On a discrete norm for polynomials

Given n+1n+1 angles 0≤θ0<θ1⋯<θn≤π0≤θ0<θ1⋯<θn≤π, we discuss various extremal problems over the class of polynomials PnPn endowed with the norm |p|n=max0≤j≤n|p(eiθj)+p(e−iθj)2|.