Equal-Phase Monotonicity for Weighted Jacobi-Radau Functions and Jacobi Lebesgue Constants
Abstract
For the Jacobi family with parameters (0,β), β≥−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≤β≤0, the positive Jacobi product formula identifies the endpoint values of the Lebesgue functions with the global Lebesgue constants, so (Λn(0,β))n≥0 is strictly increasing. At β=0 the weighted Radau functions reduce to Pm(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.
Related
- Hyperbolic Lebesgue Constants in Dimension Three
- Two-sided logarithmic estimates of the Lebesgue constants with respect to Vilenkin systems
- Lebesgue constants of Leja sequences: a universal lower bound and improved upper bounds on the real line.
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- Some weighted modular inequalities in variable Lebesgue spaces
BibTeX
@misc{castillo2026equalphase,
title = {{Equal-Phase Monotonicity for Weighted Jacobi-Radau Functions and Jacobi Lebesgue Constants}},
author = {K. Castillo and P. -C. Hang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13030},
eprint = {2609.13030},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13030},
note = {CC0},
}
RIS
TY - UNPB
TI - Equal-Phase Monotonicity for Weighted Jacobi-Radau Functions and Jacobi Lebesgue Constants
AU - K. Castillo
AU - P. -C. Hang
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.13030
UR - https://arxiv.org/abs/2609.13030
AB - For the Jacobi family with parameters (0,β), β≥−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≤β≤0, the positive Jacobi product formula identifies the endpoint values of the Lebesgue functions with the global Lebesgue constants, so (Λ_n^(0,β))_n≥0 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.
ER -