AI 中文总结
研究实线上小伍德余弦多项式\(T_n(t)\)，证明了其最大值和最小值的下界均为\(\frac{1}{60}n^{1/3}\)，核心方法未提及，主要贡献是给出该多项式最值的估计。
AI 中文摘要
我们证明，对于形如\(T_n(t)=\sum_{j = 1}^n{\cos(jt - \theta_j)}\)的每个三角多项式\(T_n\)，有\(\max_{t \in {\Bbb R}}{T_n(t)} \geq \frac{1}{60}n^{1/3}\)且\(-\min_{t \in {\Bbb R}}{T_n(t)} \geq \frac{1}{60}n^{1/3}\)。
英文摘要
We prove that $$\max_{t \in {\Bbb R}}{T_n(t)} \geq \frac{1}{60}n^{1/3} \qquad \text {and} \qquad -\min_{t \in {\Bbb R}}{T_n(t)} \geq \frac{1}{60}n^{1/3}$$ for every trigonometric polynomial $T_n$ of the form $$T_n(t) = \sum_{j=1}^n{\cos(jt-θ_j)}\,, \quad θ_j \in {\Bbb R}\,, \quad t \in {\Bbb R}\,.$$
