Asymptotic Distribution of Nodes for Near-Optimal Polynomial Interpolation on Certain Curves in R 2

Abstract

Let E\subset \Bbb R s be compact and let d n E 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 τ n ={ \bf x k,n } k=1 dnE , n=1,2,\ldots , 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 R 2 . 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.