Weierstrass' theorem in weighted Sobolev spaces (Q5931919)

It is very well known that given any compact interval \(I\), the set of all continuous (almost everywhere) functions \(C(I)\) on \(I\) is the biggest set of functions that can be approximated by polynomials in the \(L^\infty(I)\) norm. This result is the very classical Weierstrass' Theorem. There are many generalizations of this result [see e.g. the nice survey \textit{D. S. Lubinsky}, Quaest. Math. 18, No. 1-3, 91-130 (1995; Zbl 0824.41005)]. In the paper the author considers the vector of weights \(w=(w_0,w_1,\dots,w_k)\), and defines the Sobolev norm defined by NEWLINE\[NEWLINE \|f\|_{W^{k,\infty}(\Delta,w)}= \sum_{j=0}^k\|f^{(j)}\|_{L^\infty(\Delta,w)},\qquad \Delta=\bigcup_{j=0^k}\text{supp} w_j NEWLINE\]NEWLINE which leads to the weighted Sobolev space \(W^{k,\infty}(\Delta,w)\). In this context of weighted Sobolev space it is characterized the set of functions which can be approximated by polynomials for some classes of weights \(w\) with respect to the aforesaid Sobolev norm \(\|\cdot\|_{W^{k,\infty}(\Delta,w)}\). The results cover not only the bounded intervals but also some special weights for the unbounded intervals.