论文标题
Favard定理的差分类似物
A Differential Analogue of Favard's Theorem
论文作者
论文摘要
Favard的定理表征了函数的基础$ \ {p_n \} _ {n \ in \ Mathbb {z} _+} $，其中$ x p_n（x）$是$ P_ {n-1}（n-1}（n-1）$的线性组合$ p_ {0} \ equiv1 $（和$ p _ { - 1} \ equiv 0 $按惯例）。在本文中，我们探讨了该理论的差异类似物，即函数的基础$ \ {φ_n\} _ {n \ in \ Mathbb {z} _+} $，$ $φ_n'（x）$是$ $φ_{n-1} $; $φ_{n+1}（x）$ for $ n \ geq 0 $ with $φ_{0}（x）$给定（和$φ_ { - 1} \ equiv 0 $ bifundent concuttion）。我们回答有关此类功能的正交性和完整性的问题，提供表征结果，当然，还提供了许多例子，并列出了进一步研究的挑战。这项工作的动机起源于微分方程的数值解，特别是光谱方法产生高度结构化的矩阵和稳定的逐个设计方法，用于部分进化的部分微分方程。但是，我们认为，由于正交多项式，傅立叶分析和佩利（Paley） - 宇宙空间以及特殊功能的不同家族之间所产生的身份之间的有趣联系，因此我们认为该理论本身就是自己的权利。
Favard's theorem characterizes bases of functions $\{p_n\}_{n\in\mathbb{Z}_+}$ for which $x p_n(x)$ is a linear combination of $p_{n-1}(x)$, $p_n(x)$, and $p_{n+1}(x)$ for all $n \geq 0$ with $p_{0}\equiv1$ (and $p_{-1}\equiv 0$ by convention). In this paper we explore the differential analogue of this theorem, that is, bases of functions $\{φ_n\}_{n\in\mathbb{Z}_+}$ for which $φ_n'(x)$ is a linear combination of $φ_{n-1}(x)$, $φ_n(x)$, and $φ_{n+1}(x)$ for all $n \geq 0$ with $φ_{0}(x)$ given (and $φ_{-1}\equiv 0$ by convention). We answer questions about orthogonality and completeness of such functions, provide characterisation results, and also, of course, give plenty of examples and list challenges for further research. Motivation for this work originated in the numerical solution of differential equations, in particular spectral methods which give rise to highly structured matrices and stable-by-design methods for partial differential equations of evolution. However, we believe this theory to be of interest in its own right, due to the interesting links between orthogonal polynomials, Fourier analysis and Paley--Wiener spaces, and the resulting identities between different families of special functions.
