Approximation of smooth functions using Bernstein polynomials in multiple variables

Abstract

In this survey, we use (more or less) elementary means to establish the well-known 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.