On the real part of ultraflat sequences of unimodular polynomials: Consequences implied by the resolution of the phase problem
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Let Pn(z) = ∑k=0n ak,nzk ∈ ℂ[z] be a sequence of unimodular polynomials (|ak,n| = 1 for all k, n) which is ultraflat in the sense of Kahane, i.e., limn→infin; max|z|=1 |(n + 1)-1/2| Pn(z)| - 1| = 0. We prove the following conjecture of Queffelec and Saffari, see (1.30) in [QS2]. If q ∈ (0, infin;) and (Pn) is an ultraflat sequence of unimodular polynomials Pn of degree n, then for fn(t) := Re(Pn(eit))we have ||fn||Lq[0,2π] ∼ (Γ(q+1)/2)/Γ(q/2+1)√π)1/q n1/2 and ||f′n||Lq[0.2π] ∼ (Γ/(q+1/2)/(q+1)Γ(q/2+1)√π)1/q n3/2, where Γ denotes the usual gamma function, and the ∼ symbol means that the ratio of the left and right hand sides converges to 1 as n → ∞. To this end we use results from [Er1] where we studied the structure of ultraflat polynomials and proved several conjectures of Queffelec and Saffari.
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