Pointwise Remez inequality

The standard well-known Remez inequality gives an upper estimate of the values of polynomials on [ - 1 , 1 ] if they are bounded by 1 on a subset of [ - 1 , 1 ] of fixed Lebesgue measure. The extremal solution is given by the rescaled Chebyshev polynomials for one interval. Andrievskii asked about t...

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Bibliographic Details
Published in:Constructive approximation 2021-12, Vol.54 (3), p.529-554
Main Authors: Eichinger, B., Yuditskii, P.
Format: Article
Language:English
Subjects:
Analysis
Asymptotic methods
Chebyshev approximation
Mathematics
Mathematics and Statistics
Numerical Analysis
Polynomials
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Summary:The standard well-known Remez inequality gives an upper estimate of the values of polynomials on [ - 1 , 1 ] if they are bounded by 1 on a subset of [ - 1 , 1 ] of fixed Lebesgue measure. The extremal solution is given by the rescaled Chebyshev polynomials for one interval. Andrievskii asked about the maximal value of polynomials at a fixed point, if they are again bounded by 1 on a set of fixed size. We show that the extremal polynomials are either Chebyshev (one interval) or Akhiezer polynomials (two intervals) and prove Totik–Widom bounds for the extremal value, thereby providing a complete asymptotic solution to the Andrievskii problem.
ISSN:0176-4276
1432-0940
DOI:10.1007/s00365-021-09562-1