Remote Access Transactions of the American Mathematical Society
Green Open Access

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
MathSciNet review: 4237943
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let \begin{equation*} {\mathcal {K}}_n \colonequals \left \{Q_n: Q_n(z) = \sum _{k=0}^n{a_k z^k}, \quad a_k \in {\mathbb {C}}\,, \quad |a_k| = 1 \right \}\,. \end{equation*} A sequence $(P_n)$ of polynomials $P_n \!\in \! {\mathcal {K}}_n$ is called ultraflat if $(n + 1)^{-1/2}|P_n(e^{it})|$ converge to $1$ uniformly in $t \!\in \! {\mathbb {R}}$. In this paper we prove that \begin{equation*} \frac {1}{2\pi } \int _0^{2\pi }{\left | (P_n - P_n^*)(e^{it}) \right |^q \, dt} \sim \frac {{2}^q \Gamma \left (\frac {q+1}{2} \right )}{\Gamma \left (\frac q2 + 1 \right ) \sqrt {\pi }} \,\, n^{q/2} \end{equation*} 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 \begin{equation*} \frac {1}{2\pi }\int _0^{2\pi }{\left | (P_n^\prime - P_n^{*\prime })(e^{it}) \right |^2 \, dt} \sim \frac {2n^3}{3} \end{equation*} 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.


References [Enhancements On Off] (What's this?)

References

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2020): 11C08, 41A17, 26C10, 30C15

Retrieve articles in all journals with MSC (2020): 11C08, 41A17, 26C10, 30C15


Additional Information

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

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