AI 中文总结
该研究将实轴上的对称测度σ关联到单位圆测度μ，利用σ的正交多项式数据表示μ的预测相关量，结合洛梅尔多项式测度构造出具有归一化行列式极限的两参数纯原子圆测度族。
AI 中文摘要
对于实轴上每个支撑集为(-2,2)的紧子集的有限对称测度σ，我们通过将±x处的质量转移到e^(±iθ(x))（其中θ(x)=2arcsin(x/2)），在单位圆上关联一个测度μ。随后，μ的线性预测误差、Verblunsky系数和Toeplitz行列式可通过σ在单点端点x=2处的正交多项式数据表示，特别地有E_m(μ)=½ t_m||P_m||_σ²，其中t_m=P_{m+1}(2)/P_m(2)。在首系数满足某一条件下，σ的递推系数衰减会迫使t_m从t₁开始递增，这是一种与图兰（Turán）型相关的次数单调性，使得μ的首个之后的每个预测裕度都高于σ对应的系数。将σ取为洛梅尔多项式测度（其原子位于贝塞尔函数J_ν零点的缩放倒数处），可得到一个两参数纯原子圆测度族，其归一化行列式极限等于J_ν的一个值。
英文摘要
To each finite symmetric measure $σ$ on the real line, with support a compact subset of $(-2,2)$, we associate a measure $μ$ on the unit circle by transporting mass at $\pm x$ to $e^{\pm iθ(x)}$, $θ(x)=2\arcsin(x/2)$. The linear prediction errors, Verblunsky coefficients, and Toeplitz determinants of $μ$ are then expressed through the orthogonal polynomial data of $σ$ at the single edge point $x=2$. In particular $E_m(μ)=\tfrac12 t_m\|P_m\|_σ^2$ with $t_m=P_{m+1}(2)/P_m(2)$. Under a condition on the first coefficients, decay of the recurrence coefficients of $σ$ forces the $t_m$ to increase from $t_1$ onward, a Turán-type monotonicity in the degree placing every prediction margin of $μ$ past the first above the corresponding coefficient of $σ$. Taking $σ$ to be the Lommel-polynomial measures, with atoms at rescaled reciprocals of the zeros of the Bessel function $J_ν$, yields a two-parameter family of purely atomic circle measures with a normalized determinant limit equal to a value of $J_ν$.
