On the oscillations of the modulus of Rudin-Shapiro polynomials around the middle of their ranges

2022/10/10 by Tamás Erdélyi, Erdélyi, Tamás
Mathematics · #11C08 #26C10 #30C15 #41A17 #Analytic Number Theory Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.2210.04385

openalex publication_date 2022/10/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let either R_k(t) := |P_k(e^it)|² or R_k(t) := |Q_k(e^it)|², where P_k and Q_k are the usual Rudin-Shapiro polynomials of degree n-1 with n=2^k. The graphs of the trigonometric polynomials R_k on the period suggest many zeros of R_k(t)-n in a dense fashion on the period. Let \Cal N(I,R_k-n) denote the number of zeros, counted with multiplicities, of the trigonometric polynomial R_k-n in an interval I := [α,β] ⊂ [0,2π). Improving earlier results proved only for the interval I := [0,2π), in this paper we show that (n|I|)/(8π) - \frac2π (2nlog n)^1/2 - 1 ≤ N(I,R_k-n) ≤ \fracn|I|π + \frac8π(2nlog n)^1/2 , k ≥ 2 , for every interval I := [α,β] ⊂ [0,2π), where |I| = β-α denotes the length of the interval I.

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