Weierstrass’ theorem with weights

Abstract

We characterize the set of functions which can be approximated by continuous functions in the L N norm with respect to almost every weight. This allows to characterize the set of functions which can be approximated by polynomials or by smooth functions for a wide range of weights. r

Weighted Weierstrass' theorem with first derivatives

“…Theorem 2.1 is an improvement over the previous result obtained in [24,Theorem 2.1]; while the conclusions of the theorems are the same, we have completely removed the technical hypothesis on the weight required in [24]. We also characterize the set of functions which can be approximated by C 1 functions in W 1,∞ (I, w 0 , w 1 ), for a wide range of (possibly unbounded) weights w 0 , w 1 , which have a great deal of independence among them.…”

“…We have finished the proof of Theorem 2.1 following the same argument as in [24] thanks to Lemma 2.1. This is due to the fact that the hypothesis on the density of regular points that was crucial in [24] was only necessary to get approximations of f in a neighborhood of points belonging to S + 1 (w) ∪ S + 2 (w) (see [24, Lemmas 2.2 and 2.3]).…”

“…Notice that (1.1) is not the usual definition of L ∞ norm in the context of measure theory, although it is the correct one when working with weights (see, e.g., [3] and [6]). In [24] we improve the theorems in [30], obtaining sharp results for a large class of weights. Here we also study this problem both with the norm (1.1) for every weight w, and with the Sobolev norm W 1,∞ (I, w 0 , w 1 ) defined by f W 1,∞ (I,w 0 ,w 1 ) := f L ∞ (I,w 0 ) + f L ∞ (I,w 1 ) , since in many situations it is natural to consider the simultaneous approximation of a function and its first derivative.…”

“…[24], and that essentially meant that the regular points were dense in R. Items [24], since no additional hypothesis on the weight were needed there in this part of the proof.…”

“…In [30] and [24] we study the same problem with the norm L ∞ (I, w) defined by f L ∞ (I,w) := ess sup x∈I f (x) w(x), (1.1) where w is a weight, i.e. a non-negative measurable function and we use the convention 0·∞ = 0.…”