Approximation by weighted polynomials in ${\Bbb R}^k$

Maritza M. Branker¹

¹ Department of Mathematics University of Toronto Toronto, Canada M5S 3G3

Annales Polonici Mathematici, Tome 85 (2005) no. 3, pp. 261-279

Résumé

We apply pluripotential theory to establish results in $\mathbb R^k$ concerning uniform approximation by functions of the form $w^n P_n$ where $w$ denotes a continuous nonnegative function and $P_n$ is a polynomial of degree at most $n$. Then we use our work to show that on the intersection of compact sections ${\mit\Sigma} \subset \mathbb R^k$ a continuous function on ${\mit\Sigma}$ is uniformly approximable by $\theta$-incomplete polynomials (for a fixed $\theta,$ $0 \theta 1$) iff $f$ vanishes on $\theta^2 {\mit\Sigma}$. The class of sets ${\mit\Sigma}$ expressible as the intersection of compact sections includes the intersection of a symmetric convex compact set with a single orthant.

Détail

DOI : 10.4064/ap85-3-7

Keywords: apply pluripotential theory establish results mathbb concerning uniform approximation functions form n where denotes continuous nonnegative function polynomial degree work intersection compact sections mit sigma subset mathbb continuous function mit sigma uniformly approximable theta incomplete polynomials fixed theta theta vanishes theta mit sigma class sets mit sigma expressible intersection compact sections includes intersection symmetric convex compact set single orthant