Weierstrass' theorem in weighted Sobolev spaces with k derivatives

Abstract

We characterize the set of functions which can be approximated by smooth functions and by polynomials with the norm $$ \ \ {W k,\infty}_w}=\sum_ {j=0} \ (j)}\ {L \infty}_{w_j}}, $$ for a wide range of (possibly unbounded) vector weights $w=(w_0,\dots, w_k)$. We allow a great deal of independence among the weights $w=(w_0,\dots, w_k)$.