We apply pluripotential theory to establish results in concerning uniform approximation by functions of the form wⁿPₙ where w denotes a continuous nonnegative function and Pₙ is a polynomial of degree at most n. Then we use our work to show that on the intersection of compact sections a continuous function on Σ is uniformly approximable by θ-incomplete polynomials (for a fixed θ, 0 < θ < 1) iff f vanishes on θ²Σ. The class of sets Σ expressible as the intersection of compact sections includes the intersection of a symmetric convex compact set with a single orthant.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ap85-3-7,
author = {Maritza M. Branker},
title = {Approximation by weighted polynomials in $$\mathbb{R}$^k$
},
journal = {Annales Polonici Mathematici},
volume = {85},
year = {2005},
pages = {261-279},
zbl = {1092.32019},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap85-3-7}
}
Maritza M. Branker. Approximation by weighted polynomials in $ℝ^k$
. Annales Polonici Mathematici, Tome 85 (2005) pp. 261-279. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap85-3-7/