Sharp Remez Inequality

Abstract

Let an algebraic polynomial P n (ζ) of degree n be such that |P n (ζ)| 1 for ζ ∈ E ⊂ T and |E| 2π − s. We prove the sharp Remez inequalitywhere T n is the Chebyshev polynomial of degree n. The equality holds if and only ifThis gives the solution of the long-standing problem on the sharp constant in the Remez inequality for trigonometric polynomials.2000 Mathematics Subject Classification. Primary 41A17, 41A44; Secondary 30C35, 41A50.

Pointwise Remez inequality

“…In [26], the sharp constant in the Remez inequality for trigonometric polynomials on the unit circle was given. The proof was based on the following two steps:…”

“…If for fixed x 0 the extremal polynomial is a Remez polynomial, there is no additional interval outside of [−1, 1]. Intuitively, the same considerations as were used in [26] should work. However, a technical difference prevents direct applications of the principle of harmonic measure.…”

Pointwise Remez inequality

“…In [23], the sharp constant in the Remez inequality for trigonometric polynomials on the unit circle was given. The proof was based on the following two steps:…”

“…If for fixed x 0 the extremal polynomial is a Remez polynomial, there is no additional interval outside of [−1, 1]. Intuitively, the same considerations as were used in [23] should work. However, a technical difference prevents a direct applications of the principle of harmonic measure.…”