Approximation of smooth functions using Bernstein polynomials in multiple variables
Abstract
In this survey, we use (more or less) elementary means to establish the wellknown result that for any given smooth multivariate function, the respective multivariate Bernstein polynomials converge to that function in all derivatives on each compact set. We then go on to strengthen that result to obtain that any smooth function on $\mathbb R^d$ may be approximated locally uniformly in all derivatives by \emph{one} sequence of polynomials. We will use neither the axiom of choice nor the power set axiom. We will use the method of proof by contradiction.
 Publication:

arXiv eprints
 Pub Date:
 September 2016
 arXiv:
 arXiv:1609.01940
 Bibcode:
 2016arXiv160901940F
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 4102