Pointwise Remez inequality

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.

Asymptotics of Chebyshev Polynomials, V. Residual Polynomials

“…Such polynomials have been studied in numerical analysis as they have applications to the Krylov subspace iterations, see, for example, [10,15,22]. Recently they have also been used to study the Remez inequality [12]. The residual norm is given by r z 0 ,n ≡ R z 0 ,n e (1.2)…”