Lower bounds for the merit factors of trigonometric polynomials from Littlewood classes

AbstractWith the notation K≔R(mod2π),||p||Lλ(K)≔∫K|p(t)|λdt1/λandMλ(p)≔12π∫K|p(t)|λdt1/λwe prove the following result.Theorem 1. Assume that p is a trigonometric polynomial of degree at most n with real coefficients that satisfies||p||L2(K)⩽An1/2and||p′||L2(K)⩾Bn3/2.ThenM4(p)−M2(p)⩾εM2(p)withε≔1111BA12.We also prove thatM∞(1+2p)−M2(1+2p)⩾4/3−1M2(1+2p)andM2(p)−M1(p)⩾10−31M2(p)for every p∈An, where An denotes the collection of all trigonometric polynomials of the formp(t)≔pn(t)≔∑j=1najcos(jt+αj),aj=±1,αj∈R