Weighted Weierstrass' theorem with first derivatives

Abstract

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 ‖ W 1 , ∞ ( w 0 , w 1 ) : = ‖ f ‖ L ∞ ( w 0 ) + ‖ f ′ ‖ L ∞ ( w 1 ) , for a wide range of (even non-bounded) weights w j 's. We allow a great deal of independence among the weights w j 's.