m at h . C V ] 1 1 O ct 2 00 6 WEIGHTED POLYNOMIALS AND WEIGHTED PLURIPOTENTIAL THEORY

Let E be a compact subset of C and w ≥ 0 a weight function on E with w > 0 on a non-pluripolar subset of E. To (E, w) we associate a canonical circular set Z ⊂ C. We obtain precise relations between the weighted pluricomplex Green function and 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 polynormials with respect to certain exponentially decreasing weights in R . Introduction An admissible weight on a compact set E ⊂ C is a function w ≥ 0 which is strictly positive on a non-pluripolar subset of E. Associated to (E,w) is a weighted pluripotential theory involving weighted polynomials, i.e, functions of the form wp where p is a polynomial of degrees ≤ d, a weighted pluricomplex Green function VE,Q and a weighted equilibrium measure dμeq(E,w). The definitions of these concepts are given in section 1. In the one-dimesional case (N = 1) the book of Saff and Totik [SaTo] has many basic results. In the one-dimensional case, weighted polynomials arise in diverse problems – approximation theory, orthognal polynomials, random matrices, statistical physics. For an example of recent developments see [Dei]. * Supported by an NSERC of Canada Grant. Typeset by AMS-TEX 1 2 In the higher dimensional case, weighted pluripotential theory was used in [BL2] to obtain results on directional Tchebyshev constants of compact sets – the main procedure being an inductive step from circular compact sets to a weighted problem in one less variable. In this paper we further develop the relation between weighted pluripotential theory on a compact set E ⊂ C with admissible (see (1.10)) weight w and the potential theory of a canonically associated circular set Z ⊂ C (defined in (2.1)). We show that VZ , the pluricomplex Green function of Z, and dμeq(Z) the equilibrium measure of Z, are related to the weighted pluricomplex Green function and the weighted equilibrium measure of E with weight w. The main results are: Theorem 2.1. VZ = (VE,Q) ◦ L+ log |t| for t 6= 0 Theorem 2.2. L∗ (