Weighted Weierstrass' theorem with first derivatives
Let \(w:\mathbb R\to[0,+\infty]\) a measurable function and \(L^\infty(w)\) the space of measurable functions \(f:\mathbb R\to\mathbb R\) such that \[ \|f\|:\text{ess\,sup}|f(x)|w(t)\mid< +\infty,\;x\in\mathbb R, \] with the convention \(0 \cdot\infty=0\). In this paper, the authors characterize the closure of \(C(\mathbb R)\) in \(L^\infty(w)\) for any weight \(w\) and as a consequence they obtain the set of functions which can be approximated in \(L^\infty(w)\) by polynomials for any weight with compact support. Moreover, under suitable conditions, they give analogous characterizations for the set of functions which can be approximated by smooth functions in the Sobolev space \(W^{1,\infty}(w_0,w_1)\).
weighted spaces \(L^\infty\) and \(w^{1,\infty}\), approximation by polynomials
scientific article; zbMATH DE number 5173014
scientific article