发表机构
University of Coimbra; Donghua University（科英布拉大学; 东华大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对加权Jacobi-Radau函数，证明等相单调性定理，恢复并简化了Wong-Zhang与Qu-Wong关于Lebesgue常数的结论。
AI 中文摘要
对于参数为$(0,\beta)$且$\beta\geq-1/3$的Jacobi族，我们证明了相关加权Jacobi-Radau函数的连续比较定理。当两个连续函数由相同的Prüfer相位参数化时，次数较高的函数在更接近$\theta=0$处达到该相位，并且具有严格更大的振幅。特别地，所有相应相对极值的模随次数严格增加。下界$-1/3$对于此连续陈述是精确的：当$-1<\beta<-1/3$时，在足够小的正相位处振幅不等式反转。这种局部反转并不能确定离散极值不等式的最佳范围。对于$-1/3\leq\beta\leq0$，正的Jacobi乘积公式将Lebesgue函数的端点值与全局Lebesgue常数等同起来，因此$(\Lambda_n^{(0,\beta)})_{n\geq0}$是严格递增的。在$\beta=0$时，加权Radau函数退化为$P_m^{(0,-1)}$。我们由此恢复了Wong和Zhang的定理，并通过精确的全变差公式，获得了关于Legendre Lebesgue常数的Qu-Wong定理的简短证明。该表示和逐项比较共同实现了Qu和Wong提出的替代表达式方法，无需渐近展开、误差界或有限数值验证。证明基于第一作者早期预印本arXiv:2608.01404中发展的等相Prüfer架构；该架构被应用于Qu和Wong提出的问题。
英文摘要
For the Jacobi family with parameters $(0,β)$, $β\geq-1/3$, we prove a continuous comparison theorem for the associated weighted Jacobi--Radau functions. When two consecutive functions are parametrised by the same Prüfer phase, the function of higher degree attains that phase closer to $θ=0$ and has strictly larger amplitude. In particular, the moduli of all corresponding relative extrema increase strictly with the degree. The lower bound $-1/3$ is sharp for this continuous statement: when $-1<β<-1/3$, the amplitude inequality is reversed at sufficiently small positive phases. This local reversal does not determine the optimal range for the discrete extremal inequalities. For $-1/3\leqβ\leq0$, the positive Jacobi product formula identifies the endpoint values of the Lebesgue functions with the global Lebesgue constants, so $(Λ_n^{(0,β)})_{n\geq0}$ is strictly increasing. At $β=0$ the weighted Radau functions reduce to $P_m^{(0,-1)}$. We thereby recover the theorem of Wong and Zhang and obtain, through an exact total-variation formula, a short proof of the Qu--Wong theorem on Legendre Lebesgue constants. The representation and the termwise comparison together realise the alternative-expression approach proposed by Qu and Wong, without asymptotic expansions, error bounds, or finite numerical verification. The proof is based on the equal-phase Prüfer architecture developed in the first author's earlier preprint arXiv:2608.01404; it applies that architecture to the problem posed by Qu and Wong.
