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arXiv · 0912.3670

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

Vitalii V. Arestov·Alexei S. Mendelev
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Abstract

We give a solution of the problem on trigonometric polynomials $f_n$ with the given leading harmonic $y\cos nt$ that deviate the least from zero in measure, more precisely, with respect to the functional $μ(f_n)=mes\{t\in[0,2π]: |f_n(t)|\ge 1\}$. 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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Vitalii V. Arestov, Alexei S. Mendelev. 2009-12-18. Trigonometric polynomials deviating the least from zero in measure and related problems. https://arxiv.org/abs/0912.3670

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