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Uniform Limits of Sequences of Polynomials and their Derivatives

Joseph A. Ball and Thomas R. Fanney
Proceedings of the American Mathematical Society
Vol. 114, No. 3 (Mar., 1992), pp. 749-755
DOI: 10.2307/2159400
Stable URL: http://www.jstor.org/stable/2159400
Page Count: 7
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Uniform Limits of Sequences of Polynomials and their Derivatives
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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) → C(E) given by Dp = p' for all polynomials p. Let G represent the graph of D, that is $\mathscr{G} = \{(p, p'): p \text {polynomials}\}$ considered as a submanifold of C(E) × C(E). Write the interior of the set E, ∫ E as a countable union of disjoint open intervals and let Ê 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) ∈ C(E) × C(E) such that h is absolutely continuous on Ê and $k|\hat E = h'|\hat E$. 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) × C(E) i.e. for any pair (f, g) ∈ C(E) × C(E), there exists a sequence of polynomials {pn} so that pn → f and p'n → g uniformly on E, if and only if the interior ∫ E of E is empty.

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