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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The asymptotic distance between an ultraflat unimodular polynomial and its conjugate reciprocal


Author: Tamás Erdélyi
Journal: Trans. Amer. Math. Soc. 374 (2021), 3077-3091
MSC (2020): Primary 11C08, 41A17; Secondary 26C10, 30C15
DOI: https://doi.org/10.1090/tran/8313
Published electronically: February 23, 2021
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Abstract: Let

$\displaystyle {\mathcal {K}}_n \colonequals \left \{Q_n: Q_n(z) = \sum _{k=0}^n{a_k z^k}, \enskip a_k \in {\mathbb{C}}\,, \enskip \vert a_k\vert = 1 \right \}\,.$    

A sequence $ (P_n)$ of polynomials $ P_n \!\in \! {\mathcal {K}}_n$ is called ultraflat if $ (n + 1)^{-1/2}\vert P_n(e^{it})\vert$ converge to $ 1$ uniformly in $ t \!\in \! {\mathbb{R}}$. In this paper we prove that

$\displaystyle \frac {1}{2\pi } \int _0^{2\pi }{\left \vert (P_n - P_n^*)(e^{it}... ...+1}{2} \right )}{\Gamma \left (\frac q2 + 1 \right ) \sqrt {\pi }} \,\, n^{q/2}$    

for every ultraflat sequence $ (P_n)$ of polynomials $ P_n \in {\mathcal {K}}_n$ and for every $ q \in (0,\infty )$, where $ P_n^*$ is the conjugate reciprocal polynomial associated with $ P_n$, $ \Gamma $ is the usual gamma function, and the $ \sim $ symbol means that the ratio of the left and right hand sides converges to $ 1$ as $ n \rightarrow \infty $. Another highlight of the paper states that

$\displaystyle \frac {1}{2\pi }\int _0^{2\pi }{\left \vert (P_n^\prime - P_n^{*\prime })(e^{it}) \right \vert^2 \, dt} \sim \frac {2n^3}{3}$    

for every ultraflat sequence $ (P_n)$ of polynomials $ P_n \in {\mathcal {K}}_n$. We prove a few other new results and reprove some interesting old results as well.

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Additional Information

Tamás Erdélyi
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email: terdelyi@math.tamu.edu

DOI: https://doi.org/10.1090/tran/8313
Keywords: Polynomials, restricted coefficients, ultraflat sequences of unimodular polynomials, angular speed, conjugate polynomials
Received by editor(s): February 22, 2020
Published electronically: February 23, 2021
Dedicated: Dedicated to the memory of Jean-Pierre Kahane
Article copyright: © Copyright 2021 American Mathematical Society