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Uniform limits of sequences of polynomials and their derivatives


Authors: Joseph A. Ball and Thomas R. Fanney
Journal: Proc. Amer. Math. Soc. 114 (1992), 749-755
MSC: Primary 41A10; Secondary 40A30, 41A65, 47E05
DOI: https://doi.org/10.1090/S0002-9939-1992-1091175-8
MathSciNet review: 1091175
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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\vert\widehat{E} = h'\vert\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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1992-1091175-8
Article copyright: © Copyright 1992 American Mathematical Society

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