Weighted Weierstrass’ theorem with first derivatives

We characterize the set of functions which can be approximated by continuous functions with the norm f L∞(w) for every weight w. This fact allows to determine the closure of the space of polynomials in L∞(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 f W1,∞(w0,w1) := f L∞(w0) + f L∞(w1), for a wide range of (even non-bounded) weights wj ’s. We allow a great deal of independence among the weights wj ’s. © 2007 Elsevier Inc. All rights reserved.