Thomas Bloom

Weighted polynomials and weighted pluripotential theory

Let E be a compact subset of C N and w > 0 an admissible weight function on E. To (E,ω) we associate a canonical circular set Z C We obtain precise relations between the weighted pluricomplex Green function and weighted equilibrium measure of (E, w) and the pluricomplex Green function and equilibrium measure of Z. These results, combined with an appropriate form of the Bernstein-Markov inequality, are used to obtain asymptotic formulas for the leading coefficients of orthonormal polynomials with respect to certain exponentially decreasing weights in R N .

On the convergence of multivariable Lagrange interpolants

LetAdj be a triangular array in a compact setX⊂Cn. Forf analytic in a neighborhood ofX, letLd(f) denote the Lagrange interpolant tof at staged of the array. In the caseX is locally regular, we construct a continuous functionϕ satisfying the complex Monge-Ampère equation onCn−X, such that iff is analytic onϕ≤R forR>1 then, for someB>0, we have ∥Ld(f)−f∥x≤B exp(−d logR. In particular, sinceϕ≤1 onX, iff is analytic onϕ≤1, then limd ∥Ld(f)−f∥x=0.

A continuity property of multivariate Lagrange interpolation

Let {S t } be a sequence of interpolation schemes in R n of degree d (i.e. for each S t one has unique interpolation by a polynomial of total degree < d) and total order ≤ l. Suppose that the points of S t tend to 0 ∈ R n as t → ∞ and the Lagrange-Hermite interpolants, Hs t , satisfy lim t→ ∞ HS t (x α ) = 0 for all monomials x α with |α| = d + 1. Theorem: lim t→ ∞ HS t (f) = T d (f) for all functions f of class C l-1 in a neighborhood of 0. (Here T d (f) denotes the Taylor series of f at 0 to order d.) Specific examples are given to show the optimality of this result.

The Lebesgue constant for Lagrange interpolation in the Simplex

On obtient des resultats precis concernant les valeurs asymptotiques des fonctions de Lebesgue et des constantes de Lebesgue pour des points egalement espaces dans le simplexe

Sur le diamètre transfini en plusieurs variables

Resume On demontre des resultats concernant le diametre transfini des compact dans ℂn. Ces resultats mettent en lumiere la relation etroite entre le diametre transfini et des objects fondamentaux de la theorie du pluripotentiel.