Trigonometric polynomials of least deviation from zero in measure and related problems Original Research Article

We give a solution of the problem on trigonometric polynomials fnfn with the given leading harmonic ycosntycosnt that deviate the least from zero in measure, more precisely, with respect to the functional View the MathML sourceμ(fn)=mes{t∈[0,2π]:|fn(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.