AI 中文总结
本文研究具有非负切比雪夫系数的多项式经余弦缩放后的局部一致极限函数的性质，推导了相关测度的收敛性、零点与多项式次数及缩放参数的定量关系，明确了各假设条件的作用。
AI 中文摘要
设$d_k>0$，$Q_k$为次数为$r_k$、具有非负切比雪夫系数的多项式。我们研究$f_k(z)=C_kQ_k(\cos(d_kz))$的局部一致极限，其中$C_k>0$。若$f_k\to F$局部一致收敛且$F(0)>0$，则相关的正格测度弱收敛，其二阶矩收敛，且二次尾部一致可积。若额外满足$Q_k$的所有零点位于$[-1,1)$内，$d_k\to0$，且$F$的阶低于2，则对任意满足$F(A)F(-A)\ne0$的$A>0$，有$\limsup_{k\to\infty} r_kd_k^2\le 4\sum_{γ_n>A}γ_n^{-2}$，其中$\{\pmγ_n\}$是$F$的非零实零点，按重数计数。进一步，若$f_k\to F$局部一致收敛且$F(0)>0$，$Q_k$的所有零点位于$[-1,1)$内，且$F$无非零实周期、阶低于2且不属有限指数型，则$d_k\to0$，$r_kd_k\to\infty$，且$r_kd_k^2\to0$，等价于$d_k^{-1}=o(r_k)$且$r_k=o(d_k^{-2})$。边界例子说明了各假设的作用以及阶为2时的高斯边界情形。
英文摘要
Let $d_k>0$ and let $Q_k$ be a polynomial of degree $r_k$ with nonnegative Chebyshev coefficients. We study locally uniform limits of $f_k(z)=C_kQ_k(\cos(d_kz))$, where $C_k>0$. If $f_k\to F$ locally uniformly and $F(0)>0$, the associated positive lattice measures converge weakly, their second moments converge, and their quadratic tails are uniformly integrable. If additionally all zeros of $Q_k$ lie in $[-1,1)$, $d_k\to0$, and $F$ has order below two, then for every $A>0$ with $F(A)F(-A)\ne0$, $\limsup_{k\to\infty} r_kd_k^2\le 4\sum_{γ_n>A}γ_n^{-2}$, where $\{\pmγ_n\}$ are the nonzero real zeros of $F$, counted with multiplicity. If, moreover, $f_k\to F$ locally uniformly with $F(0)>0$, all zeros of $Q_k$ lie in $[-1,1)$, and $F$ has no nonzero real period, has order below two, and is not of finite exponential type, then $d_k\to0$, $r_kd_k\to\infty$, and $r_kd_k^2\to0$. Equivalently, $d_k^{-1}=o(r_k)$ and $r_k=o(d_k^{-2})$. Boundary examples show the role of the hypotheses and the Gaussian boundary at order two.
Comments14 pages, 0 figures. Zenodo v0.1.4: 10.5281/zenodo.21842188
