Pointwise Remez inequality

TL;DR

This paper extends the classical Remez inequality by characterizing extremal polynomials for fixed point values under boundedness constraints, providing a complete asymptotic solution to Andrievskii's problem.

Contribution

It identifies extremal polynomials as Chebyshev or Akhiezer polynomials and proves Totik-Widom bounds, solving Andrievskii's problem asymptotically.

Findings

Extremal polynomials are Chebyshev or Akhiezer polynomials.

Established Totik-Widom bounds for extremal values.

Provided a complete asymptotic solution to Andrievskii's problem.

Abstract

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 $1$ 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.