Chebyshev constants for the unit circle

It is proven that for any system of n points z_1, ..., z_n on the (complex) unit circle, there exists another point z of norm 1, such that $$\sum 1/|z-z_k|^2 \leq n^2/4.$$ Equality holds iff the point system is a rotated copy of the nth unit roots. Two proofs are presented: one uses a characterisati...

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Bibliographic Details
Published in:arXiv.org 2011-11
Main Authors: Ambrus, Gergely, Ball, Keith M, Erdélyi, T
Format: Article
Language:English
Subjects:
Chebyshev approximation
Rational functions
Online Access:Get full text
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Summary:It is proven that for any system of n points z_1, ..., z_n on the (complex) unit circle, there exists another point z of norm 1, such that $$\sum 1/|z-z_k|^2 \leq n^2/4.$$ Equality holds iff the point system is a rotated copy of the nth unit roots. Two proofs are presented: one uses a characterisation of equioscillating rational functions, while the other is based on Bernstein's inequality.
ISSN:2331-8422
DOI:10.48550/arxiv.1006.5153