  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.
