[论文解读] Yet Another Characterisation of Classical Orthogonal Polynomials?

K. Castillo, G. Gordillo-Núñez|arXiv (Cornell University)|Mar 9, 2026

Mathematical functions and polynomials被引用 0

一句话总结

简要：本文在 locally convex 空间上的 Maroni 对偶性基础上，将 Bochner 对经典正交多项式的分类扩展到离散线性晶格，统一了连续与离散情形，并阐明贝塞尔多项式的作用。

ABSTRACT

The NIST Handbook of Mathematical Functions (2010) and the NIST Digital Library of Mathematical Functions (2025) classify classical orthogonal polynomials through Bochner's 1929 algebraic-differential characterisation and its discretisation. Yet this classification rests on a narrow reading of Bochner's work and on a restricted notion of orthogonality that becomes inadequate once polynomials are characterised by their algebraic properties. As a result, algebraically equivalent families are treated as distinct, parameter domains are restricted, and families already implicit in Bochner's scheme are excluded. In the mid-1980s, Maroni challenged this view by extending the notion of classical orthogonal polynomials through duality theory on locally convex spaces, thereby reaching the algebraic limits latent in Bochner's framework. Yet when the notion was later enlarged to include further families, Maroni's criteria and rationale were largely set aside. To clarify this history, we revisit a less familiar line of development and use it to obtain a classification of classical orthogonal polynomials on linear lattices within Maroni's functional-analytic setting, beyond the positive-definite case. This classification recovers all known families as special cases, preserves orthogonality and the defining algebraic properties, places supposedly new families in their proper structural context, and shows that algebraically identical polynomials are often treated as distinct. Moreover, through a limit process in the weak topology of the continuous dual, we recover families implicit in Bochner's work and unify the continuous and discrete cases within a dual-topological framework. Thus, neither Bochner's classical characterisation nor its discrete analogue is modified to produce ad hoc families; both are recovered at the level of their intrinsic algebraic structure.

研究动机与目标

• 在正定测度与 OPRL 之外，激发对经典正交多项式的更广泛、代数化理解。
• 通过对偶性在局部凸空间中的作用，澄清从 Bochner 到 Maroni 与 Krall–Geronimus 的历史发展。
• 提供一个规范化、包含参数的框架，以回收已知族并澄清看似新的族。
• 展示连续与离散（线性晶格）情形如何在统一的对偶-拓扑框架中契合。

提出的方法

• 为多项式采用拓扑 LF 空间（线性诱导极限）框架，并研究该空间上的连续线性泛函。
• 以对偶术语表述 Bochner 型方程：D(phi a) = psi a，其中 phi 与 psi 为有界次数的多项式。
• 将规范性族作为在变换下的等价类来分类多项式族。
• 利用对偶性与算子的转置（平移、放缩）来连接代数性质与泛函性质。
• 证明四大主要 Bochner 家族（Hermite、Laguerre、Jacobi、Bessel）在 Maroni 的设定中出现在超越正定约束的情形。

实验结果

研究问题

• RQ1 如何在超越正定测度的对偶框架中，用函数分析的方法表征线性晶格上的经典正交多项式？
• RQ2 在 Maroni 的对偶理论中，连续与离散情形以同一代数结构统一的意义为何？
• RQ3 当参数范围不受正性限制时，经典族的规范等价类有哪些？
• RQ4 Bochner、Geronimus 与 Krall 的结果如何融入一个统一的函数分析表述？
• RQ5 代数上完全相同的多项式是否因参数域的不同而看作不同？是否可通过等价类加以澄清？

主要发现

• 在规范化、对偶-拓扑框架内，所有已知的经典族都被作为特例回收。
• 在超越正定设定下，正交性和定义性的代数性质得到保持。
• 将据称是新族放入适当的结构语境中，澄清因参数域导致的冗余。
• 在连续对偶的弱拓扑极限下，回捕捉 Bochner 工作中隐含的族。
• 连续与离散（线性晶格）情形在不修改 Bochner、Koekoek–Lesky–Swarttouw 分类的前提下实现统一。
• 离散与连续理论被证明是同一内在代数结构的不同实例。