Abstract
We give a solution of the problem on trigonometric polynomials f n with the given leading harmonic y cos n t that deviate the least from zero in measure, more precisely, with respect to the functional μ ( f n ) = mes { t ∈ [ 0 , 2 π ] : | 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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