### On the weighted approximation of continuously differentiable functions

Leopoldo Nachbin
1991 Proceedings of the American Mathematical Society
This note is an improvement of the available methods for getting results on the weighted approximation of continuously differentiable functions. We shall present what we believe to be a simplified version of the reasoning to establish some results concerning weighted approximation of continuously differentiable scalar functions on R" (see [3] ). For references to weighted approximation of continuous scalar functions on R" (see [1] and [3]). Lemma 1 reduces the search for sufficient conditions
more » ... order that a weight on R be Cm -fundamental to the finding of sufficient conditions for a weight on R to be C-fundamental. Similarly, Lemma 2 reduces the finding of sufficient conditions for a weight on R" to be Cm -fundamental to the search of sufficient conditions for a weight on R to be Cm -fundamental. We then apply Lemmas 1 and 2 together to obtain Propositions 2, 4, and 5. Fix integers n e N, n > 1, and m e N (the case m = ce will be excluded since it follows easily from all m E N ). Let K denote either R or C. Consider the algebras &>(Rn) of all K-valued polynomials on R" and Cm(R") of all continuously m -differentiable K-valued functions on R" . Write N^ = {a e N" : |q| < m} where \a\ = ax + ■ ■ ■ + an if a = (ax, ... , an) e Nn . Let Daf be the ath partial derivative of / € Cm(R") for aeN"m. Denote by 2m(Rn) the subalgebra of Cm(R") of all functions with compact support. A Cm-weight on R" is a family v = {va: a e N^} of upper semicontinuous functions va > 0 on R" . Such a weight v defines the vector space Cmv00(R") of all / £ Cm(R") such that vaDaf tends to zero at infinity for every a e N"m. Set \\f\\v = suv{va(t) ■ \Daf(t)\: t e Rn} to get a seminorm / € C'v^R") t-\\f\\v e R+ for every a e Nnm . The finite family of such seminorms makes Cmvoo(Rn) into a seminormable space, actually a