Weighted Weierstrass' theorem with first derivatives

Abstract

We characterize the set of functions which can be approximated by continuous functions with the norm $\ \ {L infty(w)}$ for every weight w. This fact allows to determine the closure of the space of polynomials in $L infty(w)$ for every weight w with compact support. We characterize as well the set of functions which can be approximated by smooth functions with the norm

$$ \ \ {W 1,\infty}(w_0,w_1)}\coloneq \ \ {L infty(w_0)}+ \ '\ {L infty(w_1)}, $$

for a wide range of (even non-bounded) weights $w_0,w_1$. We allow a great deal of independence among the weights.