APPROXIMATION © 2002 Springer-Verlag New York Inc. Asymptotic Distribution of Nodes for Near-Optimal Polynomial Interpolation on Certain Curves in R2

Abstract. Let E ‰ Rs be compact and let d En denote the dimension of the space of polynomials of degree at most n in s variables restricted to E. We introduce the notion of an asymptotic interpolation measure (AIM). Such a measure, if it exists, describes the asymptotic behavior of any scheme ¿n D fxk;ngd E n kD1, n D 1; 2; : ::, of nodes for multivariate polynomial interpolation for which the norms of the corresponding interpolation operators do not grow geometrically large with n. We demonstrate the existence of AIMs for the finite union of compact subsets of certain algebraic curves in R2. It turns out that the theory of logarithmic potentials with external fields plays a useful role in the investigation. Furthermore, for the sets mentioned above, we give a computationally simple construction for “good ” interpolation schemes. 1