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

Abstract

With the notation K≔ R ( mod 2π) , ||p|| L λ(K) ≔ ∫ K |p(t)| λ dt 1/λ and M λ(p)≔ 1 2π ∫ K |p(t)| λ dt 1/λ we prove the following result. Theorem 1. Assume that p is a trigonometric polynomial of degree at most n with real coefficients that satisfies ||p|| L 2(K) ⩽An 1/2 and ||p′|| L 2(K) ⩾Bn 3/2. Then M 4(p)−M 2(p)⩾εM 2(p) with ε≔ 1 111 B A 12. We also prove that M ∞(1+2p)−M 2(1+2p)⩾ 4/3 −1 M 2(1+2p) and M 2(p)−M 1(p)⩾10 −31M 2(p) for every p∈ A n , where A n denotes the collection of all trigonometric polynomials of the form p(t)≔p n(t)≔ ∑ j=1 n a j cos( jt+α j), a j=±1, α j∈ R.