A Differential Analogue of Favard's Theorem
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Favard's theorem characterizes bases of functions {p_n}_n∈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 ≥ 0 with p_0≡1 (and p_-1≡ 0 by convention). In this paper we explore the differential analogue of this theorem, that is, bases of functions {φ_n}_n∈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 ≥ 0 with φ_0(x) given (and φ_-1≡ 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.
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