论 $T_n^{(k)}$ 的最大临界值
经典分析与常微分方程
2017-10-18 v1
摘要
我们研究量 τ n , k : = ∣ T n ( k ) ( ω n , k ) ∣ T n ( k ) ( 1 ) , \tau_{n,k}:=\frac{|T_n^{(k)}(\omega_{n,k})|}{T_n^{(k)}(1)}\,, τ n , k := T n ( k ) ( 1 ) ∣ T n ( k ) ( ω n , k ) ∣ , T n T_n T n n n n ω n , k \omega_{n,k} ω n , k T n ( k + 1 ) T_n^{(k+1)} T n ( k + 1 ) T n ( k ) T_n^{(k)} T n ( k ) [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] τ n , k \tau_{n,k} τ n , k T n ( k ) \,T_n^{(k)}\, T n ( k ) T n ( k ) ( 1 ) \,T_n^{(k)}(1)\, T n ( k ) ( 1 ) { τ n , k } n = k + 2 ∞ \,\{\tau_{n,k}\}_{n=k+2}^{\infty} { τ n , k } n = k + 2 ∞ n n n τ n , k ≤ { τ k + 4 , k = 1 2 k + 1 3 k + 3 , n ≥ k + 4 τ k + 6 , k = 1 2 k + 1 ( 5 k + 5 ) 2 β k , n ≥ k + 6 , \tau_{n,k} \le \begin{cases} \tau_{k+4,k} = \frac{1}{2k+1}\,\frac{3}{k+3}\,, & n \ge k+4\, \\ \tau_{k+6,k} = \frac{1}{2k+1}\, (\frac{5}{k+5})^2 \beta_k\,, & n \ge k+6\,, \end{cases} τ n , k ≤ { τ k + 4 , k = 2 k + 1 1 k + 3 3 , τ k + 6 , k = 2 k + 1 1 ( k + 5 5 ) 2 β k , n ≥ k + 4 n ≥ k + 6 , β k < 2 + 10 5 ≈ 1.032 \beta_k < \frac{2+\sqrt{10}}{5} \approx 1.032 β k < 5 2 + 10 ≈ 1.032 n n n k k k τ n , k ≈ ( 2 e ) k , n ≥ k 3 / 2 ; τ n , n − m ≈ ( e m 2 ) m / 2 n − m / 2 ; τ n , n / 2 ≈ ( 4 27 ) n / 2 . \tau_{n,k} \approx \big(\frac{2}{e}\big)^k, \quad n \ge k^{3/2}; \qquad \tau_{n,n-m} \approx \big(\frac{em}{2}\big)^{m/2} n^{-m/2}; \qquad \tau_{n,n/2} \approx \big(\frac{4}{\sqrt{27}}\big)^{n/2}. τ n , k ≈ ( e 2 ) k , n ≥ k 3/2 ; τ n , n − m ≈ ( 2 e m ) m /2 n − m /2 ; τ n , n /2 ≈ ( 27 4 ) n /2 . τ k ∗ : = lim n → ∞ τ n , k \mbox 和 τ m ∗ ∗ : = lim n → ∞ n m / 2 τ n , n − m , \tau_k^{*} := \lim_{n\to\infty}\tau_{n,k} \quad \mbox{ 和 }\quad \tau_m^{**} := \lim_{n\to\infty} n^{m/2} \tau_{n,n-m}\,, τ k ∗ := n → ∞ lim τ n , k \mbox 和 τ m ∗∗ := n → ∞ lim n m /2 τ n , n − m , τ n , k \tau_{n,k} τ n , k τ n , n − m \tau_{n,n-m} τ n , n − m
引用
@article{arxiv.1710.06120,
title = {On the largest critical value of $T_n^{(k)}$},
author = {Geno Nikolov and Nikola Naidenov and Alexei Shadrin},
journal= {arXiv preprint arXiv:1710.06120},
year = {2017}
}
备注
19 pages, 2 figures