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

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

2009-12-18

arxiv.org/abs/0912.3670v1

Mathematics

Classical Analysis and ODEs

Submitted to the Journal of Approximation Theory

Scientific paper

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 $\mu(f_n)=mes\{t\in[0,2\pi]: |f_n(t)|\ge 1\}$. 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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