Uniform limits of sequences of polynomials and their derivatives

Abstract:

Let $E$ be a compact subset of the unit interval $[0,1]$, and let $C(E)$ denote the space of functions continuous on $E$ with the uniform norm. Consider the densely defined operator $D:C(E) \to C(E)$ given by $Dp = p’$ for all polynomials $p$. Let $G$ represent the graph of $D$, that is $G = \{ (p,p’):p$ polynomials} considered as a submanifold of $C(E) \times C(E)$. Write the interior of the set $E,\;\operatorname {int} E$ as a countable union of disjoint open intervals and let $\widehat {E}$ be the union of the closure of these intervals. The main result is that the closure of $G$ is equal to the set of all functions $(h,k) \in C(E) \times C(E)$ such that $h$ is absolutely continuous on $\widehat {E}$ and $k|\widehat {E} = h’|\widehat {E}$. As a consequence, the operator $D$ is closable if and only if the set $E$ is the closure of its interior. On the other extreme, $G$ is dense in $C(E) \times C(E)$ i.e. for any pair $(f,g) \in C(E) \times C(E)$, there exists a sequence of polynomials $\{ {p_n}\}$ so that ${p_n} \to f$ and ${p’_n} \to g$ uniformly on $E$, if and only if the interior $\operatorname {int} E$ of $E$ is empty.