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)‖f‖L∞(w) for every weight w  . This fact allows to determine the closure of the space of polynomials in L∞(w)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),‖f‖W1,∞(w0,w1):=‖f‖L∞(w0)+‖f′‖L∞(w1), for a wide range of (even non-bounded) weights wjwj's. We allow a great deal of independence among the weights wjwj's.