Algebraic polynomials least deviating from zero in measure on a segment

Abstract

We investigate the problem of algebraic polynomials with given leading coefficients that deviate least from zero on the segment $[–1, 1]$ with respect to a measure, or, more precisely, with respect to the functional $μ(f) = \text{mes}\left\{x ∈ [–1, 1]: ∣f (x)∣ ≥ 1 \right\}$. We also discuss an analogous problem with respect to the integral functionals $∫_{–1}^1 φ (∣f (x)∣) dx$ for functions $φ$ that are defined, nonnegative, and nondecreasing on the semiaxis $[0, +∞)$.