超平坦单模多项式与其共轭倒数多项式间的渐近距离
经典分析与常微分方程
2019-02-13 v2
摘要
令 K n : = { p n : p n ( z ) = ∑ k = 0 n a k z k , a k ∈ C , ∣ a k ∣ = 1 } . {\mathcal K}_n := \left\{p_n: p_n(z) = \sum_{k=0}^n{a_k z^k}, \enspace a_k \in {\mathbb C}\,,\enspace |a_k| = 1 \right\}\,. K n := { p n : p n ( z ) = k = 0 ∑ n a k z k , a k ∈ C , ∣ a k ∣ = 1 } . ( P n ) (P_n) ( P n ) P n ∈ K n P_n \in {\mathcal K}_n P n ∈ K n ( n + 1 ) − 1 / 2 ∣ P n ( e i t ) ∣ (n + 1)^{-1/2}|P_n(e^{it})| ( n + 1 ) − 1/2 ∣ P n ( e i t ) ∣ t ∈ R t \in {\mathbb R} t ∈ R 1 1 1 ( P n ) (P_n) ( P n ) P n ∈ K n P_n \in {\mathcal K}_n P n ∈ K n q ∈ ( 0 , ∞ ) q \in (0,\infty) q ∈ ( 0 , ∞ ) 1 2 π ∫ 0 2 π ∣ ( P n − P n ∗ ) ( e i t ) ∣ q d t ∼ 2 q Γ ( q + 1 2 ) Γ ( q 2 + 1 ) π n q / 2 \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} 2 π 1 ∫ 0 2 π ( P n − P n ∗ ) ( e i t ) q d t ∼ Γ ( 2 q + 1 ) π 2 q Γ ( 2 q + 1 ) n q /2 P n ∗ P_n^* P n ∗ P n P_n P n Γ \Gamma Γ ∼ \sim ∼ n → ∞ n \rightarrow \infty n → ∞ 1 1 1 ( P n ) (P_n) ( P n ) P n ∈ K n P_n \in {\mathcal K}_n P n ∈ K n 1 2 π ∫ 0 2 π ∣ ( P n ′ − P n ∗ ′ ) ( e i t ) ∣ 2 d t ∼ 2 n 3 3 \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} 2 π 1 ∫ 0 2 π ( P n ′ − P n ∗ ′ ) ( e i t ) 2 d t ∼ 3 2 n 3
引用
@article{arxiv.1810.04287,
title = {The asymptotic distance between an ultraflat unimodular polynomial and its conjugate reciprocal},
author = {Tamás Erdélyi},
journal= {arXiv preprint arXiv:1810.04287},
year = {2019}
}