Trigonometric polynomials deviating the least from zero in measure and related problems

2009/12/18 by В. В. Арестов, Vitalii V. Arestov, Arestov, Vitalii V. +2
Mathematics · #Algebraic and Geometric Analysis #Classical Analysis and ODEs (math.CA) #Differential Equations and Boundary Problems #FOS: Mathematics #Mathematical functions and polynomials #math.CA

paper · pdf · doi:10.48550/arxiv.0912.3670

Submitted to the Journal of Approximation Theory

arxiv created 2009/12/18 · openalex publication_date 2009/12/18 · arxiv updated 2010/01/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give a solution of the problem on trigonometric polynomials f_n with the given leading harmonic ycos nt that deviate the least from zero in measure, more precisely, with respect to the functional μ(f_n)=mes\t∈[0,2π]: |f_n(t)|≥ 1\. For trigonometric polynomials with a fixed leading harmonic, we consider the least uniform deviation from zero on a compact set and find the minimal value of the deviation over compact subsets of the torus that have a given measure. We give a solution of a similar problem on the unit circle for algebraic polynomials with zeros on the circle.

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