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

MathSciNet review:
1091175

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Abstract: Let be a compact subset of the unit interval , and let denote the space of functions continuous on with the uniform norm. Consider the densely defined operator given by for all polynomials . Let represent the graph of , that is polynomials} considered as a submanifold of . Write the interior of the set as a countable union of disjoint open intervals and let be the union of the closure of these intervals. The main result is that the closure of is equal to the set of all functions such that is absolutely continuous on and . As a consequence, the operator is closable if and only if the set is the closure of its interior. On the other extreme, is dense in i.e. for any pair , there exists a sequence of polynomials so that and uniformly on , if and only if the interior of is empty.

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DOI:
https://doi.org/10.1090/S0002-9939-1992-1091175-8

Article copyright:
© Copyright 1992
American Mathematical Society