Ragnar Sigurdsson: Polynomials of several variables with exponents in compact convex sets and associated weighted extremal functions: The Siciak-Zakharyuta and Bernstein-Walsh-Siciak theorems.

Abstract.

The lecture is a report on a joint work with Álfheiður Edda Sigurðardóttir, Benedikt Steinar Magnússon and Bergur Snorrason. For a given compact convex subset S of \({\mathbb R}^n_+\) with \(0\in S\) we define the polynomial ring as the union of the spaces ${\mathcal P}^S_m(\C^n)$ for $m=1,2,3,\dots$, consisting of all polynomials of $n$ complex variables the form $$ p(z)= \sum_{\alpha\in (mS)\cap {\mathbb N}^n a_\alpha z^\alpha $$ with the standard multi-index notation. For any given weight (external field) $q$ on a subset $E$ of $\C^n$ we associate to this class of polynomials the weighted Siciak extremal functions $\Phi^SA_{E,q,m}$, $\Phi^S_{E,q}$, and the weighted Siciak-Zakharyuta extremal functions which have their origin in classical weighted potential theory for functions of one complex variable. This is done in order to generalize two classical theorems from approximation theory.