关于Rudin-Shapiro多项式的模在其值域中部附近的振荡
On the oscillations of the modulus of Rudin-Shapiro polynomials around the middle of their ranges
AI总结：
本文研究Rudin-Shapiro多项式相关三角多项式$R_k-n$在任意子区间内的零点计数问题，改进了此前仅在全周期区间成立的结果，给出了零点个数的上下界估计。
摘要：
设$R_k(t) := |P_k(e^{it})|^2$或$R_k(t) := |Q_k(e^{it})|^2$，其中$P_k$和$Q_k$是次数为$n-1$的常规Rudin-Shapiro多项式，且$n=2^k$。三角多项式$R_k$在周期上的图像表明，$R_k(t)-n$的零点在周期上呈密集分布。设${\rm N}(I,R_k-n)$表示三角多项式$R_k-n$在区间$I := [α,β] \u2282 [0,2π)$内计及重数的零点个数。本文改进了此前仅在区间$I := [0,2π)$上证明的结果，证明对于每个区间$I := [α,β] \u2282 [0,2π)$（其中$|I| = β-α$表示区间$I$的长度），当$k \u2265 2$时，有$$\frac{n|I|}{8π} - \frac{2}π (2n\log n)^{1/2} - 1 \leq N(I,R_k-n) \leq \frac{n|I|}π + \frac{8}π(2n\log n)^{1/2}\\,.$$
英文摘要：
Let either $R_k(t) := |P_k(e^{it})|^2$ or $R_k(t) := |Q_k(e^{it})|^2$, where $P_k$ and $Q_k$ are the usual Rudin-Shapiro polynomials of degree $n-1$ with $n=2^k$. The graphs of the trigonometric polynomials $R_k$ on the period suggest many zeros of $R_k(t)-n$ in a dense fashion on the period. Let ${\Cal N}(I,R_k-n)$ denote the number of zeros, counted with multiplicities, of the trigonometric polynomial $R_k-n$ in an interval $I := [α,β] \subset [0,2π)$. Improving earlier results proved only for the interval $I := [0,2π)$, in this paper we show that $$\frac{n|I|}{8π} - \frac{2}π (2n\log n)^{1/2} - 1 \leq N(I,R_k-n) \leq \frac{n|I|}π + \frac{8}π(2n\log n)^{1/2}\,,\qquad k \geq 2\,,$$ for every interval $I := [α,β] \subset [0,2π)$, where $|I| = β-α$ denotes the length of the interval $I$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
