A Differential Analogue of Favard's Theorem

TL;DR

This paper investigates a differential analogue of Favard's theorem, characterizing function bases where derivatives relate neighboring functions, with implications for orthogonality, completeness, and applications in spectral methods for differential equations.

Contribution

It introduces and characterizes bases of functions with derivative relations akin to Favard's theorem, expanding the theory of orthogonal functions and their applications.

Findings

Characterization of differential bases with derivative relations

Analysis of orthogonality and completeness of these bases

Examples and challenges for future research

Abstract

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 $\{\varphi_n\}_{n\in\mathbb{Z}_+}$ for which $\varphi_n'(x)$ is a linear combination of $\varphi_{n-1}(x)$, $\varphi_n(x)$, and $\varphi_{n+1}(x)$ for all $n \geq 0$ with $\varphi_{0}(x)$ given (and $\varphi_{-1}\equiv 0$ 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.