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Proceedings of the American Mathematical Society

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The relationship between the zeros of best approximations and differentiability

Author: Peter B. Borwein
Journal: Proc. Amer. Math. Soc. 92 (1984), 528-532
MSC: Primary 41A50; Secondary 41A10
MathSciNet review: 760939
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Abstract: We examine the relationship between the analytic properties of continuous functions on $ [ - 1,1]$ and the location of the roots of the sequence of best polynomial approximations. We show that if the approximants have no zeros in a certain ellipse then the function being approximated must be analytic in this ellipse. We also show that the rate at which the zeros of the $ n$th approximant tend to the interval $ [ - 1,1]$ determines the global differentiability of the function under consideration.

References [Enhancements On Off] (What's this?)

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