bbb C}^N$.
(English)
[J] Ann. Inst. Fourier 53, No. 5, 1365-1385 (2003). ISSN 0373-0956; ISSN 1777-5310/e

Let $A\subset\Bbb C^N$ be an irreducible curve and let $K\subset A$ be a non-polar compact set. For $d=0,1,\dots$, let $m_d$ be the dimension of the space $P_d\vert A$ of all complex polynomials of degree at most $d$ restricted to $A$. It is known that for sufficiently large $d$ we have $m_d=dD+c$, where $D$ is the degree of $A$ and $c$ is an integer. Consider Lagrange interpolation polynomials $L_df(z)= \sum_{j=1}^{m_d}f(A_{d,j})\ell^{(d)}_j(z)$ with nodes $A_{d,j}\in K$. Let $\varLambda_d:=\Vert \sum_{j=1}^{m_d}\vert \ell^{(d)}_j\vert \Vert _K$. Assume that the nodes are chosen in such a way that $\limsup_{d\to+\infty}\varLambda_d^{1/d}\leq1$. Then $(1/m_d)\sum_{j=1}^{m_d}\delta_{A_{d,j}}\overset{\text{weak--$\ast$}}\to \longrightarrow(1/2\pi D)dd^cV^\ast_K=:\mu_K$ and $\text {supp}\mu_K\subset\partial K$, where $V_K$ is the Siciak extremal function for $K$.
[Marek Jarnicki (Kraków)]
MSC 2000:
*32U05 Plurisubharmonic functions and generalizations
31C10 Pluriharmonic and plurisubharmonic functions
41A05 Interpolation

Keywords: algebraic curve; Lebesgue constant

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