A Differential Analogue of Favard’s Theorem

Favard's theorem characterizes bases of functions $\{p_n\}_{n\in\mathbb{Z}_+}$ for which $x p_n(x)$ is a linear combination $p_{n-1}(x)$, $p_n(x)$, and $p_{n+1}(x)$ all $n \geq 0$ with $p_{0}\equiv1$ (and $p_{-1}\equiv by convention). In this paper we explore the differential analogue theorem, that is, $\{\varphi_n\}_{n\in\mathbb{Z}_+}$ $\varphi_n'(x)$ $\varphi_{n-1}(x)$, $\varphi_n(x)$, $\varphi_{n+1}(x)$ $\varphi_{0}(x)$ given $\varphi_{-1}\equiv We answer questions about orthogonality completeness such functions, provide characterisation results, also, course, give plenty examples list challenges further research. Motivation work originated in numerical solution equations, particular spectral methods rise to highly structured matrices stable-by-design partial equations evolution. However, believe theory be interest its own right, due interesting links between orthogonal polynomials, Fourier analysis Paley--Wiener spaces, resulting identities different families special functions.