雅可比零点的仿射缩放：超越Gautschi猜想的精确序关系
Affine Scaling of Jacobi Zeros: Sharp Orderings Beyond Gautschi's Conjectures
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中文总结 AI 辅助
该文解决了Gautschi关于雅可比多项式零点对次数依赖性的两个猜想，通过Liouville变换和Sturm比较确定了特定参数下的零点仿射序关系及有限次界，所得分类精确且无法扩大。
中文摘要 AI 辅助
我们解决了Gautschi关于雅可比多项式零点对次数的依赖性的两个猜想，得到了比猜想强得多的结果。这些猜想源于球面求积和超插值领域的一系列问题。通过Liouville变换和Sturm比较，我们得到了仿射比较原理，给出了重标势逐点单调性的精确阈值。我们证明，当且仅当|β|≤1/2时，存在与次数无关的平移的递增仿射序关系。对于谱尺度n+(α+β+1)/2，我们确定了两种相反序关系的精确参数区域，并表明在该区域外不存在统一的谱序关系。我们还刻画了所有等号情况，并根据贝塞尔零点推导了有限次界。所得分类是精确的，无法扩大：在所述参数区域外，相应的统一零点序关系必然不成立。
英文摘要
We settle two conjectures of Gautschi on the degree dependence of the zeros of the Jacobi polynomials $P_n^{(α,β)}$, $α,β>-1$, and obtain results substantially stronger than those conjectured. The conjectures stem from a line of questions originating in spherical cubature and hyperinterpolation. A Liouville transformation and Sturm comparison yield affine comparison principles with exact thresholds for the pointwise monotonicity of the rescaled potential. We prove that an increasing affine ordering with a degree-independent shift exists if and only if $|β|\leq1/2$. For the spectral scale $n+(α+β+1)/2$, we determine the exact parameter regions for the two opposite orderings and show that no uniform spectral ordering is possible outside them. We also characterise all equality cases and derive finite-degree bounds in terms of Bessel zeros. The resulting classifications are exact and cannot be enlarged: outside the stated parameter regions the corresponding uniform zero orderings necessarily fail.
