Chenevier正交多项式的正性与渐近性

中文摘要 AI 辅助

我们证明了Chenevier在其自守Hermite–Minkowski定理的无条件部分中对临界向量所 conjectured 的严格正性。证明确立了与(-1,1)上权重$(\arcsin x)/x$相关联的圆测度的所有Verblunsky系数的严格负性。一个正核公式和经典Schur算法给出了这些符号，而一个para-正交变换产生了每一度的正性。我们还计算了表示Schur函数负值的正密度，作为Hausdorff矩生成函数。在将其指标重标度到[0,1]后，归一化临界向量弱收敛于反正弦分布，而Chenevier的临界尺度渐近于$8\pi/n$。在临界边界处，一个非零有效积分向量对于每个容许检验函数在奇数度和零度时恰好为负。最后，我们证明了Legendre测度指数扰动的精确一阶变分公式，并推导了其端点尖点的渐近性。对于导致Chenevier权重的扰动，在Legendre测度处的导数具有$\log n/(\pi^2 n^2)$项和$n^{-2}$阶的显式常数。相应的非线性渐近性仍然是推测性的。

英文摘要

We prove the strict positivity conjectured by Chenevier for the critical vectors in the unconditional part of his automorphic Hermite--Minkowski theorem. The proof establishes strict negativity of all Verblunsky coefficients of a circle measure associated with the weight $(\arcsin x)/x$ on $(-1,1)$. A positive-kernel formula and the classical Schur algorithm give these signs, and a para-orthogonal transformation yields positivity in every degree. We also compute the positive density representing the negative of the Schur function as a Hausdorff moment generating function. After rescaling their indices to $[0,1]$, the normalized critical vectors converge weakly to the arcsine law, while Chenevier's critical scale is asymptotic to $8π/n$. At the critical boundary, a single nonzero effective integral vector is negative for every admissible test function exactly in odd degree and in degree zero. Finally, we prove an exact first-variation formula for exponential perturbations of the Legendre measure and derive its asymptotics for endpoint cusps. For the perturbation leading to Chenevier's weight, the derivative at the Legendre measure has a $(\log n)/(π^2n^2)$ term and an explicit constant at order $n^{-2}$. The corresponding nonlinear asymptotic remains conjectural.

发表机构

• Nanjing University（南京大学）

机构由 AI 辅助整理，请以论文原文为准。