A Differential Analogue of Favard’s Theorem

Favard’s theorem characterizes bases of functions {pn }n∈Z+for which xpn (x) is a linear combination of pn−1(x), pn (x), and pn+1(x) for all n ≥ 0 with p0 ≡ 1 (and p−1 ≡ 0 by convention). In this paper we explore the differential analogue of this theorem, that is, bases of functions {ϕn }n∈Z+for which ϕ0n(x) is a linear combination of ϕn−1(x), ϕn (x), and ϕn+1(x) for all n ≥ 0 with ϕ0(x) a given smooth function (and ϕ−1 ≡ 0 by convention). We answer questions about orthogonality and completeness of such functions (which are not necessarily polynomials), 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.