The L_q norm of the Rudin-Shapiro polynomials on subarcs of the unit circle

2023/11/07 by Erdélyi, Tamás
#11C08 #41A17 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2311.04395

Abstract

Littlewood polynomials are polynomials with each of their coefficients in \-1,1\. A sequence of Littlewood polynomials that satisfies a remarkable flatness property on the unit circle of the complex plane is given by the Rudin-Shapiro polynomials. Let P_k and Q_k denote the Rudin-Shapiro polynomials of degree n-1 with n:=2^k. For polynomials S we define M_q(S,[α,β]) := ( (1)/(β-α) ∫_α^β | S(e^it) |^q dt )^1/q , q gt; 0 . Let γ:= sin²(π/8). We prove that \fracγ4π(γn)^q/2 ≤ M_q(P_k,[α,β])^q ≤ (2n)^q/2 for every q > 0 and 32π/n ≤ β-α. The same estimates hold for P_k replaced by Q_k.

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